Sunday, June 7, 2009
Saturday, June 6, 2009
Thursday, June 4, 2009
Extra credit
Anyone wanting to get extra-credit toward an A, may work on Wilson's problem.
For extra credit toward a B, you could do the same sort of analysis for the matrix:
0, i hbar, o
-i hbar, 0, 0
0, 0, 0
which is the matrix of Lz in the basis: 100, 010, 001 ; which spans the 3-fold degenerate 1st excited state manifold of the 3D H.O.
These should be handed in in Sunday and should be be well-organized, clear and attractive. The key thing is to look deeply at what the eigenvectors are telling you about the nature of the spatial states. The credit begins with that part, and the clear, cogent discussion that accompanies it.
For extra credit toward a B, you could do the same sort of analysis for the matrix:
0, i hbar, o
-i hbar, 0, 0
0, 0, 0
which is the matrix of Lz in the basis: 100, 010, 001 ; which spans the 3-fold degenerate 1st excited state manifold of the 3D H.O.
These should be handed in in Sunday and should be be well-organized, clear and attractive. The key thing is to look deeply at what the eigenvectors are telling you about the nature of the spatial states. The credit begins with that part, and the clear, cogent discussion that accompanies it.
Wilson's matrix
Dear Wilson,. Here is the matrix I promised you. I hope you like it as much as I do. I think it is very beautiful.
. This is the matrix of Lz in the sub-basis: 101, 011, 110, 200, 020, 002; which spans the 6-fold degenerate 2nd-excited state manifold of the 3-dimensional harmonic oscillator. We worked on our choice for basis order, and calculating the matrix, in office hours today. Our guiding aesthetic --that influenced and guided our choice of basis-state order, was connection. We wanted to put states which "connected" next to each other as much as possible. I imagine you are pretty excited right now. I would be too in your shoes.
. If you feel inspired to do so, please solve for the eigenstates and then write a cogent summary of the nature of the states, their organization and meaning. One thing about this matrix is, I believe, that there are two states with (Lz) e.v. of zero? Do you get that too? What does that mean? How can you handle those to get maximum insight? Are some of the eigenstates related to flower states in any way? These are some of the questions I would like to see addressed in your report. I trust that you will also come up with other questions and resolutions.
. I would like to also invite everyone in class to do this, if they like. It provides an excellent opportunity to develop and exhibit your understanding of matrices, degeneracy and angular momentum in QM. Ideally I would suggest you do this by Sunday --you can hand it in at the review section-- that way you will be able to focus on reviewing 1D QM, the hydrogen atom states and degeneracies, and related topics after the Sunday review section.
Best regards,
-Zack
Summary: descent of QM
This picture, while somewhat whimsical, provides a perspective on the relationships between the different "realms" of quantum physics we have explored this quarter. On the left are the systems of higher dimensionality ( 2 and 3D); on the right, the 1D quantum systems. The key difference between those two classes (1D vs higher D) is degeneracy. In 1D there is usually only one eigenstate associated with a particular energy; in higher D there may be several eigenstates which have the same energy.This is called degeneracy. The origin of degeneracy is symmetry. The degeneracies we see in the 3D harmonic oscillator and in the hydrogen atom eigenstates arises from the apparent and less apparent symmetries of these potentials. We say that the eigenstates which have the same energy belong to a particular degeneracy manifold.
. .For the 2D harmonic oscillator, for example, the two orthogonal states 10 and 01 span the 1st excited state degeneracy manifold. We also found, through our exploration via the Lz matrix, that the two states 01 +i 10 and 01-i 10 also span this 1st excited state degeneracy manifold and, moreover, that they are Lz eigenstates with e.v.'s of hbar and -hbar.
For the 3D harmonic oscillator there are three 1st excited states: 100, 010 and 001. You can show that these are all l=1 eigenstates, meaning that they are eigenstates of the operator L^2 with e.v. 2 hbar^2, however, they are not all eigenstates of Lz. Using the Lz matrix method (i.e. calculating the matrix of Lz, finding its eigenvectors and using them to create a new basis of this 3-fold degenerate manifold) one finds that the three orthogonal states: 100+i 010, 001, and 100-i 010 also span the 1st-excited state degeneracy manifold and that they are Lz eigenstates (with e.v.'s of: hbar, 0 and -hbar, respectively.)
Let's pause on this for now and look back at what we covered in 1d QM.
1D quantum physics: highlights:
Let’s start this summary from a phenomenological perspective and focusing on solutions of the time-independent Schrodinger equation in 1 dimension. Our key focus there was on calculating quantized energies for particular potentials and examining the nature the energy eigenstates.
o Bound states are created by an attractive potential. They tend to have a characteristic length scale that depends on the strength or spatial extent of the potential.
o Confinement leads to kinetic energy. This is most evident and clearly illustrated in the nature of ground states. Kinetic energy is generally of the form hbar^2/m a^2, where a is a characteristic length scale. The 2nd derivative term in the Schrodinger equation means that there is a cost to bending the quantum state function. Psi(x) has to rise high enough that the integral of (area under) Psi^2 is 1 and then come back to zero. When this happens over a short range of x, there is a high cost in kinetic energy.
o Excited states have nodes. For a given potential, you will find that if you order eigenstates according to their energy, from lowest to highest, they will also be ordered according to the number of nodes they have.
o Some 1D potentials have an infinite number of bound states; some have a finite number of bound states
o Potentials that have a finite number of bound states also have unbound (extended) states. These are not normalizable and they are characterized by a continuous “quantum number”, usually called k or q, rather than a discreet quantum number such as n. [ For example, for a constant potential there are zero bound states and the extended states can be written as exp[ikx], where k is any real number.]
o Potentials we have studied include:
1. Attractive delta function: 1 bound state
2. Finite square well: a finite number of bound states
3. Infinite square well: infinite number of bound states
4. Harmonic oscillator: infinite number of bound states
o For a given potential, all the eigenstates together span the space of all state functions
The above list essentially talks about characteristics of the spatial eigenstates (the eigenstates of the time-independent Schrodinger equation), however one of the most important capabilities of the quantum theory is the ability to calculate and predict the time evolution of the state function.
Usually one is given the state function at a particular time, e.g., t=0. The state function at any later time can then be calculated with no uncertainty. This is done by writing the state function at t=0 as a linear superposition of energy eigenstates. The time dependence for each energy eigenstate is exp[-iEnt/hbar], where En is the energy of the nth eigenstate.. Thus one obtains the time evolution of the state function.
The belief that time evolution of the state function can be obtained in this way can be viewed as the central dogma of quantum mechanics. The analogy to classical mechanics is as follows:
* In classical mechanics one is given the initial conditions, usually position and velocity, and one then calculates the position and velocity as a function of time. The time evolution of position and velocity is governed by Newton's relation: F=ma.
* In quantum mechanics one is given the initial conditions in the form of a state function and one calculates the time evolution of this state function. The time evolution of the state function is governed by the Schrodinger equation.
There is no intrinsic uncertainty in either system. For simple, solvable systems, if the position and velocity are precisely specified at t=0, and the potential (force) is know, then the position and velocity in the future can be obtained with precision. The same is true for the state function. If the state function is precisely specified at t=0, then, for a simple, solvable system, the future state function can be obtained with precision.
Wednesday, June 3, 2009
IP 2 solution notes: finding Lz eigenstates and exploring degeneracy manifolds
Here are some notes related to a solution of the IP2 problem, which involved exploring and elucidating the degeneracy manifolds of the 2D harmonic oscillator by finding Lz eigenstates from states constructing as products of 1D HO energy eigenstates.
What are we doing here? What is this problem all about? We can step back and take a broader perspective on this whole endeavor. This is not simple, but it may be helpful. Our central conundrum is associated with the realization that whenever there is degeneracy there are an infinite number of ways to span the degeneracy manifold. If we only look at one of them, then we have no perspective on this essential aspect, which is critical to understanding QM in more than 1D.
The things we do, like getting x,y,z states from the Psi_n,l,m and, on the other hand, getting Lz eigenstates (m) from x,y states of the 2D HO, are all related to gaining perspective in this way. Without this our perspective would be very limited.
Tuesday, June 2, 2009
Problems for our final exam
Please suggest problems or problem categories that you would like to see convered on the final exam. Suggestions can be as specific or general as you like.
Saturday, May 30, 2009
Diffussion of a localized crystal state: electron dynamics in a lattice
This post is a continuation of the "thread" that started with the posts IP#3 and, more recently, "Spec.Prob.#1: subverting the dominant paradigm ".
Based on what we did in class last Thursday, we now have a sense that we can calculate time evolution of a state function without going through the tedious an unintuitive process of doing separation of variables, finding energy eigenstates and then writing the general state function as a superposition. We can instead use an alternate approach based on the formal solution of the time-dependent Schrodinger equation, H Psi(t) = i hbar d (Psi(t)/dt, which is:
Psi(t) = exp[i H t/hbar] Psi(o)
= [1 + i H t/hbar] Psi(o),
where the last line is approximate, but becomes arbitrarily accurate as t gets small.
A key question then, is when might this be useful?
An example of where it might be useful, as we discussed in class, is in the 1D crystal problem, especially the very interesting case in which an electron is initially localized at one site (at t=0) and we would like to understand the time-dependent behavior of the electron, that is, we would like to calculate how the state function of the electron changes as a function of time.
Calculating this time evolution, except for numerical errors and errors associated with the approximations one inevitably makes in dealing with infinite sums, should produce a result mathematically equivalent to a representation of the localized state as a superposition of energy eigenstates (that being the standard approach, the "dominant paradigm").
Looking at the time evolution in this new way, we can answer questions like:
What is the probability that the electron will "hop" from site i to the adjacent site i+1? or a time t?, or, to put it another way:
what is the characteristic time for an electron to hop from one sit to another?
For the two square-well problem, I think that one could get a good estimate of the time for an electron to hop (or "tunnel", as we have said in the past) from one well to another using essentially this method. (This is a problem that we discussed earlier in the quarter, but never actually solved quantitatively.) Things one would like to calculate include the characteristic tunneling time and its' dependence on the distance separating the wells. I think you might be able to do a reasonable "back of the envelope" estimate of that, i think, with the two key things being the wave-function exponential length scale outside the well, the distance between the wells, and perhaps the depth of the well (all of which will go into the overlap integral, right?).
----
In addition to being able to do most things one can do with the original methodology based on superposition of eigenstates, this formalism makes it relatively easy (possible) to introduce disorder.
For example, here is a problem involving disorder that might interesting (though, i have never done this before or seen a solution or discussion of this problem. It may be related to "localization".):
Suppose the site at which the electron starts has a different energy than all the other sites. Let say it has a larger negative alpha, and thus a lower energy (a larger, more negative e_0). Then will the electron still diffuse away or will it remain localized?
This is part of the more general question of what effect will disorder have on the propagation or diffusion of an electron in a lattice?
Based on what we did in class last Thursday, we now have a sense that we can calculate time evolution of a state function without going through the tedious an unintuitive process of doing separation of variables, finding energy eigenstates and then writing the general state function as a superposition. We can instead use an alternate approach based on the formal solution of the time-dependent Schrodinger equation, H Psi(t) = i hbar d (Psi(t)/dt, which is:
Psi(t) = exp[i H t/hbar] Psi(o)
= [1 + i H t/hbar] Psi(o),
where the last line is approximate, but becomes arbitrarily accurate as t gets small.
A key question then, is when might this be useful?
An example of where it might be useful, as we discussed in class, is in the 1D crystal problem, especially the very interesting case in which an electron is initially localized at one site (at t=0) and we would like to understand the time-dependent behavior of the electron, that is, we would like to calculate how the state function of the electron changes as a function of time.
Calculating this time evolution, except for numerical errors and errors associated with the approximations one inevitably makes in dealing with infinite sums, should produce a result mathematically equivalent to a representation of the localized state as a superposition of energy eigenstates (that being the standard approach, the "dominant paradigm").
Looking at the time evolution in this new way, we can answer questions like:
What is the probability that the electron will "hop" from site i to the adjacent site i+1? or a time t?, or, to put it another way:
what is the characteristic time for an electron to hop from one sit to another?
For the two square-well problem, I think that one could get a good estimate of the time for an electron to hop (or "tunnel", as we have said in the past) from one well to another using essentially this method. (This is a problem that we discussed earlier in the quarter, but never actually solved quantitatively.) Things one would like to calculate include the characteristic tunneling time and its' dependence on the distance separating the wells. I think you might be able to do a reasonable "back of the envelope" estimate of that, i think, with the two key things being the wave-function exponential length scale outside the well, the distance between the wells, and perhaps the depth of the well (all of which will go into the overlap integral, right?).
----
In addition to being able to do most things one can do with the original methodology based on superposition of eigenstates, this formalism makes it relatively easy (possible) to introduce disorder.
For example, here is a problem involving disorder that might interesting (though, i have never done this before or seen a solution or discussion of this problem. It may be related to "localization".):
Suppose the site at which the electron starts has a different energy than all the other sites. Let say it has a larger negative alpha, and thus a lower energy (a larger, more negative e_0). Then will the electron still diffuse away or will it remain localized?
This is part of the more general question of what effect will disorder have on the propagation or diffusion of an electron in a lattice?
Thursday, May 28, 2009
Homework #9
1. For an electron in the ground state of a hydrogen atom potential, calculate the expectation value of:
a) 1/r
b) r
c) x^2
d) the vector r = (x,y,z)
e) p^2
[what is the relationship between the expectation value of r and of x^2 ?
2. a) (Using your results from problem 1) what are the expectation values of the K.E. and potential energy, respectively. Calculate your answer in e.V. and make sure the signs are correct.
How does the K.E. compare with the P.E.? (xc) How does this relate to the virial theorem?
b) Show that the P.E. can be expressed very simply using e^2 and the characteristic length scale, a_0 = hbar^2/m e^2.
c) Show that your result for the K.E. can be expressed very simply using hbar, m and the characteristic length scale, a_0 = hbar^2/m e^2.
d) why does this make sense?
3. For many potentials, there is no intrinsic length scale associated with V(r) and yet there can be quantum length scale which emerges when a particle of mass m is localized by the potential. What are the characteristic length scales for each of the following?:
a) V(x)= -alpha delta(x)
b) V(x) = (1/2) k x^2
c) V(x) = -e^2/r (3D)
4. Using the Psi_n,l,m as a basis, what is the represention of the state Psi = A (x/2a_0) exp[-r/2a_0], where A is a normalization constant independent of r, theta and phi or x, y and z). In other words, write the state Psi = A (x/2a_0) exp[-r/2a_0] as a linear combination of Psi_n,l,m eigenstates. How many states do you need? Is this a mixed state?
5. For the 3D H.O., the 1st excited state manifold contains 3 states: which could be called 100,
010, 001; where 100 includes a term proportional to x, 010 includes a term proportional to y, ...
Show, by construction, that the 1st-excited state manifold of H can be arranged to also include real eigenstate which are proportional to x, y and z (and orthogonal to each other). What else does the 1st excited state manifold of H include? Discuss the differences and similarities of the 1st excited state manifolds of H and a symmetric 3DH.O..
6. a) Use appropriate indentities and relationships to express Psi3,2,2 + Psi3,2,-2 as a relatively simple function of the cartesian variables x, y and z.
b) Show that a 45 degree rotation of this state (around the z axis) will give another state that: is a linear combination of 2 Psi n,l,m states, and is a relatively simple function of x,y and z. What is the linear combination? What would be good names for these states?
7. Use that state from problem 4, let's call it Psi_2x, to make the state (Psi 100 + Psi_2x)/sqrt(2)
a) calculate the expectation value of x for an electron in this state? Is this a mixed state? Is x a function of time?
8. With or without doing any calculations (calculation is extra-credit of you have time), describe the nature of the trajectory of the expectation value of the vector r:
a) in the state: (Psi 100 + Psi 211)/ sqrt(2)
b) in the state: (Psi 100 + Psi 210)/ sqrt(2)
c) Are these qualitatively similar or different? Discuss?
9. a) What is the degeneracy of the 2nd excited state (manifold) of a symmetric 2-D H.O. ?
(extra credit) b) What angular momentum eigenstates are contained within this N-dimensional subspace?
----
(extra credit) a) Figure out how to make the state, B (x^2 - y^2) exp[-r/3a_0] from a combination of Psi_n,l,m and/or how to make C x y exp[-r/3a_0].
b) What is the difference (relationship) between these two states? Are they eigenstates or not? Are they degenerate?
a) 1/r
b) r
c) x^2
d) the vector r = (x,y,z)
e) p^2
[what is the relationship between the expectation value of r and of x^2 ?
2. a) (Using your results from problem 1) what are the expectation values of the K.E. and potential energy, respectively. Calculate your answer in e.V. and make sure the signs are correct.
How does the K.E. compare with the P.E.? (xc) How does this relate to the virial theorem?
b) Show that the P.E. can be expressed very simply using e^2 and the characteristic length scale, a_0 = hbar^2/m e^2.
c) Show that your result for the K.E. can be expressed very simply using hbar, m and the characteristic length scale, a_0 = hbar^2/m e^2.
d) why does this make sense?
3. For many potentials, there is no intrinsic length scale associated with V(r) and yet there can be quantum length scale which emerges when a particle of mass m is localized by the potential. What are the characteristic length scales for each of the following?:
a) V(x)= -alpha delta(x)
b) V(x) = (1/2) k x^2
c) V(x) = -e^2/r (3D)
4. Using the Psi_n,l,m as a basis, what is the represention of the state Psi = A (x/2a_0) exp[-r/2a_0], where A is a normalization constant independent of r, theta and phi or x, y and z). In other words, write the state Psi = A (x/2a_0) exp[-r/2a_0] as a linear combination of Psi_n,l,m eigenstates. How many states do you need? Is this a mixed state?
5. For the 3D H.O., the 1st excited state manifold contains 3 states: which could be called 100,
010, 001; where 100 includes a term proportional to x, 010 includes a term proportional to y, ...
Show, by construction, that the 1st-excited state manifold of H can be arranged to also include real eigenstate which are proportional to x, y and z (and orthogonal to each other). What else does the 1st excited state manifold of H include? Discuss the differences and similarities of the 1st excited state manifolds of H and a symmetric 3DH.O..
6. a) Use appropriate indentities and relationships to express Psi3,2,2 + Psi3,2,-2 as a relatively simple function of the cartesian variables x, y and z.
b) Show that a 45 degree rotation of this state (around the z axis) will give another state that: is a linear combination of 2 Psi n,l,m states, and is a relatively simple function of x,y and z. What is the linear combination? What would be good names for these states?
7. Use that state from problem 4, let's call it Psi_2x, to make the state (Psi 100 + Psi_2x)/sqrt(2)
a) calculate the expectation value of x for an electron in this state? Is this a mixed state? Is x a function of time?
8. With or without doing any calculations (calculation is extra-credit of you have time), describe the nature of the trajectory of the expectation value of the vector r:
a) in the state: (Psi 100 + Psi 211)/ sqrt(2)
b) in the state: (Psi 100 + Psi 210)/ sqrt(2)
c) Are these qualitatively similar or different? Discuss?
9. a) What is the degeneracy of the 2nd excited state (manifold) of a symmetric 2-D H.O. ?
(extra credit) b) What angular momentum eigenstates are contained within this N-dimensional subspace?
----
(extra credit) a) Figure out how to make the state, B (x^2 - y^2) exp[-r/3a_0] from a combination of Psi_n,l,m and/or how to make C x y exp[-r/3a_0].
b) What is the difference (relationship) between these two states? Are they eigenstates or not? Are they degenerate?
Tuesday, May 26, 2009
Learning and Teaching styles: letter from a former student
I realize that this class can seem frustrating at times. My goal is to teach in a way that is alive and enjoyable, and in a way that can help you learn and remember. The fraction of material that students tend to remember a year or two after a typical physics class has been studied and is surprisingly low. Here is a letter from a former student which I received a few years ago which addresses this point. It is one of a number of communications along these lines I have received from former students.
"Hi, ... I'm at Cornell and I'm finally settling back into school again now that I'm in the second semester. Anyway, I was inspired to send you this greeting right now at this very moment because I'm currently taking Stat Mech and as I was working on my homework for his class, I looked up my old homeworks from Stat Mech back in Santa Cruz. Well, you were my teacher for Stat Mech at UCSC, and of course my teacher for Quantum II, and also my waves teacher for 5B, long, long ago. It just keeps coming up over and over again that those topics are the ones I understand the most! I mean, it just can't be denied that I my understanding is WAY better in those subjects compared to others.
When I was in Quantum last semester and we got into Zeeman effect, changing basis stuff, I had this wonderful, familiar feeling of understanding, and all I had to do was go back and read some of my old homeworks from your class, and then I had it all at my fingertips again. In fact, most of the time at Cornell, I've been feeling pretty far behind all my classmates, but when it comes to the subjects that I studied with you, I actually feel like I am helping them!
When I think of Stat Mech or Quantum II or even 5B, there are only about two or three problems that I really remember from the course, but each of them involved a
serious revelation on my part. I don't actually remember any problems from any other class at UCSC!
... (two paragraphs of examples, etc deleted)...
Wow, you really had a serious impact on my
understanding of physics! Thank you."
Here is one more, that discusses a student's attitude and response, and how that changed over the quarter for this student:
"The first time I took a class from Zack it took me a long time to warm up to his teaching style. He forces kids out of their standard class comfort zone ... of regurgitative lectures and by making them participate and think in class, both of which students really seem to hate. I know i did [hate it] for the first 3/4 of Phys 105. ...
Zack has a very unique teaching style that ... gets to the fundamentals, the very foundations of what he is teaching. ... Student's who take Zack's courses do not just know the material at the end, they have a deep understanding of it...."
"Hi, ... I'm at Cornell and I'm finally settling back into school again now that I'm in the second semester. Anyway, I was inspired to send you this greeting right now at this very moment because I'm currently taking Stat Mech and as I was working on my homework for his class, I looked up my old homeworks from Stat Mech back in Santa Cruz. Well, you were my teacher for Stat Mech at UCSC, and of course my teacher for Quantum II, and also my waves teacher for 5B, long, long ago. It just keeps coming up over and over again that those topics are the ones I understand the most! I mean, it just can't be denied that I my understanding is WAY better in those subjects compared to others.
When I was in Quantum last semester and we got into Zeeman effect, changing basis stuff, I had this wonderful, familiar feeling of understanding, and all I had to do was go back and read some of my old homeworks from your class, and then I had it all at my fingertips again. In fact, most of the time at Cornell, I've been feeling pretty far behind all my classmates, but when it comes to the subjects that I studied with you, I actually feel like I am helping them!
When I think of Stat Mech or Quantum II or even 5B, there are only about two or three problems that I really remember from the course, but each of them involved a
serious revelation on my part. I don't actually remember any problems from any other class at UCSC!
... (two paragraphs of examples, etc deleted)...
Wow, you really had a serious impact on my
understanding of physics! Thank you."
Here is one more, that discusses a student's attitude and response, and how that changed over the quarter for this student:
"The first time I took a class from Zack it took me a long time to warm up to his teaching style. He forces kids out of their standard class comfort zone ... of regurgitative lectures and by making them participate and think in class, both of which students really seem to hate. I know i did [hate it] for the first 3/4 of Phys 105. ...
Zack has a very unique teaching style that ... gets to the fundamentals, the very foundations of what he is teaching. ... Student's who take Zack's courses do not just know the material at the end, they have a deep understanding of it...."
Quiz credit...
FAQ:
o So, was the quiz worth anything?
-The point of the quiz was to:
“…help you be ready for Tuesday's class, in which we will solve the H-atom problem. In fact just being aware of these questions will help you be prepared”
The real value of the quiz would be if it helped you be ready for and appreciate today’s class, especially the subtle, but important role of the quantized length scale that emerges at the very end of the calculation. It is easy to miss that after the fatigue of a long calculation.
o Can't we leave it at,"a length scale constructed from e^2, hbar, and m is hbar^2/me^2"?
-Yes
o Is there any way we can make up the quiz?
-If you post the answer to the quiz here you will get full credit for the quiz. If you also did it correctly in class you get extra credit.
o "people have been punished directly for showing up late (understandable), and indirectly for finishing homework on time…" -Jon
-Perhaps it might seem that way to you now, but from my point of view no one is being punished. Punishment is not really an issue here at all. This is about trying to create opportunities for you to learn and remember interesting and important physics.
The idea was that if people spent 1/2 hour thinking about length scales and attractive potentials they would get more out of the class.
o So, was the quiz worth anything?
-The point of the quiz was to:
“…help you be ready for Tuesday's class, in which we will solve the H-atom problem. In fact just being aware of these questions will help you be prepared”
The real value of the quiz would be if it helped you be ready for and appreciate today’s class, especially the subtle, but important role of the quantized length scale that emerges at the very end of the calculation. It is easy to miss that after the fatigue of a long calculation.
o Can't we leave it at,"a length scale constructed from e^2, hbar, and m is hbar^2/me^2"?
-Yes
o Is there any way we can make up the quiz?
-If you post the answer to the quiz here you will get full credit for the quiz. If you also did it correctly in class you get extra credit.
o "people have been punished directly for showing up late (understandable), and indirectly for finishing homework on time…" -Jon
-Perhaps it might seem that way to you now, but from my point of view no one is being punished. Punishment is not really an issue here at all. This is about trying to create opportunities for you to learn and remember interesting and important physics.
The idea was that if people spent 1/2 hour thinking about length scales and attractive potentials they would get more out of the class.
Homework due dates
How about if we all hand in HW#7 and #8 on Thursday? That way you can all get credit for the work you have done? Would that make people happy? (Or is it too late for that now (: )
Monday, May 25, 2009
Quiz tomorrow (Tuesday)
Instead of actually handing in any homework, how about if we have a quiz tomorrow at the beginning of class. Very short; based on HW8 (which has only one problem).
The quiz will ask for a length scale constructed using only e^2 (in Joules-meters), hbar and m (kg). Any quantity with units of length is correct. Assume e^2 has units of Joule-meters (i.e., so that e^2/r has units of energy). Maybe also a K.E. and P.E.. Those are really easy, right?
The quiz will ask for a length scale constructed using only e^2 (in Joules-meters), hbar and m (kg). Any quantity with units of length is correct. Assume e^2 has units of Joule-meters (i.e., so that e^2/r has units of energy). Maybe also a K.E. and P.E.. Those are really easy, right?
Special Problem #1, subverting the dominant paradigm
This is not a problem that you have to do , but, surprisingly, it is not that difficult if you understand everything we have done so far. It takes our study of 1D quantum a step further by approaching the time dependence (of the state function) in a new manner. It also uses matrix formulation. Seeing time dependence in emerge in a different way may provide a new and illuminating perspective and insight into "how QM works".
So far we have also obtained time dependence by: solving the time-independent Schrodinger equation and the writing a given state function, at t=0, as a superposition on energy eigenstates each with their own time dependence. Then the time dependence of the overall state function emerges, somewhat cryptically, from the "de-phasing" of the different exp[-i E_n t/hbar] terms.
But suppose someone came along and said:
Hmm... H Psi = i hbar d Psi/dt ....
what is the difficulty? Isn't the solution just:
Psi(t) = Psi(0) exp[ i H t/hbar]
What would you say to that? Can we apply that in some concrete and useful way???
(Pause here to reflect, argue, consider and react.)
For our crystal problem, consider the state, Psi_j = phi_0 (x-ja), which corresponds to the electron being located (only) at the jth site in the crystal. Consider the set of all such states. In the basis of these states can the Hamiltonian, H, can be written as a matrix? To the same level of approximation we used for IP#3, what is that matrix? (part a))
b) What if you were told (given) that at t=0 an electron is in the ground state of the jth potential. How could you calculate what the electron does as a function of time?
So far we have also obtained time dependence by: solving the time-independent Schrodinger equation and the writing a given state function, at t=0, as a superposition on energy eigenstates each with their own time dependence. Then the time dependence of the overall state function emerges, somewhat cryptically, from the "de-phasing" of the different exp[-i E_n t/hbar] terms.
But suppose someone came along and said:
Hmm... H Psi = i hbar d Psi/dt ....
what is the difficulty? Isn't the solution just:
Psi(t) = Psi(0) exp[ i H t/hbar]
What would you say to that? Can we apply that in some concrete and useful way???
(Pause here to reflect, argue, consider and react.)
For our crystal problem, consider the state, Psi_j = phi_0 (x-ja), which corresponds to the electron being located (only) at the jth site in the crystal. Consider the set of all such states. In the basis of these states can the Hamiltonian, H, can be written as a matrix? To the same level of approximation we used for IP#3, what is that matrix? (part a))
b) What if you were told (given) that at t=0 an electron is in the ground state of the jth potential. How could you calculate what the electron does as a function of time?
Sunday, May 24, 2009
Homework 8 & reading for Tuesday
I know you probably are getting to hate me for posting too much HW, not giving you enough warning, etc, etc., but here is another HW that will help you be ready for Tuesday's class, in which we will solve the H-atom problem. In fact just being aware of these questions will help you be prepared: [Comments and questions below would be much appreciated (and will get you credit for this assignment). How about: if you comment, then you don't have to hand it in. How does that seem? PS. These problems are probably more important, right now, than 7 and 8 from problem set #7. ]
** The quarter will be over soon. I think we have only 2 more weeks --4 more chances to meet and discuss quantum mechanics. I hope you don't miss this rare and fleeting opportunity to learn about quantum physics.
1. For a particle of mass m in the potential, V(r) = -e^2/r (in units, as Liboff uses, in which e^2/r has units of energy):
a) What is a characteristic length scale that you can construct using e^2, hbar and m?
b) How do the units of e^2 compare with the units of alpha, where alpha is the strength of a 1D delta function? (comments welcome)
[notice the similarity between the form of the characteristic lengths for this potential, -e^2/r, and the 1D delta function, -alpha delta(x).] (Please comment below)
c) What is a characteristic kinetic energy scale associated with that length scale?
d) For this potential function, what is a characteristic potential energy scale that you can make with that length scale?
2. check later please.
reading: On Tuesday we will go at a brisk pace through the process of presenting the Schrodinger eqn in 3D in spherical coordinates (Del^2) (bring your 3D glasses if you have them), separation of variables, solving for L^2 eigenstates, Ylm, solving the radial equation with the substitution u(r)=rR(r). This is pretty standard and you can read about it in any quantum book if you want to be prepared in that way.
[in Liboff, it is a little scattered: pages 367-380, more or less, treat the theta-phi part and the angular momentum eigenstates; then page 413 shows the Laplacian in spherical coordinates and pages 446 to 449, more of less, cover the hydrogen atom energies, eigenstates, degeneracies, wave function shapes and hybridization*,...
The most interested part will be the way in which we discover the quantization of energy when we solve the radial eqn. We will use the characteristic length scale in that and show how both length scale and energy are quantized together...
* hybridization refers to combining degenerate eigenstates to produce new bases of degeneracy manifolds. This provide a choice of ways to look at and use the eigenstates.
** The quarter will be over soon. I think we have only 2 more weeks --4 more chances to meet and discuss quantum mechanics. I hope you don't miss this rare and fleeting opportunity to learn about quantum physics.
1. For a particle of mass m in the potential, V(r) = -e^2/r (in units, as Liboff uses, in which e^2/r has units of energy):
a) What is a characteristic length scale that you can construct using e^2, hbar and m?
b) How do the units of e^2 compare with the units of alpha, where alpha is the strength of a 1D delta function? (comments welcome)
[notice the similarity between the form of the characteristic lengths for this potential, -e^2/r, and the 1D delta function, -alpha delta(x).] (Please comment below)
c) What is a characteristic kinetic energy scale associated with that length scale?
d) For this potential function, what is a characteristic potential energy scale that you can make with that length scale?
2. check later please.
reading: On Tuesday we will go at a brisk pace through the process of presenting the Schrodinger eqn in 3D in spherical coordinates (Del^2) (bring your 3D glasses if you have them), separation of variables, solving for L^2 eigenstates, Ylm, solving the radial equation with the substitution u(r)=rR(r). This is pretty standard and you can read about it in any quantum book if you want to be prepared in that way.
[in Liboff, it is a little scattered: pages 367-380, more or less, treat the theta-phi part and the angular momentum eigenstates; then page 413 shows the Laplacian in spherical coordinates and pages 446 to 449, more of less, cover the hydrogen atom energies, eigenstates, degeneracies, wave function shapes and hybridization*,...
The most interested part will be the way in which we discover the quantization of energy when we solve the radial eqn. We will use the characteristic length scale in that and show how both length scale and energy are quantized together...
* hybridization refers to combining degenerate eigenstates to produce new bases of degeneracy manifolds. This provide a choice of ways to look at and use the eigenstates.
Thursday, May 14, 2009
Homework #7
Added problems:
[ note: both 4's are optional. Mike or Kelsey is supposed to post something explaining the problem with 4.o]
4.o (original) Calculate the expectation value of r (the vector), in the mixed state that is an equal mix of the state 00 and one of the (correctly normalized) +-2hbar eigenstates. You know, something like: (20-02 + i 11) or its friend.
4.1 (alt.revised version) It was cool how r went around in circles in problem 3, right? Create a problem in which r goes around in circles that involves a 2hbar eigenstate (of the 2D HO).
(Problems 5 to X are probably more important to understanding H.)
5. Write Del^2 in:
a) 2D
b) 3D
(oh, i mean in both cartesian and in cylindrical (a) and spherical (b) coordinates.)
6. Can part of Del^2 in 5 be expressed in terms of L^2, where L is the appropriate angular momentum "operator"? (you can answer this with a yes or no, and by writing down the part. You don't have to prove it or derive it.)
7. What is the commutator of Lx and Ly? [hint: use the definition of L and the commutators for px and x, etc.
8. Show that if phi is an eigenstate of Lz, then (Lx+iLy)phi is also an eigenstate of Lz.
(most of the time.., i.e., unless it is zero)
-----
old:
This assignment involves polar plots of 2D H.O. eigenstates (so learning about polar plots is a good idea). If someone would outline a description here, that would be much appreciated.
If you find polar plots confusing, please feel free to do "contour plots" (plots of surfaces of constant value) instead. The main thing is to develop tools to enable you to understand and visualize the nature of a wave-function in 2 and 3 dimensions. (In 1D, we just plotted Psi(x) vs x. In 2 and 3 D we need to work a little harder on visualization techniques...) This will be essential to understanding the H-atom wave-functions.
Do those 1D H.O. states look like correct, normalized states? I think they are all right, but it is always nice to get a 2nd opinion. Problem 3 is time dependent, right?
Wednesday, May 13, 2009
H atom, electron states in crystals-IP#3
There seems to be a lot of interest in both the hydrogen atom and in electron states in crystals. I think we can do both,and it is true that we can cover angular momentum in the context of studying the hydrogen atom.
Hydrogen will be pretty elaborate: separation of spatial variables in spherical coordinates (we use spherical coordinates because of the symmetry of the potential), solving the theta-phi part (thereby getting angular momentum eig.-states), and then solving the radial part, and finally putting it all together and noticing the unusual degeneracies that occur (s and p states with the same energy...) which are related to less obvious symmetries of the 1/r potential.
With regard to crystal states, I think that some of you will appreciate the opportunity to work on them yourselves, in addition to covering that in class, so I am re-posting both the earlier notes on that and IP#3 here, with the idea class can start in on that as an interactive on-line problem right away. If you work on it i think you would get a much deeper and long-laster appreciation of this important topic, and its beautiful mathematical symmetries.
Here is a suggestion for how to obtain a concrete results. On the last page of the 3-page, handwritten notes, above, there is an expression for Eq. To evaluate that: start by assuming that the denominator is 1 (we can correct that a bit later). Eq is then equal to e0 plus the inner product of phi(x), delta-V(x) and Psi_q (x),
where phi(x) is the ground state of a single, isolated delta-function potential (it should have been called phi_0; and e0 is the energy of that isolated state. (Both of these are well-known to you.) delta-V and psi are both infinite series. That might seem scary, however, try keeping only the 3 terms n=0 and n= +-1 for Psi_q,
and only j=+-1 for delta-V (there is no j=0 term). You will see that this is justified when you do it. Combine complementary terms to get something real, and it will simplify greatly. The integrals are all effortless when you use attractive delta-function potentials!
Please post comments and questions here.
Tuesday, May 12, 2009
What should we do next?
Please comment here on what you think we should do next. (Please see poll on the right.)
Midterm Problem Poll
A poll regarding which midterm problems you may have liked is to the right. Please feel free to also comment here regarding what problems you liked or disliked, learned from or did not learn from, and why. An image of the midterm is below, for your reference.
Sunday, May 10, 2009
My thoughts on preparing for this midterm.
I think that there are different test preparation strategies and that this class is probably at the far end of a spectrum.
In this class there is an emphasis on deep understanding of a few things and mental flexibility and insight, rather than on superficial familiarity with a lot of information. I think it is relevant for studying strategy.
To make a long story short, one can prepare for an informationally intensive exam by studying shortly before and cramming information into your short term memory which you then reproduce on the test. That won't work here since: that information won't be of much value, studying and storing information in that way may reduce you mental agility.
What I would recommend is to study, by working problems, and try understand as much as possible a few days in advance (especially today), and then, by Monday evening, begin to rest and reflect in a restful and not too intensive way. (And get a good night's sleep.) That may help you assimilate your understanding and be prepared to think.
PS. Small details you forget, you can ask for during the test. (But please do prepare a good equation card and do not take advantage of that too much. Mostly, it is to hep you relax and focus on the big picture.) You want to come to this test with your "big mind".
In this class there is an emphasis on deep understanding of a few things and mental flexibility and insight, rather than on superficial familiarity with a lot of information. I think it is relevant for studying strategy.
To make a long story short, one can prepare for an informationally intensive exam by studying shortly before and cramming information into your short term memory which you then reproduce on the test. That won't work here since: that information won't be of much value, studying and storing information in that way may reduce you mental agility.
What I would recommend is to study, by working problems, and try understand as much as possible a few days in advance (especially today), and then, by Monday evening, begin to rest and reflect in a restful and not too intensive way. (And get a good night's sleep.) That may help you assimilate your understanding and be prepared to think.
PS. Small details you forget, you can ask for during the test. (But please do prepare a good equation card and do not take advantage of that too much. Mostly, it is to hep you relax and focus on the big picture.) You want to come to this test with your "big mind".
Friday, May 8, 2009
Midterm practice problems
[This is a "live post", meaning i will continue to edit both this top part and add or modify problems.]
The other thing I would mention, is that there are a lot of posts on this blog and it might be valuable for you to take the time to review them all and try to determine: which ones you understand, which ones are most important, and which ones are likely to be relevant for the midterm. Feel free to post questions, a summary, or related stuff here.
Also, on your "card" for the midterm, in addition to the Schroedinger wave eq., p as a derivative, a and a+ in terms as x and p, x in terms of a and a+, and p in terms of a and a+ (and as a derivative wi respt to x (ok, i said that twice), eig states: for inf sq well, 4 eig states for 1D HO, and the ground state for delta function, any thing else?, (and the boundary condition for the delta function) etc., please also include hbar in eV-seconds just in case there is a problem with actual numbers to work out.
I probably left some things out. This list is not comprehensive --just what popped into my head when I was wanting to mention that about hbar. Oh yeah. Length scales for each potential. (or inverse length scales).
Make sure you read all the problems and do as many as you can. You can ask questions.
Practice problems:
1. A particle of mass m (pomm) is in the ground state of an finite square well. Describe what happens if another square well of identical depth and width suddenly appears some distance away? As much detail as you can... what depends on what?...
Perhaps start with a grand statement regarding: what is the central issue of this problem, then get into details and how things depend on other things and so on.
2. A particle of mass m (pomm) is in the 1st-excited state of an finite square well. Describe what happens if another square well of identical depth and width suddenly appears some distance away? same comments as 1.
3. Consider a particle of mass m (pomm) in a 1D harmonic oscillator potential. Suppose that the particle is in a superposition state (mixed state) that is an equal combination of the mth and nth (energy) eigenstates. Calculate the expectation values of: x, x^2, p, p^2, V, T... (anything else?).
4. (More specific). For a pomm in a 1D HO, suppose state is an equal mix of |0> and |1> .
(all states are normalized). a) Calculate the expectation values of x and x^2 and thereby obtain delta x. (i hope this isn't too difficult.) b) Graph delta x (as a function of what?). Discuss, if it is interesting.
5. Suppose a pomm is initially (for t less than 0) in the ground state of a harmonic oscillator potential v=(1/2) k x^2 , and that at t=0 the potential suddenly morphs into a 2-delta function potential with two attractive delta functions of identical strength, one at x = +d/2, the other at x = -d/2 . Discuss what happens, and especially describe what you would find regarding the nature of that state function in the general vicinity of the origin (from like -3d to +3d or so) a long time later!
You may include a discussion of dependencies on strength, separation,...
b) DO the same thing for the case where one delta function is at x=0 and one is at x=+d. Does that make a difference. What might you find to be dramatically different, if anything?
Saturday afternoon additions:
6) Suppose that for t less than zero, an electron is in the ground state of an attractive delta function potential (adp) located at x=0, and that at t=0 two more adp's appear in addition to the first one), one at x= 2/k, the other at x=-2/k.
a) (quick calculation of expectation value at t=0+) Sketch the state function and V(x) at t=0+, and calculate the expectation value of the potential energy of the electron (at t=0+). How does it compare with the value of the expectation value of the potential energy of the electron before t=0?
b) (though question and essay) Describe what happens as a function of time after that. (You may focus on describing the time dependence of the "probability density".)
7. Consider a potential V(x) consisting of 2 identical attractive delta functions. When they are far apart there are two bound states, right? Show that the 2nd bound state ceases to exisit at a specific value of the ratio of the length scales k^-1 and d, where d is the separtation between the two delta functions and k^-1 is the length scale associated with the g.s. of a single delta function potential. (Is this true?) What is that ratio??
b) Describe in words several ways of looking at what happens as you approach this point from the part of parameter space in which there is an odd bound state. (to E, to k ...) (what is happening to the state???
8. (is phase important?)
Suppose you had one ensemble of sq well systems with a pomm in the state (psi_0+psi_1 )/sqrt(2), and another in the state (psi_0 + i psi_1 )/sqrt(2) . Could you distinguish these two cases via energy measurement? explain. how could you distinguish?
9, (phase) For an electron in a infinite square well, discuss the similarities and differences between the state: sqrt(2/L) Sin(2 pi x/L), (the 1st excited state), and the state which is the absolute value of that. Could you distinguish these by energy measurement (of separate ensembles of each)? Discuss that? Are the energy expectation values the same or different? Discuss other similarities or differences and end with a cogent paragraph, imagining that you ar the teacher of 139a, that explains the meaning and significance of phase in QM.
--
10. In 1D QM, which is easier: to be given the potential and asked to find the ground state, or to be given the ground state and asked to find the potential?
oh, i just remembered two more problems (closely related to each other)
11. Sketch and calculate the (transcendental) equation that would allow you to obtain the ground-state energy and wave-function for the half infinite harmonic oscillator (1D).
12. Sketch and calculate the (transcendental) equation that would allow you to obtain the ground-state energy and wave-function for the half infinite square well (1D).
(Is it clear what you are expected to do? Can anyone suggest a better way to phrase these questions?)
Do we need questions in other categories for balance? What are we leaving out?
The other thing I would mention, is that there are a lot of posts on this blog and it might be valuable for you to take the time to review them all and try to determine: which ones you understand, which ones are most important, and which ones are likely to be relevant for the midterm. Feel free to post questions, a summary, or related stuff here.
Also, on your "card" for the midterm, in addition to the Schroedinger wave eq., p as a derivative, a and a+ in terms as x and p, x in terms of a and a+, and p in terms of a and a+ (and as a derivative wi respt to x (ok, i said that twice), eig states: for inf sq well, 4 eig states for 1D HO, and the ground state for delta function, any thing else?, (and the boundary condition for the delta function) etc., please also include hbar in eV-seconds just in case there is a problem with actual numbers to work out.
I probably left some things out. This list is not comprehensive --just what popped into my head when I was wanting to mention that about hbar. Oh yeah. Length scales for each potential. (or inverse length scales).
Make sure you read all the problems and do as many as you can. You can ask questions.
Practice problems:
1. A particle of mass m (pomm) is in the ground state of an finite square well. Describe what happens if another square well of identical depth and width suddenly appears some distance away? As much detail as you can... what depends on what?...
Perhaps start with a grand statement regarding: what is the central issue of this problem, then get into details and how things depend on other things and so on.
2. A particle of mass m (pomm) is in the 1st-excited state of an finite square well. Describe what happens if another square well of identical depth and width suddenly appears some distance away? same comments as 1.
3. Consider a particle of mass m (pomm) in a 1D harmonic oscillator potential. Suppose that the particle is in a superposition state (mixed state) that is an equal combination of the mth and nth (energy) eigenstates. Calculate the expectation values of: x, x^2, p, p^2, V, T... (anything else?).
4. (More specific). For a pomm in a 1D HO, suppose state is an equal mix of |0> and |1> .
(all states are normalized). a) Calculate the expectation values of x and x^2 and thereby obtain delta x. (i hope this isn't too difficult.) b) Graph delta x (as a function of what?). Discuss, if it is interesting.
5. Suppose a pomm is initially (for t less than 0) in the ground state of a harmonic oscillator potential v=(1/2) k x^2 , and that at t=0 the potential suddenly morphs into a 2-delta function potential with two attractive delta functions of identical strength, one at x = +d/2, the other at x = -d/2 . Discuss what happens, and especially describe what you would find regarding the nature of that state function in the general vicinity of the origin (from like -3d to +3d or so) a long time later!
You may include a discussion of dependencies on strength, separation,...
b) DO the same thing for the case where one delta function is at x=0 and one is at x=+d. Does that make a difference. What might you find to be dramatically different, if anything?
Saturday afternoon additions:
6) Suppose that for t less than zero, an electron is in the ground state of an attractive delta function potential (adp) located at x=0, and that at t=0 two more adp's appear in addition to the first one), one at x= 2/k, the other at x=-2/k.
a) (quick calculation of expectation value at t=0+) Sketch the state function and V(x) at t=0+, and calculate the expectation value of the potential energy of the electron (at t=0+). How does it compare with the value of the expectation value of the potential energy of the electron before t=0?
b) (though question and essay) Describe what happens as a function of time after that. (You may focus on describing the time dependence of the "probability density".)
7. Consider a potential V(x) consisting of 2 identical attractive delta functions. When they are far apart there are two bound states, right? Show that the 2nd bound state ceases to exisit at a specific value of the ratio of the length scales k^-1 and d, where d is the separtation between the two delta functions and k^-1 is the length scale associated with the g.s. of a single delta function potential. (Is this true?) What is that ratio??
b) Describe in words several ways of looking at what happens as you approach this point from the part of parameter space in which there is an odd bound state. (to E, to k ...) (what is happening to the state???
8. (is phase important?)
Suppose you had one ensemble of sq well systems with a pomm in the state (psi_0+psi_1 )/sqrt(2), and another in the state (psi_0 + i psi_1 )/sqrt(2) . Could you distinguish these two cases via energy measurement? explain. how could you distinguish?
9, (phase) For an electron in a infinite square well, discuss the similarities and differences between the state: sqrt(2/L) Sin(2 pi x/L), (the 1st excited state), and the state which is the absolute value of that. Could you distinguish these by energy measurement (of separate ensembles of each)? Discuss that? Are the energy expectation values the same or different? Discuss other similarities or differences and end with a cogent paragraph, imagining that you ar the teacher of 139a, that explains the meaning and significance of phase in QM.
--
10. In 1D QM, which is easier: to be given the potential and asked to find the ground state, or to be given the ground state and asked to find the potential?
oh, i just remembered two more problems (closely related to each other)
11. Sketch and calculate the (transcendental) equation that would allow you to obtain the ground-state energy and wave-function for the half infinite harmonic oscillator (1D).
12. Sketch and calculate the (transcendental) equation that would allow you to obtain the ground-state energy and wave-function for the half infinite square well (1D).
(Is it clear what you are expected to do? Can anyone suggest a better way to phrase these questions?)
Do we need questions in other categories for balance? What are we leaving out?
Tuesday, May 5, 2009
Key dates: Quiz, Midterm and mid-quarter review problems
Midterm is moved to Tuesday, May 12. We will review "everything" on Thursday (the day after tomorrow).
Mid-quarter review problems 1 and 2 are due Friday at 4:30 PM. You can get extra-credit for doing a good job on these. Asking good questions and helping other people on the blog is also very much appreciated. Working together on groups off-line is also great. Maybe even better?
Mid-quarter review problems 1 and 2 are due Friday at 4:30 PM. You can get extra-credit for doing a good job on these. Asking good questions and helping other people on the blog is also very much appreciated. Working together on groups off-line is also great. Maybe even better?
Sunday, May 3, 2009
HW5 1-4, 7 solutions
In equation 24, John mentioned that that actually that should be a plus sign. (Is that what you got?)
In 7, one could also express the expectation value of x^2 in the form (1/2)(b^2) [1 + (t/tau)^2] , where tau is "characteristic time scale". What is that time scale? In thinking about that, one can get a deeper insight if you reflect on the fact that hbar/E has units of time, and fool around with trying E equal to the expectation value of the kinetic energy of an electron in this mixed state.
Homework #5, problems 5,6 &9: solutions and discussion
In 5a, the eigenvectors are obtained pretty quickly by guessing. I wouldn't expect you to converge on the eigenvectors this quickly. Probably several pages of multiplications and guessing to build your intuition is more realistic. Questions 5 and 6 are intended to build your familiarity with matrix multiplication, and eigenvector and e.v. identification, since that will be important in the latter part of this course.
In problem 9, some interesting issues arise. There is a crossover from a region of low transmission at low energy, to a region of high transmission for and incident electron of high-energy. The relationship of that to length scale (wavelength) and energy, and the phase-shift behavior are well worth understanding if you plan to take 139b and/or go to physics grad school. They care a lot about phase shifts and understanding crossover scales there. I only calculated C/A, which represents the transmission coefficient. You can do B/A, the reflectivity coefficient, by the same method and it complements C/A in an intuitive way. (What relation links them?)
Saturday, May 2, 2009
Interactive mid-quarter problems
This week I would like you to work on these interactive mid-quarter problems. I believe that these are important problems, and I think will help bring together much of what we have learned and what is important in this class, prepare us for the next stage of this class(1), and apply what we have learned to a very important class of problems(3). I hope you will be able to give them a lot of attention. This is not just another HW assignment, but rather an essential part of the learning and assessment in this class.
There are two contextual factors related to this evolution of format. One is that we are about half-way through the quarter, and i believe that your skills and basic knowledge of quantum physics formalism are developed to a point where it is possible for you to work on more more complex, open-ended and difficult problems. The other is harder for me to explain. But let me try.
In this class, and at a university in general, part of what you learn includes facts, information, methods and formalism. Another crucial part of your education involves problem solving skills, including the ability to evaluate your own progress. Problem solving tends to be straightforward when a problem is clearly defined and you are confident of what you need to do to solve it. It is less straightforward when these things are not clear, when you are wondering: what is this problem is really asking?, what basic method or starting point should I use?, or even what the answer would mean or what form the answer should take.
Yet the skill of working with poorly defined problems --resolving, refining and solving them-- is much more relevant to real world situations, including working at a technology company, doing research in graduate school, and pretty much anything else.
The idea of these problems, is for you to work on them both individually and collectively, using this site to discuss the problem, what it means, how to approach it, how to solve it, what the solution means, and so on. Evaluating your progress can be difficult on your own. There is often the fear that you may be going in completely the wrong direction, and that can be paralyzing. Evaluating direction and progress via discussion with colleagues (other students) is much different from conferring with a professor/expert. For example, you have to evaluate the quality of the advice you are getting, and ultimately, you have to decide for yourself what to do. However, it is also much less challenging than working completely alone. This is an opportunity to develop and demonstrate your understanding key concepts of QM, to learn and to help others.
There are two contextual factors related to this evolution of format. One is that we are about half-way through the quarter, and i believe that your skills and basic knowledge of quantum physics formalism are developed to a point where it is possible for you to work on more more complex, open-ended and difficult problems. The other is harder for me to explain. But let me try.
In this class, and at a university in general, part of what you learn includes facts, information, methods and formalism. Another crucial part of your education involves problem solving skills, including the ability to evaluate your own progress. Problem solving tends to be straightforward when a problem is clearly defined and you are confident of what you need to do to solve it. It is less straightforward when these things are not clear, when you are wondering: what is this problem is really asking?, what basic method or starting point should I use?, or even what the answer would mean or what form the answer should take.
Yet the skill of working with poorly defined problems --resolving, refining and solving them-- is much more relevant to real world situations, including working at a technology company, doing research in graduate school, and pretty much anything else.
The idea of these problems, is for you to work on them both individually and collectively, using this site to discuss the problem, what it means, how to approach it, how to solve it, what the solution means, and so on. Evaluating your progress can be difficult on your own. There is often the fear that you may be going in completely the wrong direction, and that can be paralyzing. Evaluating direction and progress via discussion with colleagues (other students) is much different from conferring with a professor/expert. For example, you have to evaluate the quality of the advice you are getting, and ultimately, you have to decide for yourself what to do. However, it is also much less challenging than working completely alone. This is an opportunity to develop and demonstrate your understanding key concepts of QM, to learn and to help others.
Interactive Problem #3, bound states of spatially periodic potentials
Crystalline solids, like Cu, Al, Ag, Au, C(diamond), C(graphite), Si, Ge, GaAs, CdTe, NaCl, KCl, LiF, RbI are made of atoms arranged in a crystalline structure. In such a circumstance, when the atoms are relatively close together, the electron states usually are not associated with a particular atom, but rather, extend throughout the crystal. It turns out that one can "build" these extended (non-local) crystal states from the states of individual atoms, and that this is conceptually and computationally a very succesful and useful approach.
Suppose we make a 1-dimensional crystal from an array of delta-function atoms, that is, an infinite array of attractive delta-function potentials of identical strength and equally spaced along the x axis. Using the notes from the earlier post on electron eigenstates in spatially periodic systems, see if you can find the approximate energies of all the electron eigenstates that come from the single bound state of a single, isolated attractive delta-function potential. Perhaps you can identify a "small parameter" that arises and obtain the energies to lowest order in which the energies are not all exactly equal to the starting bound state energy. (Boldness and daring may be called for here.)
b) Is there any redundancy in your states? Define a basis that spans this subspace of states. (Thoughtfulness and careful inspection may be of value here.)
c) Find a good way to present your results regarding the energies of these states. How do they compare to zero? What are possible values of the work-function for this system? How do the energies of the lattice states compare to the energy of the atomic state; how do they depend on "atom spacing", a? and so on.
Suppose we make a 1-dimensional crystal from an array of delta-function atoms, that is, an infinite array of attractive delta-function potentials of identical strength and equally spaced along the x axis. Using the notes from the earlier post on electron eigenstates in spatially periodic systems, see if you can find the approximate energies of all the electron eigenstates that come from the single bound state of a single, isolated attractive delta-function potential. Perhaps you can identify a "small parameter" that arises and obtain the energies to lowest order in which the energies are not all exactly equal to the starting bound state energy. (Boldness and daring may be called for here.)
b) Is there any redundancy in your states? Define a basis that spans this subspace of states. (Thoughtfulness and careful inspection may be of value here.)
c) Find a good way to present your results regarding the energies of these states. How do they compare to zero? What are possible values of the work-function for this system? How do the energies of the lattice states compare to the energy of the atomic state; how do they depend on "atom spacing", a? and so on.
Interactive Problem #1, mixed states
This interactive question will help us develop a deeper understand of mixed states (and eigenstates).
Suppose you are given 1000 quantum systems all prepared in the same exact mixed state. For concreteness, suppose that: it is an electron in a 1-D harmonic oscillator, and that the coefficients c_0 through c_5 may be non-zero while all other c's are zero. Suppose you want to learn about the energy of the electron and the nature of the state it is in.
What people do in practice to study the energy of electrons in bound state systems is to shine monochromatic UV or x-ray frequency light on them, so that an electron absorbs one photon in an energy conserving process; this give the electron enough energy to become free of the bound state system (i.e., if hf exceeds the "work function"). Then one detects the energy of the free electron (absorbing it into a detector) and inferring the energy of the bound electron state via conservation of energy.
Suppose that you do this with, say, 100 of your 900 mixed-state quantum systems. What will your results be? How would you display them? What would they tell you about the nature of the other 900 quantum systems you have?
PS. It might be useful to make up specific numerical values for the non-zero coefficients, and then discuss things for that specific case. This may help clarify what you are saying to others.
Suppose you are given 1000 quantum systems all prepared in the same exact mixed state. For concreteness, suppose that: it is an electron in a 1-D harmonic oscillator, and that the coefficients c_0 through c_5 may be non-zero while all other c's are zero. Suppose you want to learn about the energy of the electron and the nature of the state it is in.
What people do in practice to study the energy of electrons in bound state systems is to shine monochromatic UV or x-ray frequency light on them, so that an electron absorbs one photon in an energy conserving process; this give the electron enough energy to become free of the bound state system (i.e., if hf exceeds the "work function"). Then one detects the energy of the free electron (absorbing it into a detector) and inferring the energy of the bound electron state via conservation of energy.
Suppose that you do this with, say, 100 of your 900 mixed-state quantum systems. What will your results be? How would you display them? What would they tell you about the nature of the other 900 quantum systems you have?
PS. It might be useful to make up specific numerical values for the non-zero coefficients, and then discuss things for that specific case. This may help clarify what you are saying to others.
Interactive Problem #2, Lz eigenstates
This problem is designed to help with the transition from 1D QM to dealing with issues that commonly arise in 2- and 3-dimensional QM. This will help you to really understand angular momentum and degeneracy issues when they arise in more complex systems.
As with IP1, the concept and format of this problem, is for you to work on it both individually and collectively, using this site to discuss the problem, what it means, how to approach it, how to solve it, what the solution means, and so on. And then to turn in your own written version of it at the end of the week, which should include a pedagogic description and discussion of your results.
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IP 2.a) Calculate the matrix of the operator Lz in the basis of the 1st 6 energy eigenstates of the symmetric 2D harmonic oscillator. Can you use this matrix to find Lz eigenstates? If so, how? What do you learn from this? Explain. Discuss.****
*****see May 7 comment below
(Why the 1st 6 states?; why not 7 or 5?)
b) Are these Lz eigenstates also energy eigenstates? What energy manifold does each one belong to? Describe the nature and "content" of the degeneracy manifolds of this system. (A nuanced disussion of a is more important than b.)
c) Building on your results from this problem, what other problems can you think of? What are some possible things, related to angular momentum, that one might do next?
As with IP1, the concept and format of this problem, is for you to work on it both individually and collectively, using this site to discuss the problem, what it means, how to approach it, how to solve it, what the solution means, and so on. And then to turn in your own written version of it at the end of the week, which should include a pedagogic description and discussion of your results.
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IP 2.a) Calculate the matrix of the operator Lz in the basis of the 1st 6 energy eigenstates of the symmetric 2D harmonic oscillator. Can you use this matrix to find Lz eigenstates? If so, how? What do you learn from this? Explain. Discuss.****
*****see May 7 comment below
(Why the 1st 6 states?; why not 7 or 5?)
b) Are these Lz eigenstates also energy eigenstates? What energy manifold does each one belong to? Describe the nature and "content" of the degeneracy manifolds of this system. (A nuanced disussion of a is more important than b.)
c) Building on your results from this problem, what other problems can you think of? What are some possible things, related to angular momentum, that one might do next?
Thursday, April 30, 2009
The importance of peer-to-peer discussion
This is starting to happen already, but I have been thinking that at this point, you are starting to know and understand a lot, and it would be ideal there were much more a peer-to-peer discussion of homework problems and other issues associated with this class. Up to now, I have the sense that many questions are directed to me, more or less, and I would like to encourage you to ask and respond to each others questions freely. I can still contribute, but with a less central role in the comments and discussion. I think many issues can be addressed effectively via peer-to-peer discussion and that that has great value.
In the past there has been a high degree of correlation between active involvement in web site discussion and very high performance in a class(of the A, A+ variety). I think that, for most people, formulating questions, and discussing and explaining things to others, will tends to greatly deepen your understanding. I would like to encourage you to take advantage of this resource and contribute generously to a lively discussion with your fellow students.
(see also next post up).
In the past there has been a high degree of correlation between active involvement in web site discussion and very high performance in a class(of the A, A+ variety). I think that, for most people, formulating questions, and discussing and explaining things to others, will tends to greatly deepen your understanding. I would like to encourage you to take advantage of this resource and contribute generously to a lively discussion with your fellow students.
(see also next post up).
What we did on Thursday, wave packet propagation I
Here are some notes related to what we did in class today. This is an interactive post. What should we do next here?
Tuesday, April 28, 2009
Quiz solution
As pointed out in these solutions, the quiz used the word "probability" in a cavalier manner. Nevertheless, people did pretty well on it and seemed to know what to calculate.
The key starting point was to recognize that, for times after t=0 we need to write the state function as an expansion in energy eigenstates for the t>0 potential system. We can do this because the energy eigenstates form a basis; we want to do this because we know the time dependence of each of these eigenstates. (Also, the coefficients that arise in the expansion are relatively manageable because it is an orthonormal basis.)
In a sense, once we have clearly answered the questions: "what is a mixed state?" and "what is an eigenstate?", we are ready to move on to 3D QM. These are highly nuanced questions and are perhaps related to questions regarding the interpretation and meanings of the state function in quantum physics. We will keep working on it for a while.
Monday, April 27, 2009
Reading, for this week, next month..., and a question
Read about:
1. energy eigenstates (and momentum eigenstates) for a constant potential (free particle, V=0...).
gaussian wave-packets:
a) creating them (b(k)) and
b) time dependence (spreading and propagation).
2. Scattering in 1 dimension. what happens when a plan wave encounters a bump, e.g., a delta function?
For next month, read about angular momentum and the hydrogen atom. Chapters 9 and 10 of Liboff look pretty good for that to me. That might be a good notation to follow, i think, except maybe we could leave the 0 off of the a for the Bohr radius...
Thought question: What determines the magnitude of the Bohr radius? (That is, what sets the scale for the size of atoms?)
1. energy eigenstates (and momentum eigenstates) for a constant potential (free particle, V=0...).
gaussian wave-packets:
a) creating them (b(k)) and
b) time dependence (spreading and propagation).
2. Scattering in 1 dimension. what happens when a plan wave encounters a bump, e.g., a delta function?
For next month, read about angular momentum and the hydrogen atom. Chapters 9 and 10 of Liboff look pretty good for that to me. That might be a good notation to follow, i think, except maybe we could leave the 0 off of the a for the Bohr radius...
Thought question: What determines the magnitude of the Bohr radius? (That is, what sets the scale for the size of atoms?)
Sunday, April 26, 2009
Electron states in crystals
Here are some notes on how to find the (approximate) energy and nature of electron states in crystalline solids. (Starting in 1-dimension, of course.) There will be a part 2, where we actually calculate that thing at the end (Eq) (approximately). If you want to do that on your own in the meantime, I think you can. To get things started, I would suggest that the thing in the denominator is pretty much 1, and so you can focus on the numerator and just look for the 4 or 5 largest terms from the infinite serieses. For a concrete case, attractive delta functions are perfect, (and you know what the solution is for just one of them).
Regarding Measurement
Here are 2 papers that discuss quantum measurement issues. Note that they both assume, that measurement is an unsolved problem. On reflection, perhaps this is not surprising. Quantum systems, in our case, consist of one electron and a potential; a measurement involves a macroscopic apparatus. An idealized problem we could envision might involve a simple quantum system, like the 1D-HO or inf. sq well, for t<0, t="0">0). Then to correctly represent the measurement in the context of quantum theory, we would have to solve for the energy eigenstates of the measurement apparatus and then look at the evolution of the state function for the system. Since the measurement apparatus involves trillions of electrons and protons, this is, at the very least, a very challenging problem.
The paper by Philip Ball discusses issues and experiments related to measurement. Decoherence is an important concept here, and I would not suggest that is easy to understand or that there are simple ways to model it.
The earlier paper by P.W. Anderson uses the word "emergent" in the title. The concept of emergence is generally associated with extremely challenging and perhaps unsolvable matters that appear in science. It is related, i think, to the issue of "what is knowable", and used in the context of understanding and facing limits to reductionism in science. For example, people might say that the ineffectiveness of quantum theory in predicting the behavior of a mouse, a human (or a many-electron Hubbard model system) is not associated with any problem or incompleteness in our knowledge of the Schrodinger equation, or any question that it, in principle, governs the behavior of every electron in that mouse, but rather that there are "levels of emergence" separating phenomena at the mouse-level of complexity from the one-electron (or several electron) regime, and that these borders cannot be crossed.
As far as I noticed, neither paper mentions "wave-function collapse". Wave-function collapse, as I understand it, is something that got "invented" when people were very confused about how to understand what the Schrodinger equation was telling us. From a historical perspective, the Schrodinger equation was predicting and explaining things in a stunning level of detail. Most of these were related to energy, e.g., the H-atom energy levels, the details of their fine-structure, and their dependencies on magnetic and electric field.
My understanding, and this may not be everyone's view, is that wave-function collapse is something that got concocted in a hurry when people were desperate to "explain" the meaning of the new "quantum theory". Unlike the Pauli exclusion principle, the Heisenberg commutation relations and the normalization condition, which along with the Schrodinger wave equation form the basis of quantum theory, it has drifted along in a nether world in which it is not part of the core of quantum theory, but it has not gone away either. (Perhaps because measurement theory remains an unsolved problem?) Wave-function collapse, as I understand it, postulates a time-dependence which is different from, and exists in addition to, the time dependence which comes from the Schrodinger equation (via separation of variables, using energy as a separation parameter, solving for the energy eigenstates, and obtaining a exp{-iEt/hbar} time dependencies. I am not aware of there being any particular mathematical expression associated with wave-function collapse time dependence, though probably that is just my lack of familiarity with that area, and perhaps a number of possibilites have been proposed?
Friday, April 24, 2009
Homework #5, due Friday, 4:30 PM
Added notes: Problem 4 is particularly important; (3 is a prelude to 4). The image to the right is from a comment by Jerome about problem 4. (See comment #14 below.
The matrix problems, 5 and 6, should be computationally easy, yet very intriguing. No one is really expected to do problem 8.
1. For the 1-D harmonic oscillator:
a) Starting from the ground state wave-function, psi(x), use a+ to generate the 1st excited state wave-function and then graph it vs x.
2. Graph:
a) 2(x /a)^2 - 1
b) 2 (x/a)^3 - 3
correction added: that should say 3x instead of "3".
In what regions are each of these functions positive? negative? At what values of x do the sign changes take place?
c) Note that these are related to harmonic oscillator ground states.
3. Consider an electron moving in 2 dimensions in the potential V(x) = (1/2) k [x^2 + y^2].
a) what is the kinetic energy operator?
b) Starting with the ground state find the first three eigenstates* of H, for a particle of mass m in this potential. [Hint: If you can show that the differential equation is separable with regard to x and y, then the solutions (eigenstates) can be constructed via products of 1D HO eigenstates.]
* That is, the 3 eigenstates which have the lowest energies.
4. Using the (standard) definition of 2D angular momentum (z component), Lz= x p_y - y p_x ,
a) show that Lz can be expressed as - i hbar [ ax+ ay - ax ay+]
[hint: I think that x and y raising and lowering operators commute with each other; this is related to the separability of the diff. eq., i believe. Actually, do you need to use this? I am not sure.]
b) Calculate the expectation value of Lz for each of your eigenstates from #3
c) Being careful about commutation, calculate Lz^2 (conceptually similar to p^2, except now we are dealing with angular, rather than linear, momentum.)
d) Calculate the expectation value of L^2 (==Lz^2, in 2D) for each of your 3 states from #3. What do you learn from this? Are these expectation values zero or non-zero? Feel free to interpret speculate, propose new directions for investigation...
e) How many states are in the next "degeneracy manifold"? (That is, if you consider a 4th state, how many states are there that have the same energy and are linearly independent?) What are they? (That is, what is a spanning set for this degeneracy manifold?)
5 a) For the matrix:
| 0 1 0 |
| 1 0 1 |
| 0 1 0 |
Use intuitive methods (i.e., guess a vector and mutiply it by the matrix, ask a friend or a computer (anything NOT involving determinants)) to find the eigenvectors. Show that your putative eigenvectors are indeed eigenvectors by direct multiplication. What are their respective eigenvalues.
5 b) a) Multiply the matrix:
| 0 0 0 |
| 1 0 0 |
| 0 1 0 |
times the vector (1,0,0).
b) Do it again. (I mean with the vector that you got from a). Not the original one.) What do you learn from this?
6. a) Find the eigenvectors, and their respective eigenvalues. for each of the following matrices:
| 1 0 | | 0 1 | | 0 i |
| 0 -1 | | 1 0 | | -i 0 |
b) Express the eigenvectors of the 2nd matrix in the basis of the eigenvectors of the 3rd matrix.
Correction: Well, how about eigenvectors of the 1st matrix in the basis of the eigenvectors of the 2nd matrix. (You can do the 2nd in terms of the 3rd for extra credit, if you want.)
c) What do these matrices have in common? Do you think they like each other?? Why or why not!?
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7. Consider an electron in a 1D potential which is constant.
a) Show that exp{ikx} is an energy eigenstate with energy ___.
b) Show that it is also a momentum eigenstate with momentum hbar k.
c) Show that cos(kx) is also an eigenstate of one of these but not the other. Which one? Discuss briefly. (What is the relationship between exp{ikx} and cos(kx) ?)
d) For an intitial state, Psi(x,0), that is of Gaussian form (like the HO g.s.), calculate the time dependence of the state function, and of the expectation value of x^2.
e) For this state, show that the expectation value of p is zero, and that
f) the expectation value of p^2 is independent of time.
8) If you are really ambitious*, take the same state, multiply it by exp{ik_0 x} and then recalculate everything. I think you will likely find that the expectation of p is now finite (hbar k_0) and the the expectation value of x is now time dependent, and that you have a wave-packet moving to the right at a speed proportional to k_0/m .
*This might not be fun. Please do problem 9 first. You really don't have to do this problem.
9. Consider the 1d delta-function potential, V(x) = alpha delta(x).
Suppose that you write the part of the state function to the left of delta function as:
exp{ikx} + B exp{-ikx},
and the part of the state function to the right as
C exp{ikx},
and that you think of these terms as representing an incoming wave from the left, a reflected wave (B), and a transmitted wave (C).
a) Use the boundary conditions at the position of the delta function to calculate B and C.
b) Are B and C real or complex? What do they mean? Is there something you could graph as function of alpha that might convey some interesting information regarding them? Discuss.
c) Evaluate |B|^2 + |C|^2.
d) How are your results different for positive and negative alpha?
Homework #4 discussion related to non-local states
Zack: Today in office hours we had an interesting discussion on the integral in problem 4 (see above image) and the nature of the non-local (free) states that are involved in some of this week's HW (the t-dependent problems). Among other things, we discussed how to think about and deal with the non-local states in problems 4 and 6? (They just sort of drift away.)
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Kelsey: To continue the problem, what the above jpg basically means is that there is some probability equal to (c_1)^2 that the particle is in the ground state.
The excited states are unbound, so we don't know yet how to actually calculate their probabilities, nor do we really know much about them, except that their wavefunctions are not localized--they extend through all of space --so in any particular region their amplitude is small. [infinitely small]. So basically, the particle is either localized around the delta function, the likelihood of which is determined by (c_1)^2, or it is basically distributed through all space. [Similarly, in problem #6, the part of the initial wavefunction Psi(x,0) that is in the well that disappears at t=0 becomes essentially a superposition of unbound states. Then that part of the wave-function, after t=0, sort of spreads out and "disappears" very quickly (after t=0). The particle is therefore either localized in the ground state of the remaining well or distributed over space. I think.]
Since the probability of the particle being in the ground state, and therefore localized, is a function of c_1, which is itself a function of Psi(x, 0), the likelihood that the particle is localized is highest if the delta well is placed close to a peak of the original wavefunction Psi(x,0).
As the exponential constant k increases, so that the groundstate wavefunction of the delta potential is itself becoming concentrated in a spike around x_0, like a delta function, the probability (c_1)^2 would then seem to become an expression for the strength of the original function Psi(x, 0) at x_0, which determines the probability that the particle be found in a very small region around x_0. [ And the width of that region seems to be 4/k.]
I'm pretty sure that that is a summary of everything that was covered at office hours, but I didn't write everything down, so I might have gotten some stuff wrong.
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