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).
Thursday, April 30, 2009
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!?
---
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.)
----
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.
Thursday, April 23, 2009
Homework #4 solution notes for 1-6 (see also Discussion related to nonlocal states in #4 and 6)
Problem 6 ended a little abruptly and without comment. The result indicates that to linear order in alpha, the energy of the ground state is changed from its pure infinite-square-well value by the amount 2*alpha/L. Note that this is the expectation value of the small delta function potential in the ground state of the infinite square well (since sin (pi x/L) =1 in the center and the normaling factor is Sqrt(2/L), which gets squared in the expectation value calculation. This is a pretty simple result, and this apparent coincidence is probably something we should explore and use more if we get a chance.
Key dates: Quiz, Midterm and HW
Here is a summary of key dates:
* the due date for this week's HW is Friday (Apr 23, tomorrow)) at 4:00 PM. (my mailbox in the physics mailroom).
* There will be a Quiz this Tuesday (Apr 28)
* Our Midterm will be on Thursday, May 7
FAQ's
1) What will be on the quiz?
Well for a while this afternoon I couldn't think of anything that seemed appropriate for a quiz. Everything i could think of seemed too long, or too vague, or too qualitative; so I was about to give up on the quiz and then I thought of something that seemed pretty good.
It involves time dependence and a suddenly changing potential. I envision it with a computational part (a), an essay/discussion question of moderate length (b), and then a challenging quasi-conceptual, quasi-computational part (c) (that no one will have adequate time for haha). (Or maybe I am wrong about that last part?)
I would suggest becoming really adept in getting the actual value of indefinite integrals of Gausssians (extending to infinity), and in dealing with situations where a potential changes suddenly as a function of time.
Imagine you state function drifting through time. What would you do? Be the state function...
If i come up with any more thoughts when i write the problem, I will post more here. Feel free to post your questions and comments here. -Zack
* the due date for this week's HW is Friday (Apr 23, tomorrow)) at 4:00 PM. (my mailbox in the physics mailroom).
* There will be a Quiz this Tuesday (Apr 28)
* Our Midterm will be on Thursday, May 7
FAQ's
1) What will be on the quiz?
Well for a while this afternoon I couldn't think of anything that seemed appropriate for a quiz. Everything i could think of seemed too long, or too vague, or too qualitative; so I was about to give up on the quiz and then I thought of something that seemed pretty good.
It involves time dependence and a suddenly changing potential. I envision it with a computational part (a), an essay/discussion question of moderate length (b), and then a challenging quasi-conceptual, quasi-computational part (c) (that no one will have adequate time for haha). (Or maybe I am wrong about that last part?)
I would suggest becoming really adept in getting the actual value of indefinite integrals of Gausssians (extending to infinity), and in dealing with situations where a potential changes suddenly as a function of time.
Imagine you state function drifting through time. What would you do? Be the state function...
If i come up with any more thoughts when i write the problem, I will post more here. Feel free to post your questions and comments here. -Zack
Wednesday, April 22, 2009
Extending HW, Midterm date
Perhaps we should extend the due date for the current HW to Friday at 4:oo PM?
Also, I was thinking of Thursday May 7 as a good midterm date, but in view of that being well into the quarter, perhaps we should have a quiz on Tuesday the 28th (April) (or would the 30th be better?). What do you think?
Also, I was thinking of Thursday May 7 as a good midterm date, but in view of that being well into the quarter, perhaps we should have a quiz on Tuesday the 28th (April) (or would the 30th be better?). What do you think?
Thursday, April 16, 2009
Homework #4, final version, Due Friday 4 PM *
At this point we are getting close to the culmination of our study of 1-D QM and nearing our transition to 3-D QM, which focuses on the electron in the hydrogen atom, and angular momentum and degeneracy manifolds.
This is a really critical assignment. Part I is designed to prepare you for our future discussions regarding the nature and meaning of quantum theory; part II explores the use of raising and lowering operator formalism for the harmonic oscillator potential, and thereby introduces the matrix formalism of QM.
I hope you can spend ample time on each part. The two parts are rather different, and it is probably a good idea not to try to do them both in one day. You may want to plan to at least 2 large-time-block HW sessions in your schedule for this week (if that is how you do HW). (In other words, i think it would be confusing to do both parts at the same session.)
Part I:
1) Sketch the lowest energy 4 states of the 2 finite square well system (2 wells of identical width (nm) and depth (eV)). Count the nodes of your states and make sure they follow a systematic sequence. Do the states tend to come in pairs? How do the upper and lower pairs differ?
2) [It is essential to understand this problem before we study the nature and energy of electron states in crystals (which are systems with spatially periodic potentials).]
Sketch the four lowest energy states for a 4 finite square well system (4 wells of identical width (nm) and depth (eV), all equally spaced). How does the node counting work out? Can you do this based on just the g.s. of the single well??
3) Consider and electron in the ground state of an infinite square well of width L/2 extending from just x=0 to x=L/2 for times before t=0. Suppose that at t=0 the potential suddenly changes to an infinite square well of width L extending from x=-L/2 to x=L/2.
a) How would you approach the question of the time dependence of the electron state for t greater than zero?
b) Describe roughly what happens as a function of time. (Don't spend too much time on this one; this is just a warm-up problem.)
4) (This problem will be important for our discussion of quantum measurement (of position)).
Consider an electron in the ground state of a finite square well or harmonic oscillator or some other potential. The question of measuring what people sometimes call "the probability that the electron is in a particular region of space" could be approached in the following way:: Suppose that at t=0, the potential changes to V(x)=-alpha*delta(x-x_0). As in the previous problem, the wave-function will be unchanged between t= 0- and 0+, and, for t > 0, it will began to evolve in time in a way that is different from its time dependence before the potential changed. As in the previous problem, determining the time dependence for t > 0 involves expanding the state at t=0, Psi(x,0), in terms of the energy eigenstates of the new potential, which in this case include a single bound state (the ground state) and a continuum of positive energy states (that tend to have kinks at x=0).
a) Find the integral expression for the amplitude of the ground state coefficient in the expansion, c_1. (Evaluate it if you can. This is not required. If you do, it should be less than 1 for all k.) Convince yourself that in the limit that alpha is very very large, and thus 1/k is very very small, the value of this coefficient approaches* (2/sqrt(k)) Psi(x_0, 0),
(where k is the decay rate of the delta function g,s. wave-function (which is sqrt(k) exp{-k|x-x_0|} ). If that is the case, what is the probability that the electron is in the g.s. of the delta-function potential for t>0 (which is proportional to (c_1)^2, right?) To what other "probability" is this related?
b) Describe qualitatively the behavior of the electron state as a function of time for t less than and greater than zero.
c) What would large k reasonably mean here? If k goes to infinity, i think the probablility would go to zero, but is there an in-between region where this would work? What would be a good way to define that reqion (in terms of length scales)?
d) Discuss how this could be used to establish a position measurement on the original state.
* I think this may be correct. Perhaps someone could check, confirm or correct this (by integration) ?
5) Suppose an electron is in the g.s. of a finite square well and that at t=0 a second well, of identical width (nm) and depth (eV)) suddenly appears. What happens (with regard to the electron state)? What is the time scale for that? (That is, on what does the time scale depend?) [This is a one-electron problem. There is no 2nd electron associated with the 2nd well. The potential just changes at t=0.]
6) (optional thought problem) Suppose an electron is in the g.s. of a 2 finite square well system (2 wells of identical width (nm) and depth (eV)), and that at t=0 one well suddenly disappears. What happens?
7) a) Sketch the g.s. and 1st-excited state of a 2 delta function system (2 attractive wells of identical strength). b) Assume that the g.s. has the form A cosh(kx) in the region between the delta functions, and that the 1st exc. state has the form A sinh(kx) in the region between the 2 delta functions. Show that the transcendental equations for k-value corresponding to each state can be expressed in a form wherein tanh(kd/2) interesects 2 different functions of k, one of the form k/(2k_0 - k) and the other of the form (2k_0/k) - 1.
c) Graph those 3 functions together. Indicate which intersection corresponds to the g.s. and which to the 1st x.s.? Which has a larger k value? Which has a larger energy?
c) Show, from that graph or another one, that in the limit where d, the separation between the delta functions, becomes very large the two solutions coalesce (to what value of k?), and that as d becomes too small, the 2nd solution disappears at a critical value of d (what is that critical value of d? Where is the solution/intersection just before that is reached.). Describe, explain or discuss these results.
8) [A technique related to this problem (perturbation theory) will be relevant to our study the energy and nature of electron states in crystals. This problem also has an interesting juxtaposition of length scales.]
Consider an electron in a potential that consists of an infinite square well with a delta-function in the center.
a) For given values of V_0, L and alpha, calculate the energies of the g.s. and the first excited state. [hint: this is probably less difficult if you move the edge of the well to x=0 and put the delta function in the center of the well at L/2. That way your edge b.c. (psi=0) is automatically satisfied, and you can focus on the discontinuity issue in the center.]
b) Now let's consider the case where the delta function is weak. Then we can look at the effect of the delta-function on the energies of the infinite square well eigenstates to linear order in alpha. What inequality, related to length scales, could define "weak"?
c) For these two states, show that for small alpha (positive or negative) the difference between the actual energy and the energy of the pure inf. sq. well eigenstate approaches the expectation value of the delta-function potential in that eigenstate. [Try expanding the value of k around pi/L to linear order in alpha, and then calculate the lowest order (linear) effect of that additional small delta k term on the energy (using the relation between E and k from the Schr. eqn.)
9) Suppose {B}n is a set of orthonormal basis vectors which span an N-dimensional inner-product space. a) How many basis vectors are there? b) How would you express an arbitrary vector, V, in this space in terms of these basis vectors?
10) a) Show by matrix multiplication that (1,1) and (1,-1) are eigenvectors of the matrix:
|1, 1 |
|1, 1 |
What are their eigenvalues?
b) Find the eigenvector with the largest eigenvalue of the matrix:
|1, 1, 1, 1 |
|1, 1, 1, 1 |
|1, 1, 1, 1 |
|1, 1, 1, 1 |
What makes you believe (show) that it is an eigenvector?
What makes you believe (show) that it has the largest eigenvalue?
c) What is the trace of this matrix?
d) (extra credit) Find an eigenvector with the smallest eigenvalue.
Part II:
11. Calculate the commutator [a,a+] from the fundamental x,p commutator relationship.
12. Express x in terms of a and a+.
13. Using raising and lowering operators and Dirac formalism:
a) calculate the expectation values of x and x^2 for each (energy) eigenstate of the harmonic oscillator.
b) do the same for p and p^2.
c) graph these as a function of n.
d) Imagine doing the same calculation for the infinite square well eigenstates. It would not be fun. Don't do it!
e) How does the product delta x * delta p depend on n. (Discuss or graph it if it looks interesting.)
14. (Perhaps we should move this problem to next week. What do you think?)
For a particle moving in two dimensions one can define a and a+ operators in terms of x and px and y and py.
a) What is the angular momentum in 2D (about the axis perpendicular to the 2D plane)?
b) Express that in terms of your 2 sets of a’s and a+’s.
This is a really critical assignment. Part I is designed to prepare you for our future discussions regarding the nature and meaning of quantum theory; part II explores the use of raising and lowering operator formalism for the harmonic oscillator potential, and thereby introduces the matrix formalism of QM.
I hope you can spend ample time on each part. The two parts are rather different, and it is probably a good idea not to try to do them both in one day. You may want to plan to at least 2 large-time-block HW sessions in your schedule for this week (if that is how you do HW). (In other words, i think it would be confusing to do both parts at the same session.)
Part I:
1) Sketch the lowest energy 4 states of the 2 finite square well system (2 wells of identical width (nm) and depth (eV)). Count the nodes of your states and make sure they follow a systematic sequence. Do the states tend to come in pairs? How do the upper and lower pairs differ?
2) [It is essential to understand this problem before we study the nature and energy of electron states in crystals (which are systems with spatially periodic potentials).]
Sketch the four lowest energy states for a 4 finite square well system (4 wells of identical width (nm) and depth (eV), all equally spaced). How does the node counting work out? Can you do this based on just the g.s. of the single well??
3) Consider and electron in the ground state of an infinite square well of width L/2 extending from just x=0 to x=L/2 for times before t=0. Suppose that at t=0 the potential suddenly changes to an infinite square well of width L extending from x=-L/2 to x=L/2.
a) How would you approach the question of the time dependence of the electron state for t greater than zero?
b) Describe roughly what happens as a function of time. (Don't spend too much time on this one; this is just a warm-up problem.)
4) (This problem will be important for our discussion of quantum measurement (of position)).
Consider an electron in the ground state of a finite square well or harmonic oscillator or some other potential. The question of measuring what people sometimes call "the probability that the electron is in a particular region of space" could be approached in the following way:: Suppose that at t=0, the potential changes to V(x)=-alpha*delta(x-x_0). As in the previous problem, the wave-function will be unchanged between t= 0- and 0+, and, for t > 0, it will began to evolve in time in a way that is different from its time dependence before the potential changed. As in the previous problem, determining the time dependence for t > 0 involves expanding the state at t=0, Psi(x,0), in terms of the energy eigenstates of the new potential, which in this case include a single bound state (the ground state) and a continuum of positive energy states (that tend to have kinks at x=0).
a) Find the integral expression for the amplitude of the ground state coefficient in the expansion, c_1. (Evaluate it if you can. This is not required. If you do, it should be less than 1 for all k.) Convince yourself that in the limit that alpha is very very large, and thus 1/k is very very small, the value of this coefficient approaches* (2/sqrt(k)) Psi(x_0, 0),
(where k is the decay rate of the delta function g,s. wave-function (which is sqrt(k) exp{-k|x-x_0|} ). If that is the case, what is the probability that the electron is in the g.s. of the delta-function potential for t>0 (which is proportional to (c_1)^2, right?) To what other "probability" is this related?
b) Describe qualitatively the behavior of the electron state as a function of time for t less than and greater than zero.
c) What would large k reasonably mean here? If k goes to infinity, i think the probablility would go to zero, but is there an in-between region where this would work? What would be a good way to define that reqion (in terms of length scales)?
d) Discuss how this could be used to establish a position measurement on the original state.
* I think this may be correct. Perhaps someone could check, confirm or correct this (by integration) ?
5) Suppose an electron is in the g.s. of a finite square well and that at t=0 a second well, of identical width (nm) and depth (eV)) suddenly appears. What happens (with regard to the electron state)? What is the time scale for that? (That is, on what does the time scale depend?) [This is a one-electron problem. There is no 2nd electron associated with the 2nd well. The potential just changes at t=0.]
6) (optional thought problem) Suppose an electron is in the g.s. of a 2 finite square well system (2 wells of identical width (nm) and depth (eV)), and that at t=0 one well suddenly disappears. What happens?
7) a) Sketch the g.s. and 1st-excited state of a 2 delta function system (2 attractive wells of identical strength). b) Assume that the g.s. has the form A cosh(kx) in the region between the delta functions, and that the 1st exc. state has the form A sinh(kx) in the region between the 2 delta functions. Show that the transcendental equations for k-value corresponding to each state can be expressed in a form wherein tanh(kd/2) interesects 2 different functions of k, one of the form k/(2k_0 - k) and the other of the form (2k_0/k) - 1.
c) Graph those 3 functions together. Indicate which intersection corresponds to the g.s. and which to the 1st x.s.? Which has a larger k value? Which has a larger energy?
c) Show, from that graph or another one, that in the limit where d, the separation between the delta functions, becomes very large the two solutions coalesce (to what value of k?), and that as d becomes too small, the 2nd solution disappears at a critical value of d (what is that critical value of d? Where is the solution/intersection just before that is reached.). Describe, explain or discuss these results.
8) [A technique related to this problem (perturbation theory) will be relevant to our study the energy and nature of electron states in crystals. This problem also has an interesting juxtaposition of length scales.]
Consider an electron in a potential that consists of an infinite square well with a delta-function in the center.
a) For given values of V_0, L and alpha, calculate the energies of the g.s. and the first excited state. [hint: this is probably less difficult if you move the edge of the well to x=0 and put the delta function in the center of the well at L/2. That way your edge b.c. (psi=0) is automatically satisfied, and you can focus on the discontinuity issue in the center.]
b) Now let's consider the case where the delta function is weak. Then we can look at the effect of the delta-function on the energies of the infinite square well eigenstates to linear order in alpha. What inequality, related to length scales, could define "weak"?
c) For these two states, show that for small alpha (positive or negative) the difference between the actual energy and the energy of the pure inf. sq. well eigenstate approaches the expectation value of the delta-function potential in that eigenstate. [Try expanding the value of k around pi/L to linear order in alpha, and then calculate the lowest order (linear) effect of that additional small delta k term on the energy (using the relation between E and k from the Schr. eqn.)
9) Suppose {B}n is a set of orthonormal basis vectors which span an N-dimensional inner-product space. a) How many basis vectors are there? b) How would you express an arbitrary vector, V, in this space in terms of these basis vectors?
10) a) Show by matrix multiplication that (1,1) and (1,-1) are eigenvectors of the matrix:
|1, 1 |
|1, 1 |
What are their eigenvalues?
b) Find the eigenvector with the largest eigenvalue of the matrix:
|1, 1, 1, 1 |
|1, 1, 1, 1 |
|1, 1, 1, 1 |
|1, 1, 1, 1 |
What makes you believe (show) that it is an eigenvector?
What makes you believe (show) that it has the largest eigenvalue?
c) What is the trace of this matrix?
d) (extra credit) Find an eigenvector with the smallest eigenvalue.
Part II:
11. Calculate the commutator [a,a+] from the fundamental x,p commutator relationship.
12. Express x in terms of a and a+.
13. Using raising and lowering operators and Dirac formalism:
a) calculate the expectation values of x and x^2 for each (energy) eigenstate of the harmonic oscillator.
b) do the same for p and p^2.
c) graph these as a function of n.
d) Imagine doing the same calculation for the infinite square well eigenstates. It would not be fun. Don't do it!
e) How does the product delta x * delta p depend on n. (Discuss or graph it if it looks interesting.)
14. (Perhaps we should move this problem to next week. What do you think?)
For a particle moving in two dimensions one can define a and a+ operators in terms of x and px and y and py.
a) What is the angular momentum in 2D (about the axis perpendicular to the 2D plane)?
b) Express that in terms of your 2 sets of a’s and a+’s.
Tuesday, April 14, 2009
Ground states
Here is the page from class. More text (commentary) will be added later. Please feel free to post any queestions or comments, etc.
Friday, April 10, 2009
TA Office Hours by appointment
This quarter you can schedule office hours with your TA, John Kehayias, by appointment. If you are feeling behind, confused about something, or want to delve into something more deeply by yourself or with a group, you can email John at: kahayias at physics.ucsc.edu to schedule a meeting.
Homework #3, with solutions
Here is homework 3, which is due next Thursday. Is there anything else you can think of that we should be asking about and doing problems on? Do the problems seem interesting and not too repetitive? How is the level of difficulty? What do you think?
[final update: Saturday, 9:15 PM]
1. Calculate the characteristic size (delta x) of an electron in the ground state of:
a) an infinite square well,
b) an attractive delta-function potential,
c) a harmonic oscillator.
Where does this characteristic length scale, or size, come from in each case?
2. For each of the above cases, what is the relationship between the expectation value of the kinetic energy and the wave-function size?
3. For a potential consisting of two attractive delta-functions separated by a distance L,
a) What is the form of the ground state. Sketch the ground state.
b) Derive a relation for the wave-vector parameter* for the ground state (* the thing in the exponentials that has units of inverse length.) [hint: assume a symmetric solution, put the origin between the two, and only use the boundary conditions at one delta-function, e.g., the one at x=L/2]
c) Compare the energy and the wave-vector parameter for this state to that of the g.s. of an isolated attractive delta function.
d) How many length scales are there in this problem?
4. Sketch an intuitive guess of the ground state of 2 identical finite square wells (each of width L separated by a distance of about 5 L). Any thoughts as to what the 1st excited state may look like?
5. Write down all the eigenstates for an infinite square well of width L:
a) centered at x=0
b) centered at x=L/2
c) In what sense do these eigenstates form an orthonormal basis of a "Hilbert" space? What space do they span? What is a basis?
6. Approximately how many bound states does a finite square well of depth V_0 and width L tend to have? (within +-1)
7. For the harmonic oscillator, the first excited state is proportional to (x/a) exp{-x^2/(2a^2}.
a) Determine the normalization factor (by integration).
For an electron (one electron) in a state that is an equal mix of the g.s. and 1st excited states of a harmonics oscillator (see previous problem), write the wave-function (normalized) and calculate the expectation values of:
b) x
c) p
d) Do these depend on time? Why or why not?
----- maybe we should stop here for now? Would that be about the right length? Are we doing too many things at once?? -Zack
8. Using raising and lowering operators and Dirac formalism:
a) calculate the expectation value x^2 for each energy eigenstate of the harmonic oscillator.
b) do the same for p^2.
c) graph these as a function of n. What is the dependence on n? Explain and discuss!
d) Imagine doing the same calculation for the infinite square well eigenstates. It would not be fun. Don't do it!
e) Graph the product delta x * delta p as a function of n. (Discuss if it looks interesting.)
9. (extra credit) For a particle moving in two dimensions one can define a and a+ operators in terms of x and px and y and py.
a) What is the angular momentum in 2D (about the axis perpendicular to the 2D plane)?
b) Express that in terms of your 2 sets of a’s and a+’s.
Tuesday, April 7, 2009
Reading and digression on measurement
Pretty soon we will want to be using Dirac notation so I would suggest reading section 4.3 of Liboff, or something comparable, for that. Sections 4.4 (Hilbert spaces) and 4.5 (Hermitian objects) are also recommended.
Chapter 7 includes coverage of the harmonic oscillator. You could read that if you want. Our treatment in class should also be pretty complete. Griffiths uses similar notation. My thought was to use the Liboff notation for raising and and lowering “operators”, which can also be regarded as matrices. (In this context a matrix is a “representation” (or avatar) of an operator in a particular basis.) If the basis is well-chosen, the matrix representation can make the operator easier to understand and more intuitive.
Chapter 6 of Liboff covers time dependence. We’ll start on time dependence of quantum states in 1-D probably next week; our treatment should be self-contained, but you could read about it in advance of you like. Chapter 8 has a discussion of the finite square well, but I wouldn’t really recommend it. It seems overly complex and confusing. (Let me know if I am mistaken about that.)
Also of potential interest is section 5.1, and the subsection: Hilbert space interpretation has an interesting discussion, especially in the second to last paragraph which starts with “is there a chance…?” and includes the important phrase, "there is nothing in classical physics that is similar to this concept."
In this context, regarding measurement and its relationship to the state of a system, it's important to think about whether a measurement "leaves" the system in an eigenstate, or whether it actually destroys the system as we have conceived it, through interaction with a dissipative macroscopic system. Perhaps both are possible, depending on the nature of the measurement, but I think the latter is much more common.
Digression on measurement:
The question of what happens when a quantum particle, e.g., an electron, interacts with a macroscopic apparatus is rightfully thought to be important and significant. Though considerable effort and interest have been directed toward this issue, essentially it remains an unsolved problem. Recent papers, e.g., “Is measurement an emergent property,” P.W. Anderson PNAS, 2004?, speak to the extremely high level of difficulty and sophistication associated with this subject.
It is generally in the context of “measurement” that the concept of “wave-function collapse” arises. This may be something you have heard about somewhere? It is important to be clear that the time dependence invoked in casual discussions of wave-function collapse is distinct from that of the QM equations we study. The time dependence we will cover is based on the Schrodinger equation and comes from separation of variables. Wave-function collapse, as popularly conceived, is not, as far was we know, part of the Schrodinger equation-based formalism of QM, but rather is an addendum, an extra thing added on to address a perplexing conundrum . Ii could be described as conceptualizations or speculations regarding outcomes associated with a major class of unsolved problems.
The Schrodinger equation implies a particular sort of time dependence of all states arising from time evolving phase factors associated with each energy eigenstate (evolving at a rate proportional to its energy.) People have long been concerned that this Schrodinger wave-equation time dependence was not sufficient to describe what happens in a measurement. As I understand it, and i am no expert, wave-function collapse was hypothesized to address that concern. Although some very famous people have talked about it, it is not a tested or, as far as I know, testable theory.
I think that for a long time few people seriously considered the possibility that the time dependence of the Schrodinger equation could describe the sudden and seeming irreversible changes that occur during a measurement (involving a dissipative macroscopic object). Now, however, some people* have become less sure that time-dependence beyond that of the Schrodinger equation is needed to describe measurement. At any rate, the complexity of measurement systems, and perhaps other factors, tend to make these problems, for now, quite unsolvable.
* for example, N.D.Mermin
Chapter 7 includes coverage of the harmonic oscillator. You could read that if you want. Our treatment in class should also be pretty complete. Griffiths uses similar notation. My thought was to use the Liboff notation for raising and and lowering “operators”, which can also be regarded as matrices. (In this context a matrix is a “representation” (or avatar) of an operator in a particular basis.) If the basis is well-chosen, the matrix representation can make the operator easier to understand and more intuitive.
Chapter 6 of Liboff covers time dependence. We’ll start on time dependence of quantum states in 1-D probably next week; our treatment should be self-contained, but you could read about it in advance of you like. Chapter 8 has a discussion of the finite square well, but I wouldn’t really recommend it. It seems overly complex and confusing. (Let me know if I am mistaken about that.)
Also of potential interest is section 5.1, and the subsection: Hilbert space interpretation has an interesting discussion, especially in the second to last paragraph which starts with “is there a chance…?” and includes the important phrase, "there is nothing in classical physics that is similar to this concept."
In this context, regarding measurement and its relationship to the state of a system, it's important to think about whether a measurement "leaves" the system in an eigenstate, or whether it actually destroys the system as we have conceived it, through interaction with a dissipative macroscopic system. Perhaps both are possible, depending on the nature of the measurement, but I think the latter is much more common.
Digression on measurement:
The question of what happens when a quantum particle, e.g., an electron, interacts with a macroscopic apparatus is rightfully thought to be important and significant. Though considerable effort and interest have been directed toward this issue, essentially it remains an unsolved problem. Recent papers, e.g., “Is measurement an emergent property,” P.W. Anderson PNAS, 2004?, speak to the extremely high level of difficulty and sophistication associated with this subject.
It is generally in the context of “measurement” that the concept of “wave-function collapse” arises. This may be something you have heard about somewhere? It is important to be clear that the time dependence invoked in casual discussions of wave-function collapse is distinct from that of the QM equations we study. The time dependence we will cover is based on the Schrodinger equation and comes from separation of variables. Wave-function collapse, as popularly conceived, is not, as far was we know, part of the Schrodinger equation-based formalism of QM, but rather is an addendum, an extra thing added on to address a perplexing conundrum . Ii could be described as conceptualizations or speculations regarding outcomes associated with a major class of unsolved problems.
The Schrodinger equation implies a particular sort of time dependence of all states arising from time evolving phase factors associated with each energy eigenstate (evolving at a rate proportional to its energy.) People have long been concerned that this Schrodinger wave-equation time dependence was not sufficient to describe what happens in a measurement. As I understand it, and i am no expert, wave-function collapse was hypothesized to address that concern. Although some very famous people have talked about it, it is not a tested or, as far as I know, testable theory.
I think that for a long time few people seriously considered the possibility that the time dependence of the Schrodinger equation could describe the sudden and seeming irreversible changes that occur during a measurement (involving a dissipative macroscopic object). Now, however, some people* have become less sure that time-dependence beyond that of the Schrodinger equation is needed to describe measurement. At any rate, the complexity of measurement systems, and perhaps other factors, tend to make these problems, for now, quite unsolvable.
* for example, N.D.Mermin
Thursday, April 2, 2009
Homework 2 with solutions
------
1. (delta-function potential, g.s.)
Assume that there is a negative-energy solution for a particle of mass m under under the influence of an attractive delta function potential; using only exponentials that diminish as "you" move away from the origin, determine the exact, normalized ground state. (Ground state generally means lowest energy state. In this case it is also the symmetric, node-free state centered at the mirror symmetry point of the potential).
2. (delta-function potential, size) Calculate the expectation value of x^2 for a particle in the state from problem 1. Take its square root; it should have units of length, right? Give this characteristic length scale (which should depend on alpha, hbar and m, I think) a name*, and then rewrite your g.s. using this length scale parameter in both the exponent and the normalization pre-factor. (Your state should now look really simple.)
*may I suggest a short simple name like a_delta (a subscript delta)?
3. (delta-function potential, calculating "V") Calculate the expectation value of the potential energy for a particle in the ground state of an attractive delta-function potential. What is the difference between potential energy as a function of x and the expectation value of the potential energy? What is the meaning of each? [Computationally, this problem is very easy once you know what to do. It is really about understanding the concept of expectation value and what that means.]
PS. (added Sunday 4-5, 7:40 PM) For K.E. of a particle in this state, see problem 9. If 1-3 were too easy, you may like 9 better.
4. (infinite square well) a) What is the (normalized) g.s. wave-function and energy for a particle of mass m experiencing an infinite square well potential of width L?
b) Calculate the expectation value of p for a particle in this state.
c) Calculate the expectation value of p^2 for a particle in this state.
d) Using your result from c), what is the expectation value of the kinetic energy for a particle in the g.s. of an infinite square well.
*** The above problems, 1, 2, 3 and 4, are especially important problems. I would recommend doing them first and trying to finish them by Tuesday, if possible. They help establish some basic concepts that we can discuss and explore. Also, plan to allow enough time for 5- 9, as they may be somewhat computationally difficult. 5a and 8b are especially important to understand. Also, maybe 5b.
I know we haven't actually covered expectation value calculations in this class, but I am hoping you know what to do? If not, please feel free to post your questions here. Also, this is really a peer-to-peer format and anyone can answer anyone's questions.
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5. (finite square well) Consider a particle of mass m in a finite square well. Suppose that someone has numerically determined the parameters of the ground state and that thus you are given k, alpha, B and A (as well as m, hbar and L). (Treat them all as known".) Write down and evaluate the expressions (integrals) for the expectation value of:
a) potential energy
b) kinetic energy
c) momentum
6. With reference to the preceding problem:
a) How does the expectation value of the potential energy compare with that of a particle of the same mass in the g.s. of an infinite square well of the same width?
b) What is the sign of the expectation value of the kinetic energy contribution from the region outside the well?
How does the expectation value of the kinetic energy contribution from inside the well compare with that of a particle of the same mass in the g.s. of an infinite square well of the same width?
7. Momentum expectation values don't seem to depend explicitly on the potential function, V(x), so let's try working with a state without thinking about potentials for a moment.
a) Find the normalization factor for the quantum state:
e{ikx} e{-(x^2/2a^2)
[ I think it is 1/(pi^(1/4) sqrt(a)]
Perhaps someone can check that and post here? Thanks.]
b) Calculate the expectation value of the momentum of a particle in this state.
(Funny how that doesn't have an explicit m in it? I suppose m must have an influence on the momentum in some way?)
8. For a particle in the state: e{-(x^2/2a^2)
find the normalization factor (same as above i think) and then
a) calculate the expectation value of the momentum.
b) calculate the expectation value of the momentum squared.
c) calculate the expectation value of the x^2.
d) What are delta x and delta p for this state? (You know, like x^2-x^2 and p^2 - p^2..., but with the brackets in the right place.)
9. (extra credit-difficult: delta-function potential, K.E.)
Calculate the expectation value of the kinetic energy for a particle of mass m in the g.s of an attractive delta-function potential (as in problem 1-3). Show that the K.E. is indeed positive, as it must be, despite any indications to the contrary. [Be careful around x=0]
10. (extra-credit) Consider an attractive delta function potential in the center of an infinite square well. Assume that L, the size of the infinite sq well is finite, but very large compared to the characteristic size of the ground state of the delta-function potential by itself. Solve for the ground state for this "confined problem", where you can assume the g.s. wave function goes to zero at the boundaries of the well. Look at the nature of the solution in the limit L-->infinity. (I think to solve this you need both exponentials. The growing one will have a small (i think negligible) effect at x=0, but will be essential to making the w-f go to zero at the boundary (x=L/2). This gives us a sense of what the growing exponential can do and what happens to it as L gets really large...
Wednesday, April 1, 2009
Outline and road map for 139a:
This post is intended to give you a idea of where we are going and help you prepare for each class and get a sense of the big picture and the details of what we are hoping to accomplish and cover in this class. I will continue to add to it as time goes on.
As mentioned previously, this class weaves together 3 themes which could be loosely characterized as:
1) quantum formalism, postulates, mathematical underpinnings and structure,
2) quantum physics in 1-dimension (mostly bound states),
3) quantum physics in more than 1-d (2 & 3), focusing particularly on the hydrogen atom with its interesting and unusual degeneracies, etc.
We will weave the formalism into 2) and 3) and deal with concrete examples for each new concept. Quantum physics provides the underpinning for most of physics, chemistry and more and, in terms of its impact both technology and fundamental understanding it is primarily the story of the electron. (Why?) That will be our approach. We endeavor to develop an integrated mathematical and intuitive understanding of this amazing, low-mass creature which, when confined, acquires significant kinetic energy. This simple and unusual trait plays a central role in many quantum phenomena of great importance.
We will start with the wave equation for electrons, which was discovered by Edwin Schrodinger in the 1920's and is now commonly know as the Schrodinger equation. We will never loose site of the essence of its being a wave equation. With that and other starting postulates we will do separation of variable, talk about time dependence, normalization and expectation values.
On Thursday, April 2, we will begin our investigations of 1-D QM. Specifically we will briefly study momentum eigenstates for free electrons, and then focus on the ground states of 4 different 1-D potentials and begin calculating expectation values of things like p, x, p^2, x^2, V and T (the kinetic energy).
The 4 potentials will be: infinite square well, harmonic oscillator, attractive delta-function and finite square well. (Note that two of them are actually limiting cases of the 4th one.) We will use the Sch.-wave eqn to solve for the g.s. in each case with appropriate boundary conditions. Interestingly, I think the boundary conditions appear quite differently in each case. You may want to look at the delta-function potential, and how the ground state emerges for that from the unusual boundary condition at the position of the delta-function. In general, it is the combination of boundary conditions, the wave equation and the normalization requirement that lead to the quantized energy levels from which quantum mechanics gets its name.
Then I think we should we look at momentum eigenstates and free-particle (gaussian) wave-packets, which are discussed, e.g., on page 160 of Liboff. This will help us understand confinement and differences between non-local and local (bound) states.
After that we can examine the excited bound states of our 1-D potentials. This will include exploring the use of raising and lowering operators for the harmonic oscillator. (Why are raising and lowering operators only used for the H.O.? Could they be developed and used for other postentials?)
With this concrete background we will be in a position to delve into the structure, formalism and meaning of quantum theory. Linear algebra, hermitian matrices, basis, linear expansion, superposition and inner-products will all be critical concepts here; we will explore their relevance to quantum theory. At this point we can also talk a bit about measurement and "non-commuting observables". x and p are examples of non-commuting observables. More examples will arise in a natural wasy when we begin to study 2 and 3-dimensional quantum systems, which have degenerate energy eigenstates arising from their symmetries, something that is largely absent in 1-d.
....
As mentioned previously, this class weaves together 3 themes which could be loosely characterized as:
1) quantum formalism, postulates, mathematical underpinnings and structure,
2) quantum physics in 1-dimension (mostly bound states),
3) quantum physics in more than 1-d (2 & 3), focusing particularly on the hydrogen atom with its interesting and unusual degeneracies, etc.
We will weave the formalism into 2) and 3) and deal with concrete examples for each new concept. Quantum physics provides the underpinning for most of physics, chemistry and more and, in terms of its impact both technology and fundamental understanding it is primarily the story of the electron. (Why?) That will be our approach. We endeavor to develop an integrated mathematical and intuitive understanding of this amazing, low-mass creature which, when confined, acquires significant kinetic energy. This simple and unusual trait plays a central role in many quantum phenomena of great importance.
We will start with the wave equation for electrons, which was discovered by Edwin Schrodinger in the 1920's and is now commonly know as the Schrodinger equation. We will never loose site of the essence of its being a wave equation. With that and other starting postulates we will do separation of variable, talk about time dependence, normalization and expectation values.
On Thursday, April 2, we will begin our investigations of 1-D QM. Specifically we will briefly study momentum eigenstates for free electrons, and then focus on the ground states of 4 different 1-D potentials and begin calculating expectation values of things like p, x, p^2, x^2, V and T (the kinetic energy).
The 4 potentials will be: infinite square well, harmonic oscillator, attractive delta-function and finite square well. (Note that two of them are actually limiting cases of the 4th one.) We will use the Sch.-wave eqn to solve for the g.s. in each case with appropriate boundary conditions. Interestingly, I think the boundary conditions appear quite differently in each case. You may want to look at the delta-function potential, and how the ground state emerges for that from the unusual boundary condition at the position of the delta-function. In general, it is the combination of boundary conditions, the wave equation and the normalization requirement that lead to the quantized energy levels from which quantum mechanics gets its name.
Then I think we should we look at momentum eigenstates and free-particle (gaussian) wave-packets, which are discussed, e.g., on page 160 of Liboff. This will help us understand confinement and differences between non-local and local (bound) states.
After that we can examine the excited bound states of our 1-D potentials. This will include exploring the use of raising and lowering operators for the harmonic oscillator. (Why are raising and lowering operators only used for the H.O.? Could they be developed and used for other postentials?)
With this concrete background we will be in a position to delve into the structure, formalism and meaning of quantum theory. Linear algebra, hermitian matrices, basis, linear expansion, superposition and inner-products will all be critical concepts here; we will explore their relevance to quantum theory. At this point we can also talk a bit about measurement and "non-commuting observables". x and p are examples of non-commuting observables. More examples will arise in a natural wasy when we begin to study 2 and 3-dimensional quantum systems, which have degenerate energy eigenstates arising from their symmetries, something that is largely absent in 1-d.
....
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