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.
The characteristic length scale is the square root of the expectation value of x^2, right?
ReplyDeleteyes
ReplyDeleteHints and Clarification:
ReplyDeleteFor problem 2, please express your K.E.'s in the form: [hbar^2/(2m(delta x)^2] times a number.
What is that number for each system?
P.S. I think the numbers are pretty much something like, 1/4,1/3, 1/2. Which goes with which systems? What do you learn from that?
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For 3, you'll get a transcendental eqn that can be represented as a graph as a function of q, where q is the exponetial decay factor. It helps to define q_0= m alpha/hbar^2 , which is the "q" from the single delta function problem, and concentrate on the region q less than 2q_0. (why?) [Look for an intersection of your hyper tanh function and the 1/q -like stuff.]
For 3, are we going to get something like Cosh[kx] in between the two delta functions, and if so, is this "k" the same as the decay constant for the exponentials outside the delta functions?
ReplyDeleteFor 3, I was wondering if the initail setup is something like
ReplyDeletephi(x) = A(exp(-k|x+L/2|) + exp(-k|x-L/2|))
where A is some constant?
is this the right approach?
Mike, Those are both key questions that are critical to starting this problem. I think symmetry considerations do imply that i the center the solution must be of the form Acosh(kx), as you susggest. Regarding the 2nd question, would the k's be the same, what do you think is the basis for answering that? (anyone)
ReplyDeleteGabriel, Sorry i didn't see your comment this morning. What you suggest is a great approach. It is very intuitive and gives us a sense of how to build molecular-type states from atomic states, which is extremely important. See how far you can get with it. It may be computationally more difficult than using cosh(kx) in the center. I am not sure. What you suggest is especially good when the 2 potentials are not too close together, which is the most interesting region and the most relevant to molecule formation.
ReplyDeleteIt this particuar csse, 2 delta functions, we made be able to get an exact solution more easily from the cosh(kx) approach, but in general what you suggest is a powerful method and you can take a wave-function like that and calculate T and V expectation values and minimize their sum to get a ground state.
I plugged in phi(x) for the outer regions (phi(x) = A*Exp(-k|x|) and got -hbar^2 k^2/2/m = E;
ReplyDeleteFor phi(x) in the middle (phi(x) = B*Cosh(bx)), I get -hbar^2 b^2/2/m = E;
These two together seem to show that b^2 = k^2, and since cosh is even, we can take the positive b, giving b=k.
Does that make sense?
For problem 3 I tried to integrate the Time-Independent Schrodinger equation with the appropriate expression for V(x) which is just the sum of two delta functions that i put at L/2 and -L/2. I integrate from -infinity to infinity and I get that the Psi(L/2) = -Psi(-L/2)... in other words it seems as though I get that Psi is actually an odd function. Is this correct? Mike how exactly did you get that the value for Psi in between was cosh?
ReplyDeleteFor number 7, is it alright if we use other values of the wavefunction? The value of the first excited state seems to differ considerably from the value given by the book. The values I would like to use are given by this link (they are essentially the same as given by Griffiths):
ReplyDeletehttp://hyperphysics.phy-astr.gsu.edu/hbase/quantum/imgqua/hoscwf4.gif
Adi, Those wave-functions seem fine. I think they are different ways of expressing the same thing and if you plug in the values for a and alpha you should find they are identical. If not, that would mean i made a mistake. The wave-functions at that link look very elegant! The answer should be exactly the same.
ReplyDeleteJustin, integrating the Schrodinger eqn is very advanced and difficult. I am not sure exactly what you did, but I would recommend bailing on that approach and follow more what Mike is doing.
Mike, whether or not that makes sense to me is not so important. The critical thing is that it comes directly from the Schrodinger eqn, which is the ultimate authority and arbiter in this class.
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ReplyDeleteI suppose I made some assumptions I can't make. Thanks I'll try something else.
ReplyDeleteThis comment has been removed by the author.
ReplyDeletealready normalized.
ReplyDeleteJust to clarify the terminology, when problem 5 asks for the eigenstates of the infinite square well, is it asking for the wave functions, e.g. sin(x), etc?
ReplyDeleteFor problem 6, can we just use the expression for the energy of the infinite square well to approximate the energy of the finite well? Or is another approach required?
Heather: Yes, just the wave functions (normalized).
ReplyDeletefor 6, that sounds like a good approach.
Justin, regarding using cosh in the middle, I just thought that it should be some function that was symmetric, and looked somewhat like an exponential, which is cosh.
ReplyDeleteIt is 3:20 am and I am still having fun with this homework. Thanks to all those people who contribute here on the blog, in the library, and wherever I can find you.
ReplyDeletehttp://pics.livejournal.com/amexist/pic/0001a2dy
I think that problems 1-7 are enough for this assignment. I would like to see raising and lowering operators in action before attacking #8
ReplyDeleteYeah, I agree with Gabriel about problem 8.
ReplyDeleteThere is a section in the book about raising and lowering operators. It starts on page 193. They're called creation and annihilation operators.
ReplyDelete