One of the the key goals of this homework assignment is to learn about quantum degeneracy, that is, the occurrence of orthogonal states (wave-functions) which have the same energy (and hence the same time dependence).
1.
a) Write the (normalized wave-function of an electron in the) ground state of the 2-dimensional harmonic oscillator.
b) What is its energy?
2.
a) Write the 1st excited states of the 2-dimensional harmonic oscillator.
b) What is their energy?
c) How many are there?
3.
a) Write the 2nd excited states of the 2-dimensional harmonic oscillator.
b) What is their energy?
c) How many are there?
4 & 5 have been deleted because of concern that that might take be an unnecessary distraction that would take time away from more important problems.
Bonus Problem: Come up with a (simple) way to illustrate the nature of some or all of these states; something that illustrates where the wave-function (at t=0) tends to be large, and where it is not. Perhaps think about including some way to distinguish where the wave function is positive and where it is negative (phase). You can use sketches, color, surfaces of constant amplitude, polar plots, 2-dimensional contour plots or whatever you come up with.
6.
a) What is the normalized wave-function of an electron in the ground state of a e^2/r potential in 3 dimensions (i.e., a hydrogen atom). (You can include the 1/(4*pi*epsilon0 factor as you like.)
b) What is its energy?
7.
a) What are some normalized wave-function of an electron in the 1st excited states of a e^2/r potential in 3 dimensions (i.e., a hydrogen atom). (You can include the 1/(4*pi*epsilon0 factor as you like.)
b) What is the energy of those states?
c) How many are there?
8. extra credit. advanced.
For an electron in a 2DHO potential in the state which is an equal mix (normalized) of the ground state, Psi00, and the first excited state Psi10, calculate the expectation values of x, and y, respectively, as a function of time.
9. extra credit. very advanced.
For an electron in a 2DHO potential in the state which is an equal mix (normalized) of the ground state, Psi00, and the first excited state (Psi10 + i Psi01)/sqrt(2), calculate the expectation values of x, and y, respectively, as a function of time. What do you observe? What is this calculation trying to tell you? (This normalized mixed state is [Psi00 + (Psi10 + i*Psi01)/sqrt(2)]/sqrt(2).)
10. extra credit. Very, very advanced.
For an electron in the 1st excited state (Psi10 + i*Psi01)/sqrt(2), calculate the (z component of the) angular momentum using Lz = X Py - Y Px. (see 1st comment below)
10-background. an optional extra problem. Because of an interest expressed in momentum expectation values, here is something you can do for fun that may provide a good warm-up problem for 10.
-Calculate the expectation values of p and of p^2 for the ground state of the infinite square well.
(aside: what is p? How do you calculate its expectation value?)
-If you are feeling ambitious and you like derivatives, do the same for the ground state of the harmonic oscillator.
Any suggestions for problems are most welcome and can be posted here. I mean that! (and though I might be surprised, it would be a very pleasant surprise).
Please post your questions here. The only bad questions are the ones that go unasked :( . I am sure if you have a question, a lot of other people are wondering the same thing. Also, you can become a follower of this site. Some people use made up whimsical profiles, some use their real name. It is entirely up to you. This is your education. I encourage you to make the most of it.
We should calculate the angular momentum for the mixed first excited state of the 2d-SHO, with the (��10±i��01)/sqrt(2) combination that you showed us in class.
ReplyDeleteWe would do this by applying the angular momentum operator (in the z direction) to our combined state ��(x,t) and the angular momentum of the state would be the eigenvalue of the operator.
Mathematically, Lz=XPy-YPx, or in position space, Lz=-i♄(x*∂/∂y-y*∂/∂x). What we want to do then is to apply this operator to our wave function �� and (if �� in an angular momentum eigenstate) Lz*��=c*�� where c is just a number which would be that angular momentum of our state.
Does this look right?
Thank you,
Kevin Hambleton
Sorry if the symbols are unreadable. This site don't seem to like psi or h-bar.
DeleteLooks right to me.
Deletegood suggestion. i used that for problem 10. Did I understand your ideas correctly?
DeleteYes of course! I know were not really covering angular momentum operators yet, but I think this is a very insightful example of quantized angular momentum
DeleteShould I be getting a general equation for the energy values of the harmonic oscillator, or do I need to power through the schrodinger equation for each one? I attempted to derive it at ground state, and came to E= -1/2kr^2*[1+(1/2m)], but I'm not sure where to go from there to apply it to the excited states. Also, how does the 2nd excited state work? I feel there should be only two seperate wave functions that can give the same energy, but I'm not too confident in my reasoning.
ReplyDeleteNevermind the first part, found the mistake that happened during the calculus.
DeleteWell, for the 1DHO the energies are:
Delete(1/2) hbar*w (gs)
(3/2) hbar*w (1st
(5/2) hbar*w (2nd
....
(where w=sqrt(k/m) .
For the 2DHO gs the energy is hbar*w .
That's a start.
What do you think the energy is for the 1st excited state?
Regarding the 2nd excited states, you might be able to make one by combining states from the 1DHO (like what we did for the 1st ex state).
So the energy for the 1st excited state of the 2DHO might be 3*hbar*w where w=sqrt(k/m). Is k left as a variable?
DeleteLooking through the problems, I could not see the point of 4 and 5. Also, I think those could be a bit of a distraction from our main focus, which is related to understanding the states and the degeneracies of the hydrogen atom.
ReplyDeleteI suggest we delete 4 and 5 from this problem set.
In question 3, is Psi(1,1) orthogonal to Psi(2,0) and Psi(0,2)? Can we express the 2nd excited state as a combination of all 3 different states? If so, how? Immediately I think of x^2+xy+y^2 prop to PSI(2nd ex. st.).
ReplyDelete1) yes, those 3 are mutually orthogonal.
ReplyDelete2) well any normalized combination of those three states is also a 2nd excited state. One can think of the "2nd excited state degeneracy manifold" as a subspace which is spanned by those 3 orthogonal (and normalized) states.
This is a deep subject. There is no right answer to the question "what is the 2nd excited state?", because there are many answers, many 2nd excited states (for the 2DHO).....
For problem 8, when you say "equal mix (normalized)" of Psi00 and Psi10, does this mean the mixed wave function is just equal to the sum of the normalized ground state and 1st excited state, or is there an additional normalization factor that I have to find?
ReplyDeletegood question. Since the mixed state must be normalized, it follows that there must be an additional normalization factor of 1/sqrt(2). Is that what you were thinking?
DeleteI think the second excited state of the 1D-SHO is written incorrectly here. I'm pretty sure it should read [2/(a*sqrt(pi))]^(1/2)*(1/2-x^2/a^2)*e^(-x^2/(2a^2)).
ReplyDeleteSo I think the (x^2/a^2-1) should be replaced with (x^2/a^2-1/2)
The energies in the answers to problems 1 & 2 were not included in the solutions. What are they?
ReplyDeleteFor problem 3 also.
ReplyDelete