Exploring Quantum Physics Problems with a Charged Particle in a Confined Box

[William Mitzger]

I'm a high school student at East Mecklenburg HS in North Carolina. I'm working on some quantum physics problems as part of a non-graded extra assignment for a higher-level physics course offered at my high school. Our teacher has said we can ask anyone, besides fellow students, for assistance as long as we can solve the problems and explain the solutions. The assignment isn't for a grade, and is more to see if any of us can do it than anything else. If you have the time, I hope you can offer some insight into these questions:

--------------------------------------------------------------------------------------------------
Q1) A particle of mass "m" and charge "q" is confined to a one dimensional box whose sides at x=0 and x=a. A measurement of the energy shows that the particle is in the second (first excited) energy eigenstate. At time t=0 the wall at x=a is suddenly moved to x=b with b>a/

a) find the relation between a and b so that the energy is unchanged.
b) find the probability that the energy remains unchanged.

Q2) An atom with no permanent magnetic moment is said to be diamagnetic and has an induced diamagnetic moment mu=-d(DELTA E)/dB where DELTA E is the change in energy when the atom is placed in a magnetic field B. Find the induced diamagnetic moment for the hydrogen atom in its ground state when a weak magnetic field is applied. (Note the derivative is a partial derivative.)

Q3) A particle of mass "m" and charge "q" is confined to a one-dimensional box whose sides are at x=0 and x=a are conductors. Thus, when a voltage is applied to the two walls, the particle experiences a uniform electric field of amplitude E_0. If this voltage is applied at t=0, when the particle is in the ground state, find an expression giving the probability that after a long time the particle has made a transition to the nth excited state. Evaluate this expression for n=2.

Q4) The two dimensional oscillator Hamiltonian (see hamiltonian.pdf). When perturbation of H'=lambda*(m*omega^2)(xy) with lambda much less than one is applied, find the first order energies and the corresponding appropriate zero order energy eigenstates for the three degernerate states |2,0>, |0,2>, and |1,1>.

Q5) A plane rotator consists of a mass "m" rigidly constrained to move at a distance "R" from a fixed origin, i.e., a bead of mass "m" on a frictionless circular wire of radius "R." Schrodinger's equation for the rigid plane rotator is (see attached pdf). I=mR^2 is the rotation about the z-axis.

a)Find the energy eigenvalues and normalized energy eigenfunctions.
b)at time t=0, the wave function of the rotator is PSI(PHI, 0)=A*cos^2(PHI). Find PSI(PHI, t).
c)The particle is given a charge of +q and a weak static uniform electric field is applied along the x axis. Find the first non-zero corrections to the energy eigenvalues.
d)Consider the extreme case of a very large uniform electric field acting on the rigid plane rotator. Devise a physically plausible approach for this situation and obtain the zero order energy eigenvalues as well as the first order corrections.
e) Instead of a constant electric field, an electric field pulse of the form E(t)=E_0 * e^(-gamma*t) is applied starting at time t=0. If the rotator is initially in an energy eigenstate, find the first order probability that after a long time it has made a transition to another energy state as a result of the electric field pulse.
f)The electric field is removed and a static uniform magnetic field is applied along the z-axis. Find the energy eigenvalues.

Email me at emperor_bob2000@hotmail.com and I can send you the PDF's with the equations. they're almost unreadable if typed out.

 

[HallsofIvy]

Well, since YOU say it's homework- here it goes

 

[William Mitzger]

Oh, if you're going to move it... could you move it into "advanced physics" since it's related to quantum, which I don't think many people would call introductory.

 

[berkeman]

Welcome to the PF, William. Those are indeed advanced questions for high school. What kind of physics classes do they offer at your school?

One of the most important rules here on the PF is that you must show your own work in order for us to help you. We do not provide answers (espeically not via e-mail to you), but we can provide hints and tutorial advice when we see your work.

So which problems do you want to start with first? Show us the relevant equations and your attempt at a solution and we'll go from there.

 

[William Mitzger]

oh I'm sorry, I forgot to post my reasoning for the problems so far.

Here's that:

1.) I believe the first one requires the application of the sudden approximation. I'm not entirely sure how to go about that though, any suggestions or examples would be appreciated greatly. I have a general expression for the transition probabilities, but I'm not sure if using the sudden approximation is required.
I'm guessing the change in size of the box would have to be some multiple of the original length? Any suggestions on how to show that?

2.) For two I'm confused where I would need to begin. Should this be attacked like a perturbation? Should I examine the basics of "hydrogen in a magnetic field?" Any particular relationships you suggest examining?

3) This seems to be a sudden approximation perturbation. I'm not sure what hamiltonians I should be using and what the perturbation is exactly. Help setting this problem up would be appreciated. Is this not correct? Should sudden approximation not be considered since even though the change is instant, the question asks for behavior at large t's? If so, is it an adiabatic perturbation, or should I just apply general time dependent perturbation theory?

4.) I'm not entirely fluent with the dirac notation. My largest problem here is what to use for the |n_x, n_y>. What exactly does this represent? It's just PSI for those particular quantum numbers right? If so, what type of wave function would be appropriate? Should I try to approach this in matrix form?

5.) I believe (a) is fairly straightforward, although I'm not sure what Hamiltonian I should be using.
For (b) I think I should merely multiply PHI by the time dependence. I'm not sure what that is,
but does something like: e^-iwk seem right? How do you go about determining the time dependent portion?
For (c) I think I should use perturbation theory and H' should be something on the oder of qE. Any suggestions would be appreciated.
(d) is where I'm really confused. My teacher hasn't been in town to answer any questions so I'm not sure what is meant by a physically plausible approach. If this means a physically plausible means to create and test the situation, then I've got a few ideas how to answer it. If he means a plausible use of perturbation or variation of parameters, I'm a little shakier. Could you treat the particle in the strong magnetic field as the 0th order situation, and then consider the rotation to be the angular momentum components to be the perturbation in the case where the magnetic field is very strong?
For (e) I suspect I should have to time dependent perturbation (neither sudden nor periodic). This doesn't look too convoluted, but any suggestions or examples would be helpful.
For (f) I'm not entirely sure how to proceed. Should I apply the sudden approximation to the results of (e)? How do you go about doing that? Do you apply the perturbation to each possible state of (e)?
Thanks. It's hard to show actual work because of all the greek letters and integrals, but I'd appreciate some advice.

 

[Coto]

Wow. Highschool physics assignment? I can imagine why your teacher told you that these problems were extra and it was more for fun. These are problems that could be potentially put into the 1st quantum mechanics course (in my university, that's 3rd year.)

Concepts breach on abstract algebra and deeper electrostatics (diamagnetism-- in fact, the technical aspects of diamagnetic and paramagnetic theory is delayed till a further course and there's only a quick derivation of process thinking of it classically).. Anyways, none of the questions are overly difficult, however they do all require knowledge of these subject areas to be fully understood.

As for (1), the traditional method for showing something like this is to start by writing the solution to schrödingers equation at t=0 in the well (i.e first excited state ==> write down the first eigenstate. Infinite square well has a simple equation for that.)

Recall that the full solution can be written down in the form:
$$\Psi (x,t) = \sum_{n=0} c_n \cdot \psi_n(x) e^{-iE_nt/\hbar}$$

The c_n's are the key to this problem. Remember that |c_n|^2 can be thought of the probability that a measurement of the energy will return E_n.

Recall the formula for c_n:
$$c_n = \int^{\infty}_{-\infty} \Psi^*(x,0)\psi_n(x) dx$$
Work from here and you should be able to find the desired relations for a and b. I haven't worked the problem out myself, however this is what I would try.

 

[Coto]

Number 2 is quite easy actually.. write down the energy of the hydrogen atom in its ground state (trivial). The trick is how does the magnetic field affect the hydrogen atom? This can be related by the hamiltonian:
$$H = -\gamma B\cdot S$$
for a spin 1/2 charged particle. B, S vectors.

Manipulating this to find your energy of the particle, you can then find delta(E) and finally take the partial w.r.t. B.


A quick glance at #3 seems like it'd be a combination of the ideas for #1 and #2. Hopefully this puts you on a successful track. A quick glance at the question you have about #4: In general I found that if given a question in vector notation.. do the question in vector notation. It's usually a hint saying: "Trust me, use vector notation, its simple."

Finally, #5 is 100% introductory QM. It's a classic problem that you would solve for infinite square well, finite square well, free particle, etc, etc, etc. Look at an example for how they go about that type of question with these examples, and answers flow directly from it. Unfortunately it seems you don't have an intro book on QM, which would make finding these examples much more difficult.

 

[Coto]

I revise my opinion about #5 after reading c, d, e, etc. eheheh.

By the sounds of it though, this sounds a lot like a Larmor precession idea, using principles of magnetic resonance. Or at least a contrived example of magnetic resonance using a rigid classical setup in an electric field.