Tough QM questions

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  • #1
AKG
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Well, I doubt they're tough, but I have no idea what's even being asked, or how I should go about solving these things. The questions are:

Use the bBohr quantization rules to caluclate the energy levels for a harmonic oscillator, for which the energy is p²/2m + mw²r²/2; that is, the force is mw²/r. Restrict yourself to circular orbits. What is the analog of the Rydberg formula? Show that the correspondence principle is staisfied for all values of the quantum number n used in quantizing the angular momentum.

Any general clarifications as to what is even being asked would be helpful. The second question is:

The classical energy of a plane rotator is given by E = L²/2I where L is the angular momentum and I is the moment of inertia. Apply the Bohr quantization rules to obtain the energy levls of the rotator. If the Bohr frequency condition is assumed for the radiation in transitions from states labeled by n1 to states labled by n2, show that (a) the correspondence principle holds, and (b) that it implies that only transitions Delta-n = +/- 1 should occur.
 

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nrqed
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AKG said:
Well, I doubt they're tough, but I have no idea what's even being asked, or how I should go about solving these things. The questions are:

Use the bBohr quantization rules to caluclate the energy levels for a harmonic oscillator, for which the energy is p²/2m + mw²r²/2; that is, the force is mw²/r. Restrict yourself to circular orbits. What is the analog of the Rydberg formula? Show that the correspondence principle is staisfied for all values of the quantum number n used in quantizing the angular momentum.

Any general clarifications as to what is even being asked would be helpful.
The Bohr quantization rule corresponds to setting the magnitude of the angular momentum to be a multiple of h bar. For a circular orbit, this means that you set [itex] m v r = n \hbar [/itex]. To find the analogue o fthe Rydberg formula, just consider the transition between states with different energies which would give you the energy of an emitted photon.


The correspondence principle states that in the limit of large n, one should recover the classical results.

The second question is:

The classical energy of a plane rotator is given by E = L²/2I where L is the angular momentum and I is the moment of inertia. Apply the Bohr quantization rules to obtain the energy levls of the rotator. If the Bohr frequency condition is assumed for the radiation in transitions from states labeled by n1 to states labled by n2, show that (a) the correspondence principle holds, and (b) that it implies that only transitions Delta-n = +/- 1 should occur.
The steps are the same as in the previous problem. To show that [itex] \Delta n = \pm 1 [/itex], you simply think about what would be teh classical result for, say, the frequency of light emitted by the rotator (imagine that the rotator is charged, say). Then you will see that the quantum result gives the classical result only when the n's differ by one (otherwise the emitted photon would have twice the classical frequency, or three times, etc etc)

Pat
 
  • #3
AKG
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Thanks for trying, but I don't understand any of what you wrote. What makes it difficult is that the book does a terrible job explaining it (it seems to assume I should already understand this stuff) and the prof just teaches out of the book in a messier, less-intelligible way. I doubt you have the patience to explain what you're talking about, (if you do, that would be great) so do you have any links that could explain the basics so I can understand what you're saying? For example:

The Bohr quantization rule corresponds to setting the magnitude of the angular momentum to be a multiple of h bar. For a circular orbit, this means that you set [itex] m v r = n \hbar [/itex]. To find the analogue o fthe Rydberg formula, just consider the transition between states with different energies which would give you the energy of an emitted photon.
Angular momentum of what? I dont' know what the Rydberg formula is, nor what is meant by the transition between states with different energies. I have a feeling it has something to do with electrons moving from higher to lower energy states, but like I said, I really don't get this stuff.

As for the second bit, I'm lost, I'll just ask for now: what's a rotator?
 
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I, too, am doing this problem set, and have similar questions.
I suppose the main one is how does one show classically the frequency of light emitted by a charged rotator?
 
  • #5
nrqed said:
The Bohr quantization rule corresponds to setting the magnitude of the angular momentum to be a multiple of h bar. For a circular orbit, this means that you set [itex] m v r = n \hbar [/itex]. To find the analogue o fthe Rydberg formula, just consider the transition between states with different energies which would give you the energy of an emitted photon.


The correspondence principle states that in the limit of large n, one should recover the classical results.

EDIT: I figured out the rest of the question just by going over what you said again. It really helped out a lot.
 
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  • #6
nrqed
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Ingwe said:
I, too, am doing this problem set, and have similar questions.
I suppose the main one is how does one show classically the frequency of light emitted by a charged rotator?
Sorry, I did not have access to the Internet for a few days.

Indeed, that's a key point. But the answer is simple: the frequency of the emitted light is simply the frequency of the rotator.

Pat
 

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