A quantum mechanics question.

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Minerva
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A quantum mechanics question.

Postby Minerva » Wed Jun 03, 2009 9:54 am UTC

This is a school textbook question. I'm not expecting anybody to feed me the answer; I'll give you what I think the answer is thus far, and if I could get a bit of a nudge in the direction of correctness if it's not correct, or confirmation that it is correct, that would great.

Consider an electron in a hydrogen atom in the quantum state: [imath]\psi = \sqrt{\frac{1}{6}} \psi_{321} + \sqrt{\frac{1}{2}} \psi_{320} + \sqrt{\frac{1}{3}} \psi_{3 2 -1}[/imath]

(Where the subscripts are the quantum numbers n, l, m respectively for the basic states of a hydrogen atom, such that [imath]\psi_{nlm} = R_{nl} Y^{m}_{l}[/imath].)

The energy of the electron, L2 and LZ are measured. What are the value(s) of these quantities that can be measured, and what is the expectation value of LZ?

Now;
Energy: The energy of the electron is determined by the quantum number n, and there is no ambiguity in the measured value of the energy, since n =3 in all of the terms in the wavefunction. E = - 13.6 eV * (1/32), which is equal to about -1.51 eV.

L2: The eigenvalue of L2 is l(l+1)[imath]\hbar^{2}[/imath], and l = 2 for all of the terms, therefore, the measured value of L2 is simply equal to 6[imath]\hbar^{2}[/imath].

The eigenvalue of LZ is [imath]\hbar[/imath]m, so we have three different possible values corresponding to each of the three terms, LZ = [imath]\hbar[/imath] with probability 1/6, LZ = 0 with probability 1/2 and LZ = -[imath]\hbar[/imath] with probability 1/3.

Add up those three terms to get <LZ>, and we have (1/6 - 1/3)[imath]\hbar[/imath] = -(1/6)[imath]\hbar[/imath].

Does that all sound like it's on the right track?

Thanks.
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Certhas
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Re: A quantum mechanics question.

Postby Certhas » Wed Jun 03, 2009 11:59 am UTC

looks absolutely correct.
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danpilon54
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Re: A quantum mechanics question.

Postby danpilon54 » Wed Jun 03, 2009 4:33 pm UTC

Looks good to me too
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Re: A quantum mechanics question.

Postby Minchandre » Wed Jun 03, 2009 6:02 pm UTC

I'll jump on the train of "Everything's a-okay"

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Minerva
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Re: A quantum mechanics question.

Postby Minerva » Thu Jun 04, 2009 6:33 pm UTC

OK, thanks :)

Looks like I'll have to up the ante a bit so I don't get too confident. :)

Here's another question which I've got absolutely no idea about:

Suppose a spin-[imath]\frac{1}{2}[/imath] particle experiences a Hamiltonian of the form [imath]\hat{H} = k(S_{Z} + \lambda S_{X})[/imath], for small [imath]\lambda[/imath] and some constant k.

Determine an appropriate basis to express the problem to 0-th order, and describe the qualitative effect of each of the terms of the Hamiltonian on these basis states, and determine the correction, to first order, to the eigenstates and eigenvalues of energy.

OK. So, an appropriate basis to express the problem to zero order? OK, we put lambda = 0, so we simply have the Hamiltonian proportional to Sz. So, can't we just express this Hamiltonian in the z basis?

Describe the qualitative effect of each of each of the terms in the Hamiltonian on these basis states?
Well, I really don't understand what this is asking for.

I'm trying to visualise what kind of physical system this could possibly be. Let's suppose it's something like spin precession in a magnetic field. Suppose the direction of the magnetic field is in the ZX plane, perturbed slightly off the Z-axis. That makes sense, doesn't it? Thus, k quantifies the field strength and/or the gyromagnetic ratio, and lambda quantifies the angular perturbation of the magnetic field off the Z axis in the ZX plane.

OK, so I think that's a physical interpretation of the Hamiltonian, consistent with the Hamiltonian. But what is the interpretation of these terms in terms of the effect of the Hamiltonian on these basis states? What basis states?

Determination of the first order correction to the eigenstates and eigenvalues of energy? So, the first order correction to the eigenstates and eigenvalues of the Hamiltonian. But I don't really know where to begin determining these. Sure, there are plenty of discussions of basic perturbation theory for some small potentials in the textbook(s), but I've not seen discussions of anything that looks like a spin matrix, in the context of perturbation theory.

Could anyone give me a little bit of help with this question?

Thanks.
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ThinkerEmeritus
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Re: A quantum mechanics question.

Postby ThinkerEmeritus » Thu Jun 04, 2009 8:46 pm UTC

Minerva wrote:OK, thanks :)

Looks like I'll have to up the ante a bit so I don't get too confident. :)

Here's another question which I've got absolutely no idea about:

Suppose a spin-[imath]\frac{1}{2}[/imath] particle experiences a Hamiltonian of the form [imath]\hat{H} = k(S_{Z} + \lambda S_{X})[/imath], for small [imath]\lambda[/imath] and some constant k.

Determine an appropriate basis to express the problem to 0-th order, and describe the qualitative effect of each of the terms of the Hamiltonian on these basis states, and determine the correction, to first order, to the eigenstates and eigenvalues of energy.

OK. So, an appropriate basis to express the problem to zero order? OK, we put lambda = 0, so we simply have the Hamiltonian proportional to Sz. So, can't we just express this Hamiltonian in the z basis?


Yep. It would help in the next step if you wrote the basis down, presumably in matrix form.

Describe the qualitative effect of each of each of the terms in the Hamiltonian on these basis states?
Well, I really don't understand what this is asking for.


A good response would be to evaluate [imath]k Sz \psi1[/imath] and [imath]k Sz \psi2[/imath] and interpret the results in words, and then do the same for
[imath]k \lambda[/imath] S2.

I'm trying to visualise what kind of physical system this could possibly be. Let's suppose it's something like spin precession in a magnetic field. Suppose the direction of the magnetic field is in the ZX plane, perturbed slightly off the Z-axis. That makes sense, doesn't it? Thus, k quantifies the field strength and/or the gyromagnetic ratio, and lambda quantifies the angular perturbation of the magnetic field off the Z axis in the ZX plane.

OK, so I think that's a physical interpretation of the Hamiltonian, consistent with the Hamiltonian. But what is the interpretation of these terms in terms of the effect of the Hamiltonian on these basis states? What basis states?


The matrix-style eigenstates of Sz is presumably what is wanted. The physical interpretation of the Hamiltonian you have quoted is correct, but may be more than needed. However, if you were asked to find an exact solution, that interpretation would be invaluable.

Determination of the first order correction to the eigenstates and eigenvalues of energy? So, the first order correction to the eigenstates and eigenvalues of the Hamiltonian. But I don't really know where to begin determining these. Sure, there are plenty of discussions of basic perturbation theory for some small potentials in the textbook(s), but I've not seen discussions of anything that looks like a spin matrix, in the context of perturbation theory.

Could anyone give me a little bit of help with this question?

Thanks.


The formulae for perturbation theory will be in terms of matrix elements of the perturbation Hamiltonian. What are the matrix elements of a matrix?
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