And yet another thing: even though the electrons doesn't interact, they still have to obey the Pauli exclusion principle, right? So we can't consider them independently after all, or...?
I see. Thanks.
Oh, just one more thing: how do you construct the state vector for the whole system from the individual single particle states? Is it like a tensor product (I'm unsure of the mathematics here.)
Hi,
I was just reading up on some QM, and I was wondering this: under what circumstances can you treat a bunch of electrons as occupying "single particle states", and when do you have to use one wavefunction depending on all the coordinates, psi(r1, r2, r3, ...)? Hope you know what I mean...
Hi,
How can you infer from these equations,
a = b_{max}(b_{max}+\hbar) \quad \text{and} \quad a = b_{min}(b_{min}-\hbar),
that b_{max} = -b_{min}? It is used in the derivation of the angular momentum eigenvalues...
I'm reading in Sakurai's 1st chapter that this follows from the "associative axiom":
\langle\beta|\cdot\left(X|\alpha\rangle\right) = \left(\langle\beta|X\right)\cdot|\alpha\rangle
so we might as well write \langle\beta|X|\alpha\rangle. I know this is basic stuff, but I thought this...
Okay, I've read and re-read the section on tensor operators and the Wigner-Eckart theorem in Sakurais book, but I'm still confused. Could anyone explain to me how to think about vector and tensor operators and the significance of the Wigner-Eckart theorem? :confused:
Thanks.
Thanks. I must say I like the Baker-Hausdorff-way better, although the wave mechanics approach is more straightforward (and probably what we were supposed to do).
My answer, obtained by the Baker-Hausdorff theorem, was correct, it has just been confirmed :)
Thanks, guys.
Physicsmonkey: Is the wave mechanics approach related to the fact that exp(-ipa/h) is the translation-in-space operator?
Thanks everybody. I'm still having problems, though.
I've found the Baker-Hausdorff theorem in my book and I'll try to apply that (I don't have my papers here right now), but I'm also interested in the "wave mechanics" approach. My problem is, I'm bad at working with exponentials of...
We have a particle in a harmonic oscillator potential. The eigenstates are denoted {|0>,|1>,...,|n>,...}. Initially the particle is in the state |s> = exp(-ipa)|0>, where p is the momentum operator.
I need to find <x> as a function of time using the Heisenberg picture. The problem is, how do...
Thanks, guys. I understand 2) and 3) now. There's still the issue about 1), though...
I know he means units, but as far as I know only numbers can have units. We need to extend the concept of units if we are to apply to operators, aren't we? Or am I missing something?
I have some basic questions concerning operators. What is actually meant by the following:
1) The dimensionality of an operator? E.g., what does it mean to say that the operator K has the dimension of 1/length (an example from Sakurai's book)? Operators act on abstract mathematical states to...
Hmm, I think it's because we were able to factor the Hamiltonian perfectly, but I don't know, I was wondering the same thing...
I also have a quick question for you (I asked this in a another thread, but I'm still thinking about it). Suppose you have a hydrogen electron in the mixed state...
I don't know if I'm doing it right, but I get the same result as perturbation theory - E = (n+1/2)hw - ma^2/2 - (that's a good sign, isn't it :)), so perturbation theory yields the exact result in this case?
I've tried completing the square, but I don't know how to go on.
The energy of a classical particle in an electromagnetic field is just the sum of the energy associated with the electric and magnetic field, respectively, right?
E = \frac{q_1q_2}{4\pi\epsilon_0r^2}-\mu\cdot B
Well, it could be solved analytically, but I don't know how. I'm not particularly familiar with solving such differential equations. I've tried solving it numerically in Matlab, and this confirms my result, but it's not quite satisfying.
I don't know much about magnetic interactions in QM...
Hi,
I'm new to this subject, so bear with me. We consider the harmonic oscillator with a pertubation:
\hat{H}' = \alpha\hat{p}.
(What kind of a perturbation is that anyway, it's not a disturbance in the potential, what does it correspond to physically.)
Now I have to calculate the...
No no, I think you misunderstand. The EXPECTATION value is not the result of a single measurement. This is what I mean: Can you have a particle in a state, which is a superposition of eigenstates of L_z, but which - at some point in time (say t=0) - has different EXPECTATION values for L_x and...
Hi,
Say you have an electron in the hydrogen atom. Can this electron be in a state that is a superposition of the usual energyeigenstates (which are also eigenstates of L_z) AND have different expectation values for L_x and L_y? Are x and y symmetric in the sense that their expectation values...
Ok, thanks.
I've thought about something else. What if the ground state for H_{l+1} is degenerate (so that there are two groundstates), let's call this other ground state |\psi_0'\rangle; then it could be that a_{(l+1)-}\psi_0'\rangle yields an acceptable wavefunction (not the 0-function) and...