- #1
skynelson
- 58
- 4
I'm working on the time-dependent Schrodinger equation, and come across something I don't understand regarding notation, which is not specific to TDSE but the Schrodinger formalism in general. Let's say we have a non-trivial potential. There is a stage in the development of the TDSE where we write the coefficient for the ## n##th energy eigenstate as
$$ \frac{\partial c_n(t)}{\partial t} = \frac{-i}{\hbar} \sum_k c_k(t) <n|\hat{V}(t)|k>e^{i\omega_{nk}t},$$
in other words, the time dependence of the ##n##th coefficient depends on each of the other ##k## coefficients as well as the potential, ##<n|\hat{V}(t)|k>##.
My question is about the potential, ##V_{nk} \equiv <n|\hat{V}(t)|k>##. This is a matrix element representing a transition amplitude between the ##k##th energy eigenstate and the ##n##th energy eigenstate.
Can somebody please provide an example of the form this would explicitly take? It seems there are notoriously few potentials that are analytically solvable in the Schrodinger equation, so I am having trouble understanding what ##V_{nk}## would look like. I guess you could say I am unclear on what the ##\hat{V}## matrix looks like.
One example I found is for the potential $$\hat{V}(t)=2 \hat{V} cos(\omega t).$$
However, this doesn't help since it is only explicit about the time dependence. (e.g. the result is ##<n|\hat{V}(t)|k> = 2 V_{nk} cos(\omega t)##, but I want to have an example of ##V_{nk}##.)
Thank you!
P.S. I have seen this worked out for the example of the quantum harmonic oscillator energy eigenstates, using creation and annihilation operators, but that formalism is so unique I think it would be helpful see another example as well.
P.P.S. The particle-in-a-box, again, seems not so helpful, since the form of the potential is non-analytical, just step functions.
$$ \frac{\partial c_n(t)}{\partial t} = \frac{-i}{\hbar} \sum_k c_k(t) <n|\hat{V}(t)|k>e^{i\omega_{nk}t},$$
in other words, the time dependence of the ##n##th coefficient depends on each of the other ##k## coefficients as well as the potential, ##<n|\hat{V}(t)|k>##.
My question is about the potential, ##V_{nk} \equiv <n|\hat{V}(t)|k>##. This is a matrix element representing a transition amplitude between the ##k##th energy eigenstate and the ##n##th energy eigenstate.
Can somebody please provide an example of the form this would explicitly take? It seems there are notoriously few potentials that are analytically solvable in the Schrodinger equation, so I am having trouble understanding what ##V_{nk}## would look like. I guess you could say I am unclear on what the ##\hat{V}## matrix looks like.
One example I found is for the potential $$\hat{V}(t)=2 \hat{V} cos(\omega t).$$
However, this doesn't help since it is only explicit about the time dependence. (e.g. the result is ##<n|\hat{V}(t)|k> = 2 V_{nk} cos(\omega t)##, but I want to have an example of ##V_{nk}##.)
Thank you!
P.S. I have seen this worked out for the example of the quantum harmonic oscillator energy eigenstates, using creation and annihilation operators, but that formalism is so unique I think it would be helpful see another example as well.
P.P.S. The particle-in-a-box, again, seems not so helpful, since the form of the potential is non-analytical, just step functions.