Time-Dependent Perturbations in Schrodinger Equation

[Gear300]

From what I see, the time dependence in potentials do not change the spatial aspect of the wave function. They contribute a time dependence to the population of what were originally stationary states. If this is the case, then does that imply that the time independent bases are the ones we generally observe?

 

[Amok]

What do you mean by 'changing the spatial aspect of the wavefunction'?

 

[jeppetrost]

I think the trick is, that any wave-function can be constructed from the set of eigenfunctions for some hamiltonian (since these are always complete).
This means that what ever is your wave function, you can write it in terms of eigenfunctions of eg. the harmonic oscillator or what ever other potential you're working with. Now, when you do perturbations, the changes are small, so we write the new wavefunction as a combination of the solutions to the time-independent potential.
I hope this wasn't too cryptic.

 

[Amok]

Even if the perturbation is not small you can write the "new" wf as combination of the solutions of the time-independent problem as long as you allow the coefficients to depend on time. This is a general result (I think).

 

[Gear300]

I'm a little confused on what we're observing though. For a perturbation, we should be able to make some statement on the population transitions depending on what basis we choose, and this should reflect in experiment. If you were to change basis from the unperturbed stationary states to some other orthonormal bases, the population transitions might have a different behavior. Essentially, what I'm asking is whether it is appropriate to say that we observe transitions between the stationary states rather than some other set of states.

 

[Matterwave]

Usually you set up your problem so that your initial state is an eigenstate of the unperturbed Hamiltonian. Due to the perturbation, that state is no longer an eigenstate and will evolve in time. We use time-dependent perturbation theory to find the properties of that time evolution. You don't actually observe the "state" of the particle. The wave-function is not an observable. What you do measure is the energy of the particle.

 

[Gear300]

Though, if we applied a perturbation to the hydrogen atom and did a spectral analysis while the perturbation is on, we should see discrete lines. So, in this sense, isn't it like saying that the stationary states are the ones we observe?

 

[SpectraCat]

Though, if we applied a perturbation to the hydrogen atom and did a spectral analysis while the perturbation is on, we should see discrete lines. So, in this sense, isn't it like saying that the stationary states are the ones we observe?

Yes, that is one of the postulates of quantum mechanics ... that for any single measurement, you can only measure eigenvalues of the observable you are measuring. In the case of a perturbed system, you observe the eigenvalues of the perturbed system. However that doesn't mean that the system was in an eigenstate to start with ... expanding the wavefunctions in the unperturbed basis is a convenient mathematical trick that is consistent with the postulates of quantum mechanics and allows one to predict the eigenvalues of the perturbed systems with arbitrary accuracy. The results of single measurements of non-stationary states cannot be predicted (other than to say you will always observe one of the eigenvalues), however the average result of many repetitions of the same measurement on identically prepared non-stationary states can be predicted (provided you do enough repetitions) .. we call this the expectation value, and it doesn't have to correspond to an eigenvalue of the system.

Does that help?

 

[Gear300]

Yes, that helps. I found it a little odd that time-dependence in the Hamiltonian does not seem to change the observed energy eigenbasis.

 

[SpectraCat]

Yes, that helps. I found it a little odd that time-dependence in the Hamiltonian does not seem to change the observed energy eigenbasis.

Well ... the eigenvalues of a time-dependent Hamiltonian *do* change with time, in the sense that for any specific value of t, you will have a different set of eigenvalues. What may be confusing you is that we typically *choose* not to represent it that way when we work out the mathematics. It is typically much easier to treat the time-dependent terms in the Hamiltonian as time-dependent perturbations to some well-understood time-independent problem.

 

[Gear300]

Well ... the eigenvalues of a time-dependent Hamiltonian *do* change with time, in the sense that for any specific value of t, you will have a different set of eigenvalues.

I figured that was the case at first, but in the example I gave, the spectral lines don't move around (or at least I do not think that is the case). It seemed apparent to me that the only basis we observe are the eigenstates of the time-independent hamiltonian.

 

[jeppetrost]

But do remember that you calculate the energychanges due to the perturbation too. The eigenenergies do certainly change - time dependent or not.