Does the time dependent hamiltonian have stationary states?

[nebula009]

It doesn't seem like a time dependent hamiltonian would have stationary states, am I wrong? I've run into conflicting information.

 

[atyy]

Where have you seen that there are stationary states?

 

[nebula009]

Some notes I have been given have some kind of typo in them I think. There is a suggestion that time dependent Schrödinger equations can be solved by expanding initial states in terms of stationary states, and that seemed like a problem. I wanted to make sure my intuition is correct, but I should have clarification on the issue from the author of the notes shortly as well.

 

[atyy]

It's a terminology thing. The time-dependent Schroedinger equation (TDSE) can contain a Hamiltonian that has no explicit time dependence.

For example, the TDSE given in http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/scheq.html has a Hamiltonian with no explicit time dependence because the potential U is a function of x only.

When the Hamiltonian has no explicit time dependence, we can treat the wavefunction as a product of two functions, one a function of space, the other of time. This is called separation of variables, and results in an equation called the time-independent Schroedinger equation (TISE) for the part of the wavefunction that is only a function of space. The stationary states are given by solving the TISE, substituting back into the TDSE and seeing that in some sense those states don't change in time. A general solution of the TDSE in which the Hamiltonian has no explicit time dependence is formed by a superposition of TDSE solutions, each oscillating with its own frequency.

If U is a function of x and t then attacking the TDSE by separation of variables using functions of space only and time only doesn't work, and the situation is usually treated by time-dependent perturbation theory. http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/Time_Dep_PT.htm

 

[Ken G]

And not to add anything to that excellent answer, but just in case it's not clear, the answer is that the time-dependent SE does admit solutions that are stationary states-- they are the energy eigenstates of any time-independent Hamiltonian. You can easily see that any such eigenstate will have a trivial time dependence of simply advancing its phase by the factor e^-iEt/hbar, and this is called "stationary" because a global phase factor like that does not (by itself) induce any changes in the observables.

 

[nebula009]

thanks all! It did turn out to be a terminology thing. I had thought that the TDSE meant that the Hamiltonian must be time dependent, but in the context that the confusion arose we are talking about the state vector evolving in time, with the phase factor mentioned.

Actually I'm not sure why I thought this anymore. I mistakingly read/heard something and conflated the two categories. Thanks again for the answers because I did become independently interested in the possibility of eigenstates for an H(t).