Are probabilities for Eigen Vectors of Hamiltonians unaffected as time evolves?

[ask_LXXXVI]

Let a state vector $$\psi$$ is eigen vector for a Hamiltonian H which governs the Schrodinger equation (in its general form)of a system. Then, will probability distribution of $$\psi$$ w.r.t any observable remain unchanged as time evolves?

 

[Vanadium 50]

What does it mean for something to be an eigenvector of the Hamiltonian?

 

[ask_LXXXVI]

sorry.Perhaps I bungled up and used some wrong terminology.

I will first give an ex.
Consider a free particle moving in 1-D and in absence of any potentials.
thus H = p²/2m , where p is momentum operator.

.


Now consider a $$\psi$$ (x,t) given by

[PLAIN]http://img225.imageshack.us/img225/3446/image042.png

The $$\psi$$ (x,t=0) is an eigen vector for the momentum operator.And as can be seen is also an eigen vector for the H operator in this case.

And as it turns out the probability distribution of this state vector wrt x or p remains unchanged with respect to time. As the time exponential is just a phase.


My question is in a general case , if the state vector (for t=0 ,i.e $$\psi$$(x,t=0))is an eigen vector of an observable and also of the Hamiltonian , then will probability distribution of this state vector remain unchanged over time wrt any observable.

 

[Vanadium 50]

Right, and let me ask my question again. What does it mean for something to be an eigenvector of the Hamiltonian?

 

[ask_LXXXVI]

I might be wrong but,Isn't the Hamiltonian a hermitian operator? So why can't there be eigen vectors for it.

In the example I gave
The hamiltonian is of the form
($$\frac{\partial }{\partial x}$$)² .

So isn't the state vector (corresponding to a definite momentum p),$$\psi$$(x) = exp(ipx/ħ) an eigen vector for it?

 

[Andeplane]

An eigenvector of the hamiltonian is a state that satisfies the time-independent SL.

And yeah, all you get from the time evolution operator is a phase that will cancel out in any expectation value calculation and probability density calculation.

If you have a system where the solution cannot be separated (one spatial-part and one time-part) I guess we have a different story. Or if the hamiltonian is timedependent.

 

[ask_LXXXVI]

Yes. I was asking about the simpler cases where Hamiltonian is time independent and solutions are separable in time and spatial parts.

 

[Vanadium 50]

I understand why you are struggling. I have asked you the same question twice, and instead of answering it, you have guessed at what I must mean and answered some other question.

 

[ask_LXXXVI]

I understand why you are struggling. I have asked you the same question twice, and instead of answering it, you have guessed at what I must mean and answered some other question.

oh, so is that the very definition of Hamiltonian , that its eigen vectors are such that their probabilities are unaffected over time ?
Sorry, as I didn't get your question .

 

[Andeplane]

I have no idea what the confusion are right now, but i'll try :p

Say you have a system (potential and boundary conditions etc) so that the SE (Schrödinger Equation) is separable into a time-dependent part and a spatial-dependent part. Then, if the hamiltonian is time-independent (as you were talking about), the time evolution operator is just given as
$U(t-t_0) = \exp\Big(-\frac{i}{\hbar} \hat H (t-t_0)\Big)$
which for the eigenstates of the hamiltonian only would (as you suggest) add a global phase which is not possible to measure, and doesn't change any expectation values or other physical property (the phase will get canceled).

A linear combination of eigenstates with different energies would of course be affected.

 

[Andeplane]

oh, so is that the very definition of Hamiltonian , that its eigen vectors are such that their probabilities are unaffected over time ?
Sorry, as I didn't get your question .

The definition of the hamiltonian in QM is the operator that "controls" the time evolution. Usually the hamiltonian represents the energy, but sometimes this is not the case. An example of this is a time-dependent rotating magnetic field (which gives a time-dependent hamiltonian) where you can do a unitary transformation to remove the time-dependency of the hamiltonian. The 'new' hamiltonian does not have the systems energies as its eigenvalues, but rather is the one defining the time evolution operator.

One could maybe draw some connections to classical mechanics and fictive forces in rotating systems, but I'm not sure about that!