Future of a particle in a box - I'm fundamentally confused

[timn]

Future of a particle in a box -- I'm fundamentally confused

Homework Statement

A particle is somewhere in the right half of a one-dimensional infinite potential well with sides at x=-a/2 and x=a/2. The particle's wave function is constant over x, i.e.

\psi(x) = \sqrt{2/a}

for 0<x<a/2 and zero for all other x.

Will the particle remain localized at later times?

Homework Equations

The Attempt at a Solution

I don't even understand the question. How could you call this particle localized, when all we know about its position is a probability function? Does "remain localized" mean that the probability function doesn't change, or that we know how it will change?

What would happen with the particle if it was just left there? Would the wave function change, and if so, why?

A "free" particle in a box has the wave function http://en.wikipedia.org/wiki/Particle_in_a_box#Wavefunctions" -- why doesn't this one? Is it because we already have some information on its position?

Sorry if the questions are many.

 

[vela]

The problem is asking you, will the particle stay confined to the right half of the box?

You're confusing the state of the system with the eigenstates of the system. The eigenstates are special in that a system in an eigenstate will remain in that eigenstate. That's why they're sometimes referred to as stationary states. Generally, the system will be in a superposition of eigenstates, and the state will evolve as determined by the Schrodinger equation.

 

[timn]

So since the state of the system is not a single eigenstate, but a superposition of at least two, it will change with time.

Your answer made me a lot wiser -- thank you!

 

[jeebs]

The eigenstates are special in that a system in an eigenstate will remain in that eigenstate. That's why they're sometimes referred to as stationary states.

I was under the impression that if you make a measurement, you will measure the eigenvalue corresponding to the system being in a particular eigenstate, but you could only assume the system to again be found in that eigenstate immediately after the initial measurement?

when you say "system in an eigenstate will remain in that eigenstate", do you mean indefinitely?

 

[vela]

You're right. I should have explicitly said energy eigenstates, which is what I was talking about. The system won't necessarily stay in eigenstates of other observables indefinitely.

 

[timn]

You're right. I should have explicitly said energy eigenstates, which is what I was talking about. The system won't necessarily stay in eigenstates of other observables indefinitely.

What is special about energy eigenstates? How could I know if an eigenstate of a quantity is such that the particle stays in it indefinitely?

Also, are other eigenstates still called stationary states, even if they aren't stationary?

 

[Matterwave]

Are you learning qm yourself, or are you taking a course? Most of these questions are standard questions that should be addressed in any QM class.

 

[timn]

Matterwave, I'm learning on my own by reading Gasiorowicz's Quantum Physics, which I know is used for introductory QM courses. I'd love a suggestion for a way to find the answers to these standard questions without bothering the helpful people over here.

 

[vela]

It's because the Hamiltonian is the generator of time evolution. A stationary state is an eigenstate of the Hamiltonian.

 

[timn]

Thanks vela! I forgot that the Hamiltonian is the energy operator. Much clearer now.