One dimensional potential well

[jimmycricket]

Homework Statement

A particle of mass m is confined in a one dimensional well by a potential V. The energy eigenvalues are
$$E_{n}=\frac{\hbar^2n^2\pi^2}{2mL^2}$$
and the corresponding normalized eigenstates are
$$\Phi_{n}=\sqrt{\frac{2}{L}}sin(\frac{n\pi x}{L})$$
At time t=0 the particle is in the ground state. Find the probability that the particle is between x=0 and x=L/6.
Explain why this probbility does not depend on time.

Homework Equations

The Attempt at a Solution

I have found the probability to be $$\frac{1}{6}-\frac{\sqrt{3}}{4\pi}$$
My question is why the probability does not depend on time. Is it because the particle is in the ground state?

 

[ShayanJ]

The answer lies in the form of time dependence of the ground state(Or any other energy eigenstate).
I can explain it to you but I don't feel good about just handing the answer to you. I would do it if it was hard to answer the question but here things are simple and you should just think about the form of time dependence and the process of calculating probabilities.

 

[jimmycricket]

I appreciate it may be a simple concept to you but I don't know where to look for this information. I don't want an answer handed on a plate but maybe some hints for further reading that could lead me to it myself

 

[ShayanJ]

You mean you don't know about time-dependent Schrodinger equation and the process of separation of variables to get time-independent Schrodinger equation? Its a bad idea to ignore these in a QM course!
Anyway, you should just study the things I mentioned from your textbook. Then it will be clear(and relatively easy) to you too.

P.S.
If you had an introduction to those things, you didn't learn them well, so you should again go and study them.

 

[jimmycricket]

thanks. I'll go and read up a bit

 

[jimmycricket]

would you suggest any particularly good texts to read?

 

[ShayanJ]

All introductory textbooks on QM cover that. But I think Quantum Mechanics:Concepts and Applications by Nouredin Zettili is a good choice.