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First off sorry for the badly worded title.
1. Homework Statement
Beginning of Question:
Consider a single quantum particle of mass M trapped in the infinite square well potential, V(x), given by
V(x)= 0 if 0 < x < L
infinity otherwise
The wave function for a particle in the n-th energy level is: Ψn(x) = √(2/L) sin(nπx/L)
a.) I found the expected position and momentum of a particle in the n-th energy level.
b.) I calculated the expectation value for the energy of a particle in the n-th energy level using the hamiltonian.
Bit of Question I'm stuck on:
c.)
Suppose that the particle initially starts in the lowest energy level and the potential is instantaneously changed to:
V(x) = 0 if 0 < x < L/2
infinity otherwise
Find the probability that the particle ends up in the lowest energy level of the new potential.
[/B]
I'm not exactly sure what to do here. I assume it must be something along the lines of:
1st - Finding the lowest allowed energy level
2nd - Finding the probability that the particle would be in this state.
I was thinking that I might be able to use a method along these lines:
Renormalise the wave equation first to account for the change in potential?
Then repeat what I did in part b.) to find the expectation value of the energy of the particle in the n-th energy level?
I would surely then be able to find the lowest expected value for the energy?
And then I would be able to find the probability that a particle is in that state?
Or have I got totally the wrong idea here? It seems as though I'm ignoring the fact that the potential changed instantaneously.
1. Homework Statement
Beginning of Question:
Consider a single quantum particle of mass M trapped in the infinite square well potential, V(x), given by
V(x)= 0 if 0 < x < L
infinity otherwise
The wave function for a particle in the n-th energy level is: Ψn(x) = √(2/L) sin(nπx/L)
a.) I found the expected position and momentum of a particle in the n-th energy level.
b.) I calculated the expectation value for the energy of a particle in the n-th energy level using the hamiltonian.
Bit of Question I'm stuck on:
c.)
Suppose that the particle initially starts in the lowest energy level and the potential is instantaneously changed to:
V(x) = 0 if 0 < x < L/2
infinity otherwise
Find the probability that the particle ends up in the lowest energy level of the new potential.
Homework Equations
The Attempt at a Solution
.[/B]
I'm not exactly sure what to do here. I assume it must be something along the lines of:
1st - Finding the lowest allowed energy level
2nd - Finding the probability that the particle would be in this state.
I was thinking that I might be able to use a method along these lines:
Renormalise the wave equation first to account for the change in potential?
Then repeat what I did in part b.) to find the expectation value of the energy of the particle in the n-th energy level?
I would surely then be able to find the lowest expected value for the energy?
And then I would be able to find the probability that a particle is in that state?
Or have I got totally the wrong idea here? It seems as though I'm ignoring the fact that the potential changed instantaneously.