Particle in a box momentum uncertainty

[Vaal]

1.
this is for a particle in a box (infinite potential well):
For the wave functions calculate (as a function of n)
$$\Delta$$x
$$\Delta$$p
$$\Delta$$x$$\Delta$$p

Homework Equations

h = h bar below

u⁺_n(x)=(1/$$\sqrt{a}$$ )cos((n-1/2)(pi)x/a)
u^-_n(x)=(1/sqrt{a})sin(n(pi)x/a)
E^-_n=h²/(2m)(n(pi)/a)²
E⁺_n=h²/(2m)((n-1/2)(pi)/a)²

The Attempt at a Solution

ok i think the Delta x is easy, just find the variance of u⁺_n(x)² and u^-_n(x)² as a function of n (correct me if i am wrong)

where i am lost is on the momentum, i can obviously get it from the energy but i don't see how to determine its variance or even how it would change within one state. i would just use the uncertainty principle but that seems to be what they want you to show in the next step (also it is an inequality not an exact equation)

thanks for any help and sorry if this is an absurd question, it has been awhile since i have done any of this sort of stuff.

 

[vela]

Find $\langle \hat{p}^2 \rangle$ where

$$\hat{p} = \frac{\hbar}{i}\frac{\partial}{\partial x}$$

 

[Vaal]

Ok, I think I figured out the first part described above, although I can't seem to make the numbers come out right.

I am lost on the next problem, even more so than the last.

My well currently has walls at +/- a and a particle is in the ground state. the walls are suddenly moved to +/- b with b>a,
what is the probability the particle will be in ground state of new potential? & what is probability of the particle being in first excited state?

i realize the ground state of the new potential has less energy so the particle is more likely to be at a higher state than it was in the old potential but i have no idea how to make it quantitive and get the exact probabilities. any help would be greatly appreciated. thanks

 

[vela]

My well currently has walls at +/- a and a particle is in the ground state. the walls are suddenly moved to +/- b with b>a,
what is the probability the particle will be in ground state of new potential? & what is probability of the particle being in first excited state?

i realize the ground state of the new potential has less energy so the particle is more likely to be at a higher state than it was in the old potential but i have no idea how to make it quantitive and get the exact probabilities. any help would be greatly appreciated. thanks

Let the particle be in the state $|\psi\rangle$ and let $|\phi_n\rangle$ for n=1, 2, 3, … denote the eigenstates of the new Hamiltonian. You can express $|\psi\rangle$ in terms of the new eigenstates:

$$|\psi\rangle = c_0|\phi_0\rangle + c_1|\phi_1\rangle + \cdots$$

where the c_n's are constants, possibly complex. You need to find what the c_n's are for n=0 and n=1. The probability of finding the particle in a particular state is equal to the modulus squared of the corresponding constant.