Modified Harmonic Oscillator probabilities

[ma18]

Homework Statement

The e-functions for n=0,1,2 e-energies are given as

psi_0 = 1/(pi^1/4 * x0^1/2)*e^(x^2/(2*x0^2)

psi_1 =...

psi_2 =...

The factor x0 is instantaneously changed to y= x0/2. This means the initial wavefunction does not change.

Find the expansions coefficients of the initial state to the three lowest states of the modified potential. What energy measurements would you find and with what probability.

Homework Equations

Prob En = c_n ^2

En = hbar*omega*(n+0.5)

The Attempt at a Solution

I can find the new energies:

En = hbar*omega*(n+0.5) = m*omega^2*x_0^2*(n+1/2)

Putting in y = x_0 instead we get:

En = m*omega^2*y^2*(n+1/2)
= 1/4 m*omega^2*x_0^2*(n+1/2)
= 1/4 hbar*omega*(n+0.5)For the expansion coefficients I am not sure what to put into the formula:

cn = integral from -inf to inf: phi* * psi dx

I know that the modified eqn to psi_0 is:

psi_m0 = sqrt(2)*e^2 * psi_0

But if I put this in I just get sqrt(2)*e^4 and that doesn't make sense for the probs (>1)

 

[BiGyElLoWhAt]

the e-functions for n=0,1,2 e-energies are given as

psi_0 = 1/(pi^1/4 * x0^1/2)*e^(x^2/(2*x0^2)

psi_1 =...

Psi_2 =...

The factor x0 is instantaneously changed to y= x0/2. This means the initial wavefunction does not change.

Yes, assuming that $\dot{x}_0$ = 0. Otherwise the phase shift will change.

Find the expansions coefficients of the initial state to the three lowest states of the modified potential. What energy measurements would you find and with what probability.

Homework Equations

prob en = c_n ^2

en = hbar*omega*(n+0.5)

The Attempt at a Solution

i can find the new energies:

En = hbar*omega*(n+0.5) = m*omega^2*x_0^2*(n+1/2)

putting in y = x_0 instead we get:

En = m*omega^2*y^2*(n+1/2)
= 1/4 m*omega^2*x_0^2*(n+1/2)
= 1/4 hbar*omega*(n+0.5)


for the expansion coefficients i am not sure what to put into the formula:

Cn = integral from -inf to inf: Phi* * psi dx

Is this a typo? $\Psi^* \Psi$

i know that the modified eqn to psi_0 is:

Psi_m0 = sqrt(2)*e^2 * psi_0

but if i put this in i just get sqrt(2)*e^4 and that doesn't make sense for the probs (>1)

Honestly I'm not sure what you're trying to do, if that was supposed to be a complex conjucate (that's what it looks like to me) Your psi functions are real, so... Again, though, I really don't know what your trying to do. I'm not going to guarantee to be able to help you, but could you give a clearer explanation of the problem?

It almost looks as thought you're trying to normalize a function to solve for the coefficients, but I'm not sure.

 

[TSny]

I know that the modified eqn to psi_0 is:

psi_m0 = sqrt(2)*e^2 * psi_0

This isn't correct. You can't factor out e² like that.