Harmonic potential energy well

1. Jun 11, 2009

cleggy

1. A particle in a harmonic potential energy well is in a state described by the initial wave function

Ψ(x, 0) = (1/√2) (ψ1(x)+ iψ3(x))

where ψ1(x)and ψ3(x) are real normalized energy eigenfunctions of the harmonic oscillator with quantum numbers n =1 and n = 3 respectively.

(a)
Write down an expression for Ψ(x, t) that is valid for all t> 0. Express your answer in terms of ψ1(x), ψ3(x)and ω0, the classical angular frequency of the oscillator.

(b)
Find an expression for the probability density function at any time t> 0. Express your answer in terms of ψ1(x), ψ3(x), ω0 and t.Use the symmetry of this function to show that the expectation value,<x> = 0 at all times.

2. Relevant equations

3. The attempt at a solution

I have reached Ψ(x, t) = (1/√2) (ψ1(x)exp(-3iw$$_{}0$$t/2+ iψ3(x)exp(-7iw$$_{}0$$t/2)

for part (b) I'm not sure how to calculate the probability density function at any time t>0 ?

I know that the probability density needs to be even function of x and so therefore being symmetrical about the centre of the well at all times.

Then using the sandwich integral to calculate <x>, it will yield zero as the integrand is an odd function.

2. Jun 11, 2009

Cyosis

What is the definition of the probability density function?

3. Jun 11, 2009

cleggy

it's just the |Ψ|^2 is it not? I don't know how to do the math

4. Jun 11, 2009

Cyosis

That's correct, do you now that $\psi^* \psi=|\psi|^2$, with $\psi^*$ the conjugated wave function?

5. Jun 11, 2009

cleggy

so the conjugate of Ψ(x, t) is (1/√2)(ψ1(x)exp(+3iwt/2) + iψ3(x)exp(7iwt/2)) ?

6. Jun 11, 2009

cleggy

The exponentials disappear and then |Ψ|^2 = (1/2)(ψ1(x) - ψ3(x)).

Am I on the right tracks here?

7. Jun 11, 2009

Cyosis

That doesn't look correct to me. Write out the whole expression and show your steps, start with showing what you got for$\psi^*$.

8. Jun 11, 2009

cleggy

i get $$\Psi$$$$\ast$$=(1/√2)(ψ1(x)exp(+3iwt/2) + iψ3(x)exp(7iwt/2)) for

9. Jun 11, 2009

Cyosis

That is not correct. Note that there is an i in front of $\psi_3$.

10. Jun 11, 2009

cleggy

$$\Psi$$$$\ast$$=(1/√2)(ψ1(x)exp(+3iwt/2) - iψ3(x)exp(7iwt/2))

11. Jun 11, 2009

Cyosis

Looking good. Now multiply and be careful when working out the brackets.

12. Jun 12, 2009

cleggy

Right so I should have

|$$\Psi$$|^2 = 1/2 |$$\psi$$1|^2 + |$$\psi$$3|^2

+ 2$$\psi$$1$$\psi$$3sin(2wot)

13. Jun 12, 2009

Cyosis

Almost correct, don't forget that the entire expression is multiplied by 1/2 not just the first term.

Now write down the expression for <x>.

14. Jun 12, 2009

cleggy

that should have been

|$$\Psi$$|^2 = [1/2][|$$\psi$$1|^2 + |$$\psi$$3|^2

+ 2$$\psi$$1$$\psi$$3sin(2wot)]