What is the Time Evolution of a Particle in a Harmonic Potential Energy Well?

[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.

Homework Equations

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.

 

[Cyosis]

What is the definition of the probability density function?

 

[cleggy]

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

 

[Cyosis]

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

 

[cleggy]

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

 

[cleggy]

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

Am I on the right tracks here?

 

[Cyosis]

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

 

[cleggy]

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

 

[Cyosis]

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

 

[cleggy]

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

 

[Cyosis]

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

 

[cleggy]

Right so I should have

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

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

 

[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>.

 

[cleggy]

that should have been

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

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