- #1

razidan

- 75

- 1

## Homework Statement

A bound particle is in a superposition state:

[tex] \psi(x)=a[\varphi_1(x)e^{-i\omega_1t}+\varphi_2(x)e^{-i\omega_2t}] [/tex]

Calculate [itex] <x> [/itex] and show that the position oscillates.

## Homework Equations

[tex] <x>=\int_{-\infty}^{\infty} \psi(x) x \psi^*(x) \mathrm{d}x [/tex]

## The Attempt at a Solution

[itex] <x>=\int_{-\infty}^{\infty} \psi(x) x \psi^*(x)\mathrm{d}x=\int_{-\infty}^{\infty} x[|\varphi_1(x)|^2 +x|\varphi_2(x)|^2+2\varphi_1(x)\varphi_2(x)\cos(\tilde{\omega}t) ]\mathrm{d}x [/itex]

where [itex] \tilde{\omega}=\omega_1 - \omega_2 [/itex],

I have assumed that [itex] \varphi_1(x) [/itex] and [itex] \varphi_2(x) [/itex] are real functions (is this a valid assumption? I think it is, because if there were some imaginary component it could just go into the phase).

Here I got stuck.

what I think i need to do is:

1)say that [itex]\int_{-\infty}^{\infty} x|\varphi_1(x)|^2 \mathrm{d}x [/itex] is zero (same for the second state). I think it's true because of a mathematical trick. maybe odd function over a symmetric interval type of thing? but I am not sure if i can say [itex] \varphi(x) [/itex] has a defined parity,

2) define the oscillation amplitude [itex]A=\int_{-\infty}^{\infty} 2x\varphi_1(x)\varphi_2(x)\mathrm{d}x [/itex].

If this is true, what guarantees that the integral is finite?

Is this correct? is it too generalized and there was anything more specific i can do?

I feel like I'm being stumped by mathematical properties and not the physics :/

Thanks

R.