How Can We Show that <x> = ∫ Φ* (-ħ/i) ∂Φ/∂p dp in Quantum Mechanics?

[NeoDevin]

Homework Statement

Show that

$$<x> = \int \Phi^* \left(-\frac{\hbar}{i}\frac{\partial}{\partial p} \right) \Phi dp$$

Homework Equations

$$\Phi(p,t) = \frac{1}{\sqrt{2\pi\hbar}} \int^{\infty}_{-\infty} e^{\frac{-ipx}{\hbar}} \Psi(x,t)dx$$

$$\Psi(x,t) = \frac{1}{\sqrt{2\pi\hbar}} \int^{\infty}_{-\infty} e^{\frac{ipx}{\hbar}} \Phi(p,t)dp$$

The Attempt at a Solution

I started out with

$$<x> = \int^{\infty}_{-\infty} \Psi^* x \Psi dx$$

Using the above equation for $\Psi(x,t)$ (and it's conjugate) gives:

$$\Psi^* (x,t) = \frac{1}{\sqrt{2\pi\hbar}} \int^{\infty}_{-\infty} e^{\frac{-ipx}{\hbar}} \Phi(p,t)dp$$

and

$$x\Psi(x,t) = \frac{x}{\sqrt{2\pi\hbar}} \int^{\infty}_{-\infty} e^{\frac{ipx}{\hbar}} \Phi(p,t)dp$$

$$= \frac{1}{\sqrt{2\pi\hbar}} \int^{\infty}_{-\infty} x e^{\frac{ipx}{\hbar}} \Phi(p,t)dp$$

$$= \frac{1}{\sqrt{2\pi\hbar}} \int^{\infty}_{-\infty} \frac{\partial}{\partial p} e^{\frac{ipx}{\hbar}} \Phi(p,t)dp$$

$$= \frac{1}{\sqrt{2\pi\hbar}} \left[ \left( e^{\frac{ipx}{\hbar}} \Phi(p,t) \right) \bigg|^{\infty}_{-\infty} -\int^{\infty}_{-\infty} e^{\frac{ipx}{\hbar}} \frac{\partial}{\partial p}\Phi(p,t)dp \right]$$

$$= -\frac{1}{\sqrt{2\pi\hbar}} \int^{\infty}_{-\infty} e^{\frac{ipx}{\hbar}} \frac{\partial}{\partial p} \Phi(p,t) dp$$

Substituting into the original equation for $<x>$ then gives

$$<x> = \int^{\infty}_{-\infty}\left( \frac{1}{\sqrt{2\pi\hbar}} \int^{\infty}_{-\infty}e^{\frac{-ipx}{\hbar}} \Phi^* (p,t) dp \right) x\Psi(x,t) dx$$

$$= \frac{1}{\sqrt{2\pi\hbar}} \int^{\infty}_{-\infty} \Phi^* (p,t) \int^{\infty}_{-\infty} e^\frac{-ipx}{\hbar} (x\Psi(x,t))dx dp$$

$$= -\frac{1}{2\pi\hbar} \int^{\infty}_{-\infty} \Phi^* (p,t) \int^{\infty}_{-\infty} e^{\frac{-ipx}{\hbar}} \int^{\infty}_{-\infty} e^{\frac{ipx}{\hbar}} \frac{\partial}{\partial p} \Phi(p,t) dp dx dp$$

$$= -\frac{1}{2\pi\hbar} \int^{\infty}_{-\infty} \Phi^* (p,t) \int^{\infty}_{-\infty} \frac{\partial}{\partial p} \Phi(p,t) \int^{\infty}_{-\infty} e^{\frac{-ipx}{\hbar}} e^{\frac{ipx}{\hbar}} dx dp dp$$

$$= -\frac{1}{2\pi\hbar} \int^{\infty}_{-\infty} \Phi^* (p,t) \int^{\infty}_{-\infty} \frac{\partial}{\partial p} \Phi(p,t) \int^{\infty}_{-\infty} dx dp dp$$

I'm pretty sure I messed up somewhere, since that integral is infinite...

Any help would be appreciated.

 

[NeoDevin]

I also tried the reverse, starting with the expression you're supposed to get for <x>, and working back from there using similar methods... but it gives me the same problem.

 

[jpr0]

Your calculation seems fine up until the point where you substitute your expression for $$x\Psi(x,t)$$. You should be integrating over two dummy variables in your final expression, say $p$ and $p^{\prime}$, but you have written both dummy variables as the same variable $p$. The expression you derived for $$x\Psi(x,t)$$ in terms of an integral over $p$, change $p$ to $p^{\prime}$ and everything should work out.

 

[NeoDevin]

So then for the second last line we end up with
$$<x> = -\frac{1}{2\pi\hbar} \int^{\infty}_{-\infty} \Phi^* (p,t) \int^{\infty}_{-\infty} \frac{\partial}{\partial p'} \Phi(p',t) \int^{\infty}_{-\infty} e^{\frac{-i(p-p')x}{\hbar}} dx \ dp' \ dp$$

Is that integral doable?

 

[Dick]

You should recognize the x integral as a delta function.

 

[NeoDevin]

Oh, right... Duh. Got it now