How Does the Momentum Operator Act in the Matrix Element <pm|p|pn>?

[nigelscott]

Can somebody explain how you get:

<p_m|p|p_n> = (1/2π)∫e^-ip_mx/h (-ih∂/∂x) e^ip_nx/h dx
...
= ih∂δ(p_m - p_n)/∂x

Conceptually, I am having a problem with how the inner product is formed when a derivative is involved (i.e. the ... steps)

 

[USeptim]

Hi Nigelscott,

The derivative operator just makes a derivative over ∂x over either p_n(x) or p_m(x). The inner product just makes the multiplication: ∂p_n(x)/∂x * p_m(x) and this has to be integrated over all the space and multiplied by -i2π/h in order to get <p_m|p|p_n>.

Best regards.
Sergio

 

[nigelscott]

OK. But the differential gives (ip_n/h)e^ip_nx/h ... what happens to the p?

 

[MisterX]

The equation in the initial post is incorrect. It should be $$\langle p_m | x | p_n\rangle = i \hbar \delta^\prime (p_m - p_n) $$

Where
$$ \delta^\prime(p) = \frac{d}{dp} \delta(p)$$

 

[Matterwave]

I'm a little confused, since eigenstates are ortho-normal, and $|p\rangle$ is an eigenstate of p, wouldn't the answer simply be $$\langle p_m|p|p_n\rangle = p_n\langle p_m|p_n\rangle = p_n\delta(p_m-p_n)$$?

 

[MisterX]

As for a derivation - first what are the eigenstates of p? We can choose a scheme such that

$$\langle x \mid x' \rangle = \delta(x-x') $$
$$\mid p \rangle = \sqrt{\frac{1}{2\pi}} \int \mathop{dx} e^{i\mathbf{p} \cdot \mathbf{x}/\hbar}\mid x \rangle $$

note that these states are not normalized; $\langle p \mid p' \rangle = \delta(p-p')$, however $\left| \langle p \mid p' \rangle \right|^2$ is not defined under this scheme. Continuing
$$\langle p | x | p'\rangle =$$ $$\frac{1}{2\pi} \int\int \mathop{dxdx'} e^{-i\mathbf{p} \cdot \mathbf{x}/\hbar} x e^{i\mathbf{p'} \cdot \mathbf{x'}/\hbar} \langle x \mid x' \rangle $$
$$\frac{1}{2\pi} \int_{\infty}^\infty \mathop{dx} e^{-i\mathbf{p} \cdot \mathbf{x}/\hbar} x e^{i\mathbf{p'} \cdot \mathbf{x}/\hbar} $$
$$=\frac{1}{2\pi} \int_{\infty}^\infty \mathop{dx} x e^{i(\mathbf{p'} - \mathbf{p}) \cdot \mathbf{x}/\hbar}$$
If you're astute you may already recognize this is going to be related to the derivative of the Dirac delta. For example what we have is a scaled and shifted version of the Fourier transform of $1 x$ which is going to be related to the derivative of the Fourier transform of $1$. Alternately, you could try integrating by parts.

 

[Matterwave]

@MisterX I'm seeing a "p" sandwiched in between the kets in the OP, not an "x", hence my confusion.

 

[MisterX]

The OP was incorrect. Matterwave is correct regarding <p_m |p |p_n > with the "normalization" chosen above.

 

[nigelscott]

Thanks to all. I believe Matterwave has put me on the right track. I was trying to evaluate the integral containing the partial derivative. If I evaluate the integral using the OP p and then substitute the partial derivative at the end everything falls into place.

 

[MisterX]

Thanks to all. I believe Matterwave has put me on the right track. I was trying to evaluate the integral containing the partial derivative. If I evaluate the integral using the OP p and then substitute the partial derivative at the end everything falls into place.

Except maybe it's not correct. The sign is wrong for one. What does it mean to take the derivative with respect to x of a delta function that clearly involves only p's? Does the derivative apply only to the delta function, or to the expression that might come after it?

$$\langle p_m|p|p_n\rangle = p_n\langle p_m|p_n\rangle = p_n\delta(p_m-p_n)$$
That's it. The p_n does not become a derivative. It's a value, and does not mean the momentum operator.

What was the source of this problem?