Matrix element of momentum

In summary: The problem was with the equation in the initial post. It should be $$\langle p_m | x | p_n\rangle = i \hbar \delta^\prime (p_m - p_n) $$
  • #1
nigelscott
135
4
Can somebody explain how you get:

<pm|p|pn> = (1/2π)∫e-ipmx/h (-ih∂/∂x) eipnx/h dx
...
= ih∂δ(pm - pn)/∂x

Conceptually, I am having a problem with how the inner product is formed when a derivative is involved (i.e. the ... steps)
 
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  • #2
Hi Nigelscott,

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

Best regards.
Sergio
 
  • #3
OK. But the differential gives (ipn/h)eipnx/h ... what happens to the p?
 
  • #4
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
$$ \delta^\prime(p) = \frac{d}{dp} \delta(p)$$
 
  • #5
I'm a little confused, since eigenstates are ortho-normal, and ##|p\rangle## is an eigenstate of p, wouldn't the answer simply be $$\langle p_m|p|p_n\rangle = p_n\langle p_m|p_n\rangle = p_n\delta(p_m-p_n)$$?
 
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Likes nigelscott and bhobba
  • #6
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.
 
  • #7
@MisterX I'm seeing a "p" sandwiched in between the kets in the OP, not an "x", hence my confusion.
 
  • #8
The OP was incorrect. Matterwave is correct regarding <p_m |p |p_n > with the "normalization" chosen above.
 
  • #9
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.
 
  • #10
nigelscott said:
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 pn does not become a derivative. It's a value, and does not mean the momentum operator.

What was the source of this problem?
 

What is the matrix element of momentum?

The matrix element of momentum is a quantity used in quantum mechanics to describe the transition between two quantum states. It represents the probability amplitude for a particle to transition from one state to another, and is related to the momentum operator in quantum mechanics.

How is the matrix element of momentum calculated?

The matrix element of momentum is calculated by taking the inner product of the initial and final quantum states, and then applying the momentum operator. The resulting value is a complex number that represents the amplitude of the transition.

What is the significance of the matrix element of momentum?

The matrix element of momentum is an important quantity in quantum mechanics because it allows us to calculate the probability of a particle transitioning between two states. It is also used in calculations of scattering amplitudes and decay rates.

How does the matrix element of momentum relate to the uncertainty principle?

The uncertainty principle states that we cannot simultaneously know the exact position and momentum of a particle. The matrix element of momentum plays a role in this principle by giving us information about the probability of a particle having a certain momentum, which in turn affects the uncertainty in its position.

Can the matrix element of momentum be measured experimentally?

Yes, the matrix element of momentum can be measured experimentally through various techniques such as scattering experiments or particle detectors. By measuring the momentum of particles before and after a transition, we can calculate the matrix element and compare it to theoretical predictions.

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