Expectation value of momentum for free particle

[Joker93]

Hello!
Could somebody please tell me how i can compute the expectation value of the momentum in the case of a free particle(monochromatic wave)? When i take the integral, i get infinity, but i have seen somewhere that we know how much the particle's velocity is, so i thought that we can get it from the expectation value of its momentum.

Also, Wikipedia states(here: https://en.wikipedia.org/wiki/Free_particle#Measurement_and_calculations -- just scroll down a bit) that one CAN do the calculation via the integral and get hbar*k.

Thanks!

 

[A. Neumaier]

The result depends on the state in which you compute the expectation. In a Gaussian state the result is finite.

 

[Joker93]

The result depends on the state in which you compute the expectation. In a Gaussian state the result is finite.

Hello! No, no, i am talking about a monochromatic wave(single k).

 

[A. Neumaier]

Hello! No, no, i am talking about a monochromatic wave(single k).

For an unnormalizable wave function such as $|k\rangle$ you cannot talk about expectations, let alone compute them!

 

[Joker93]

For an unnormalizable wave function such as $|k\rangle$ you cannot talk about expectations, let alone compute them!

So, why does Wikipedia(the link i have provided) state that it is hbar*k?

 

[A. Neumaier]

So, why does Wikipedia(the link i have provided) state that it is hbar*k?

You must calculate the expectation for wave packets (such as Gaussians), and then take the limit of infinite spread in position, and zero spread in momentum.

 

[Joker93]

You must calculate the expectation for wave packets (such as Gaussians), and then take the limit of infinite spread in position, and zero spread in momentum.

Gaussian with infinity variance gives a plane wave or a constant(in space) function?

 

[A. Neumaier]

Gaussian with infinity variance gives a plane wave or a constant(in space) function?

$\psi(x)=e^{\pm ik\cdot x-x^2/2\sigma^2}$ gives (after normalization, and with the correct sign) a moving Gaussian wave packet with momentum $\langle p\rangle=\hbar k$ and approaches a plane wave in the limit $\sigma\to\infty$

 

[Joker93]

$\psi(x)=e^{\pm ik\cdot x-x^2/2\sigma^2}$ gives (after normalization, and with the correct sign) a moving Gaussian wave packet with momentum $\langle p\rangle=\hbar k$ and approaches a plane wave in the limit $\sigma\to\infty$

Thanks!