QM: Probability of measuring momentum

[WWCY]

Hi all,

My question is in reference to the following paper: https://arxiv.org/pdf/1202.1783.pdf

In equation 3.8, the author computes an order-of-magnitude approximation of probability of measuring negative momentum from the following wavefunction:
$$
\Psi_k =\sum_{k=1,2} \frac{B_k}{\sqrt{4\sigma ^2 + 2it}} \exp (ip_k (x - \frac{p_k}{2}t) - \frac{(x - p_k t)^2}{4\sigma ^2 + 2it})
$$

Equation 3.8 is:
$$\int_{-\infty}^{0} dp \exp[-200(p-0.3)^2]$$

From what I understand, one should first take a Fourier-Transform of the above wavefuction, find $|\Psi_k (p,t)|^2$ and then take an integral from $-\infty$ to $0$ to get an expression that allows such a computation.

However, I'm not so sure what the term "order-of-magnitude approximation" entails. Would it be right to assume that the computation steps I mentioned above only considers terms of the highest order of magnitude?

Thanks in advance for any assistance.

 

[Stephen Tashi]

My question is in reference to the following paper: https://arxiv.org/pdf/1202.1783.pdf

In equation 3.8, the author computes an order-of-magnitude approximation of probability of measuring negative momentum from the following wavefunction:

$$
\Psi_k =\sum_{k=1,2} \frac{B_k}{\sqrt{4\sigma ^2 + 2it}} \exp (ip_k (x - \frac{p_k}{2}t) - \frac{(x - p_k t)^2}{4\sigma ^2 + 2it})
$$

The article you cite says eq 3.8 is for the particular case

p1= 0.3, p2= 1.4, σ= 10, A1= 1.8, A2= 1. (3.5)

and:

It is important to check that this probability backflow cannot be explained by the tiny probability of having negative momentum which comes from this gaussian state. An order of magnitude estimate will suffice here. We have two gaussian wavepackets centered about different momenta. Consider the wavepacket centered around p = 0.3. The probability that a measurement of the momentum of this state would yield a negative answer is given
approximately by

Prob$( p < 0) = \int_{-\infty}^{0} dp \exp[-200(p-0.3)^2] ~ 10^{-10} \ \ \ (3.8)$

The article integrates a gaussian centered at p1 over negative values of p. Does that make sense as computation for Prob( p < 0) if we ignore the component of the wave that involves p2?

 

[WWCY]

Thanks for your response.

The article integrates a gaussian centered at p1 over negative values of p. Does that make sense as computation for Prob( p < 0) if we ignore the component of the wave that involves p2?

I suppose not, since we are leaving out an entire part of the wavefunction in our computation. However the authors state that this was an order-of-magnitude estimate. Is this a valid line of reasoning?