QM: Probability

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Homework Statement


I am supposed to find probability of staying in x < 0 for a superposition of two Gaussians. The wavefunction is something along the lines of:

Screen Shot 2017-11-05 at 5.43.08 PM.png


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The Attempt at a Solution


Usually, the step involved in finding probabilities for 1 particle is just to perform the integral of ##|\Psi|^2## between 2 points (##-\infty## to 0 i believe) . However, I believe this wavefunction is the sum of 2 separately normalized gaussians. I'm not sure how I should proceed.

Advice is greatly appreciated!
 

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  • #2
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You can still calculate this integral. Just calculate the integral of both separately and add them.
 
  • #3
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You can still calculate this integral. Just calculate the integral of both separately and add them.
Thanks for the response!

But if I choose to do both integrals from -infinity to infinity, and sum them, does that not give me a probability of 2?
 
  • #4
Ray Vickson
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Homework Statement


I am supposed to find probability of staying in x < 0 for a superposition of two Gaussians. The wavefunction is something along the lines of:

View attachment 214386

Homework Equations




The Attempt at a Solution


Usually, the step involved in finding probabilities for 1 particle is just to perform the integral of ##|\Psi|^2## between 2 points (##-\infty## to 0 i believe) . However, I believe this wavefunction is the sum of 2 separately normalized gaussians. I'm not sure how I should proceed.

Advice is greatly appreciated!
The probability that ##x < 0## at time ##t## is ##\Pr(x < 0|t) = P(t) = \int_{-\infty}^0 |\psi(x,t)|^2 \, dx##, so you need to write out the four terms of ##|\psi|^2 = \psi^* \, \psi## and then integrate them separately. It will be complicated and unpleasant.

However, I am not quite sure what the statement "staying in ##x < 0##" means. Interpreted literally, it is asking for the probability that ##x<0## for all ##t##, and this is very different from asking that ##x < 0## for any single value of ##t##. Is that really what is wanted?
 
  • #5
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The probability that ##x < 0## at time ##t## is ##\Pr(x < 0|t) = P(t) = \int_{-\infty}^0 |\psi(x,t)|^2 \, dx##, so you need to write out the four terms of ##|\psi|^2 = \psi^* \, \psi## and then integrate them separately. It will be complicated and unpleasant.
Will this not give me something unphysical? In the problem, these Gaussians were normalized separately, with A being their normalization constants, if I chose to do the integral from ##-\infty## to ##\infty##, would it not give me something > 1?

However, I am not quite sure what the statement "staying in ##x < 0##" means. Interpreted literally, it is asking for the probability that ##x<0## for all ##t##, and this is very different from asking that ##x < 0## for any single value of ##t##. Is that really what is wanted?
Yep, I was asked to investigate how the probability for the region ##-\infty## to 0 evolved with time, so I do need an expression for all ##t##. To provide some context, all of this is part of a project about quantum backflow, and I'm tasked to do the "simpler" calculations like the ones above.

Thanks for the assistance!
 
  • #6
haruspex
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staying in x<0" means. Interpreted literally, it is asking for the probability that x<0 for all t
Quite. I wonder if it was intended to ask for the probability at t=∞.
these Gaussians were normalized separately
Then you do not have enough information. You need to know the proportions in which to combine them before renormalising.
 
  • #7
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If the given wavefunction is properly normalized, then the normalization constants take care that the total integral over the squared wave function is 1. This means ##|A_1|^2 + |A_2|^2 = 1##. An integral over a smaller x range will then give a value smaller than 1.

If the question asks for ##t \to \infty##, you'll need a different approach. Hint: Look at the momentum distribution.
 
  • #8
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Hi all, thanks for the responses, I think it's best for me to upload the paper I'm referring to.

https://arxiv.org/pdf/1301.4893.pdf

All of the stuff that I'm referring to is on pages 4 and 5 of the document. In particular, I'm looking to recover analytic expressions for Figures 1. (Probability Current) and 2. (Probability from ##-\infty<x<0##).

For probability current, I'm assuming that it's just a case of plugging the wavefunction into the Current equation.

I'd really appreciate it if someone could look through it and advise me on the interpretation!
 
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  • #9
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Then you do not have enough information. You need to know the proportions in which to combine them before renormalising.
Do you mean something like ##\frac{1}{\sqrt{2}} \psi_1 + \frac{1}{\sqrt{2}} \psi_2##?
 
  • #10
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I initially made a post in the HW section about this topic but I believe that my question was poorly formulated, hence I'm attempting to ask a better one here, since it isn't exactly HW.

I am studying the probability current, and probability of a sum of 2 gaussians. The type of wavefunction is as shown :
Screen Shot 2017-11-05 at 5.43.08 PM.png

where ##\hbar = m = 1##

The ultimate goals are to:
a) Obtain an analytic expression for Probability Current at x = 0
b) Obtain an analytic expression for probability from ##-\infty < x < 0## for all ##t## - to study how the probability of the two wavepackets remaining at ##x<0## changes with time.
c) To obtain plots similar to Figures 1 (Probability Current) and 2 (Probability) in this paper (pages 4 and 5 respectively): https://arxiv.org/pdf/1301.4893.pdf

What I intend to do:

I was given an initial Gaussian of: ##\Psi(x,0) = e^{ipx}e^{\frac{-x^2}{2\sigma ^2}}##.

Normalizing, finding ##\widetilde{\psi}## and then ##\Psi(x,t)## for 1 Gaussian gives,
$$\Psi_k = \frac{\sqrt{\sigma}}{\pi^{1/4} \sqrt{\sigma ^2 + it}}exp(ip_k (x - \frac{p_k}{2}t) + \frac{(x - p_k t)^2}{2(\sigma ^2 + it)}) $$
then summing over 2 Gaussians will give,
$$\Psi_k =\sum_{k=1,2} \frac{\sqrt{\sigma}}{\pi^{1/4} \sqrt{\sigma ^2 + it}}exp(ip_k (x - \frac{p_k}{2}t) + \frac{(x - p_k t)^2}{2(\sigma ^2 + it)}) $$
which is not dissimilar to the one above, except for the fact that I should have started with ##\Psi = e^{ipx}e^{\frac{-x^2}{4\sigma ^2}}##, I believe.

For a: I intend to place the Sum of Gaussians into the Probability Current equation, and chug through the algebra to derive the expression, evaluating the expression at x = 0.

For b: This is the part where I am not too sure about what to do. This is a sum of separately normalized wavefunctions, simply performing ##\int_{-\infty}^{0}|\Psi(x,t)|^2 dx## does not seem the right thing to do as I believe such functions don't seem to be orthogonal, and more fundamentally ##\int_{-\infty}^{\infty}|\Psi(x,t)|^2 dx## might ##>1## (?).

In my earlier post, some commenters seemed to question my interpretation of a) and b). I too am slightly unsure and I hope someone can take the time to skim through the noted sections of the paper and provide me with some advice, or some notes I should go through before attempting to solve the problem.

Any assistance is greatly appreciated!
 

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  • #11
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Why do you feel each Gaussian must have equal weighting? The coefficients for each term could quite general as long as the whole this is normalised.
 

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