Probability of Staying in x<0 for Superposition of 2 Gaussians

[WWCY]

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:

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!

 

[mfb]

You can still calculate this integral. Just calculate the integral of both separately and add them.

 

[WWCY]

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?

 

[Ray Vickson]

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?

 

[WWCY]

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!

 

[haruspex]

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.

 

[mfb]

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.

 

[WWCY]

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!

 

[WWCY]

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 $\frac{1}{\sqrt{2}} \psi_1 + \frac{1}{\sqrt{2}} \psi_2$?

 

[WWCY]

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 :

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!

 

[Jilang]

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.