Traveling quantum Gaussian wave packet

[boudreaux]

Homework Statement

Solve the Schrodinger equation and show the probability density is as follows

Relevant Equations

The Schrodinger equation for a free particle is

$$ih(\partial \Psi/\partial t) = -\frac{\hbar}{2m} \partial^2 \Psi/ \partial x^2$$

Consider an initial state described by the wavefuntion

$$\Psi(x,0) = \sqrt {(\pi_0(x))}exp(iQx)$$

where Q is a constant and $$\pi_0$$ is a normalized gaussian distribution function with zero mean and variance $$(\sigma_0)^2$$

$$\pi_0(x) = (1/\sqrt{(2\pi)\sigma_0})*exp(\frac{-x^2}{ 2(\sigma_0)^2})$$

Solve the schrodinger equation and show that the probability density $$\pi(x,t) = |\Psi(x,t)|^2 $$ at t>0 is given by

$$\pi(x,t) = (1/\sqrt{(2\pi)\sigma(t))}exp( (-\frac{(x-vt)^2) }{ (2\sigma(t)^2) })$$

What are the formulas $$\sigma(t)$$ and v?

Hint: might be easier to use $$\tau = 2m(\sigma_0)^2/\hbar , l = 2Q(\sigma_0)^2 $$

I tried plugging Psi into the right of the Schrödinger equation but can't get anything close to the solution or anything that is usable. How should I solve this?

 

[PeroK]

This level of homework is a bit tough without Latex:

https://www.physicsforums.com/help/latexhelp/

 

[boudreaux]

updated with LaTeX!

 

[DEvens]

First, be careful that you are not trying to put $\Psi(x,0)$ into the Schrödinger equation. That is the $t=0$ boundary condition. You need $\Psi(x,t)$.

Recall that $| \Psi |^2$ means multiply $\Psi$ by its complex conjugate. So that means that the $\exp (i Q x)$ factor does not appear in $\pi (x,t)$. It's just a phase, so when you do the abs-square you get 1.

So maybe $\Psi(x,t)$ can be obtained by the obvious means of putting the $t$ back into the $\Psi(x,0)$ formula?

$$ \Psi(x,t) = \sqrt{\pi(x,t)} \exp(i Q x)$$

This does give you back the correct $\Psi(x,0)$. So now, if you put this in the S.E., and turn the crank, you should get something that involves an equation in $\sigma(t)$. And the $(t)$ part is trying to suggest that it is only a function of time. Meaning the $x$ parts of the S.E. should be identically solved by the form of $\Psi(x,t)$. Check that is true. If it's not, can you guess what has to be added to $\Psi(x,t)$ to make it true? Remember that $\pi(x,t)$ can't change, so the additional stuff has to be just a phase.

When you struggle through that, you should be able to pull out a functional form for $\sigma(t)$ and a relation between $v$ and $Q$.

The hint is telling you that the equation might be simpler if you change the time coordinate to the indicated parameter. It may mean that a bunch of the constants in front of the $\Psi$ are cancelled, meaning your equations are a lot easier to write.

 

[PeroK]

Homework Statement:: Solve the Schrödinger equation and show the probability density is as follows
Relevant Equations:: The Schrödinger equation for a free particle is

$$ih(\partial \Psi/\partial t) = -\frac{\hbar}{2m} \partial^2 \Psi/ \partial x^2$$

Consider an initial state described by the wavefuntion

$$\Psi(x,0) = \sqrt {(\pi_0(x))}exp(iQx)$$

where Q is a constant and $$\pi_0$$ is a normalized gaussian distribution function with zero mean and variance $$(\sigma_0)^2$$

$$\pi_0(x) = (1/\sqrt{(2\pi)\sigma_0})*exp(\frac{-x^2}{ 2(\sigma_0)^2})$$

Solve the Schrödinger equation and show that the probability density $$\pi(x,t) = |\Psi(x,t)|^2 $$ at t>0 is given by

$$\pi(x,t) = (1/\sqrt{(2\pi)\sigma(t))}exp( (-\frac{(x-vt)^2) }{ (2\sigma(t)^2) })$$

What are the formulas $$\sigma(t)$$ and v?

Hint: might be easier to use $$\tau = 2m(\sigma_0)^2/\hbar , l = 2Q(\sigma_0)^2 $$

I tried plugging Psi into the right of the Schrödinger equation but can't get anything close to the solution or anything that is usable. How should I solve this?

How do you solve the Schroedinger equation (SDE) for any potential? What's the general method?

$v$ must be related to $Q$ in some way, which you may have to work out as part of the solution. Similarly, $\sigma(t)$ represents the standard deviation of the Gaussian at time $t$. Note that the solution remains a Gaussian, but its standard deviation changes with time, and this is something else that should come out of your solution.

Note that $\exp{iQx}$ is not a phase factor, as it includes the variable $x$. Do you know or can you guess what this factor represents? Hint: what does $v$ often represent.

Can you intepret the solution? Before you solve it, it might be useful to see whether you can figure out what the solution means.

This problem, I would say, is less than easy! If you are unsure of what you are doing with the SDE, I suggest you find some less complicated examples before returning to this.

If you do press ahead, get your algebra hat on.

 

[PeroK]

$$\pi_0(x) = (1/\sqrt{(2\pi)\sigma_0})*exp(\frac{-x^2}{ 2(\sigma_0)^2})$$

I'd check that expression. I think it should be:$$\pi_0(x) = \frac{1}{\sigma_0\sqrt{2\pi}}\exp(\frac{-x^2}{ 2(\sigma_0)^2})$$