- #1
boudreaux
- 9
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- 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 Schrodinger equation but can't get anything close to the solution or anything that is usable. How should I solve this?