Spreading of a Free-Particle Gaussian Wave Packet

[derelictee]

Homework Statement

It can be shown that the position probability density for the one-dimensional, free-particle Gaussian wave packet can be expressed as shown below, where m is the mass of the particle and L is the position uncertainty at time t=0, vgr = hbar *k0/m, k0 is the average wave number as discussed in class. Let x, t and P(x,t) be measured, respectively, in units of L, T=2mL^2/hbar, A=1/sqrt(2pi*L^2), then the position probability can be written as P(x,t)= sqrt(1/(1+t^2)) exp(-(x-vt)^2/(2 (1+t^2)), where v= 2 mLvgr/hbar. In order to demonstrate the spreading of the wave packet, use Mathematica to calculate <x> in units of L at time t./>

Homework Equations

<x> = integral from -infinity to infinity of x times psi squared

The Attempt at a Solution

I am both a mathematica and quantum mechanics novice, so I'm not sure where my mistake lies. Since psi[x,t]^2 = P[x,t], I integrated P(x,t)= sqrt(1/(1+t^2)) exp(-(x-vt)^2/(2 (1+t^2)) from -infinity to infinity with respect to x and got the answer Sqrt[2Pi]vt. That's not one of the answer choices. The answer choices are: 0, vt, sqrt(1+t^2 (1+v^2)), or sqrt(1+t^2 ). Where did I go wrong? [Note: I don't really understand all of that "in units of L" bit in the question.] Thanks in advance.

 

[fzero]

$$P(x,t)$$ is not properly normalized.