Question

# The initial wave function of a free particle is: Ψ(x,0) = A, for |x| = 0,...

The initial wave function of a free particle is:

Ψ(x,0) = A, for |x|

= 0, otherwise

where a and A are positive real numbers. The particle is in a zero (or constant) potential environment since it is a free particle

a) Determine A from normalization.

b) Determine φ(p) = Φ(p,0), the time-zero momentum representation of the particle state. What is Φ(p,t)? Sketch φ(p). Locate the global maximum and the zeros of φ(p). Give the expression for the zeros (i.e., for the location of the zeros).

c) Determine the momentum space probability density |Φ(p, t)|2 and show then that Φ(p, t) is normalized in momentum space. (You can use a table integral.) Sketch |Φ(p,t)|2 and locate the global maximum and the zeros. Give the expression for the zeros.

d) Crudely estimate σx and σp for time zero. Are the results consistent with the Heisenberg uncertainty principle?

e) Is it possible to find the group velocity? If so, what is it?

#### Homework Answers

Answer #1

x-position space

p-momentum space

v^g- group velocity

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