Homework #02

Problem #1

Consider the Gaussian Distribution ρ(x)= Ae^(-λ(x-a)^2), where A, a, and λ are positive real constants (Look up any integral needed)

[A]  Use equation 1= ∫ ρ(x) dx (limits on integral are negative infinity to positive infinity) to determine A.

[B]  Find , , and σ.

[C]  Sketch the graph of ρ(x).


Problem #2

At time t=0 a particle is represented by the wave function…
Ψ (x,0) = {A(x/a), if 0≤x≤a,
  {A((b-x)/(b-a)), if a≤x≤b,
  {0,  otherwise
Where A, a, and b are constants.

[A]  Normalize Ψ (that is, find A, in terms of “a” and “b”)

[B]  Sketch Ψ (x,0) as a function of “x”.

[C]  Where is the particle most likely to be found at t=0?

[D]  What is the probability of finding the particle to the left of “a”?  Check your result in the limiting cases of b=a and b=2a.

[E]  What is the expectation value of “x”?