Physics 471 Problem Set 2 Fall 2004
6. (Griffiths 1.4) At time t= 0 a particle is represented by the wave function
Ψ(x, 0) =
Ax/a, if 0 ≤x≤a
A(b−x)/(b−a),if a≤x≤b
0,otherwise
where A, a and bare constants.
(a) Normalize Ψ(x, 0) (that is, find Ain terms of aand 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=aand b= 2a.
(e) What is the expectation value of x?
7. (Griffiths 1.5) Consider the wave function
Ψ(x, t) = Ae−λ|x|e−iωt ,
where A,λand ωare positive constants.
(a) Normalize Ψ.
(b) Determine the expectation values of xand x2.
(c) Find the standard deviation of x. Sketch the graph of |Ψ|2as a function of xand
mark the points (< x > +σ) and (< x > −σ) to illustrate the spread in x. What is
the probability that the particle would be found outside this range?
8. (Griffiths 1.9) A particle of mass mis in the state
Ψ(x, t) = Ae−a[(mx2/¯h)+it],
where Aand aare positive constants.
(a) Find A.
(b) (Omit this part.)
(c) Calculate the expectation values of x,x2,pand p2.
(d) Find σxand σp. Is their product consistent with the uncertainty principle?
9. Determine hxiand hpifor the state
ψ(x) = 1
(a2π)1/4e−x2/2a2+ikx ,
where aand kare real.
10. (Griffiths 2.4) Calculate < x >,< x2>,< p >,< p2>,σxand σp, for the nth stationary
state of the infinite square well. Check that the uncertainty principle is satisfied. Which
state comes closest to the uncertainty limit?
