15 Suppose that we replace the usual probability rule of quan-
tum mechanics with one of the formP ∝ |〈Ψ|Ψ〉|M, withM >2.
SupposeM= 4. Show, by considering the example in discussion
question B on p. 917, that this leads to nonconservation of proba-
bility.16 In example 12 on p. 984, we defined unnormalized wavefunc-
tions for the traveling-wave solutions to the Schr ̈odinger equation in
a “quantum moat,” and calculated the inner product〈ccw|cw〉= 0
to verify that the counterclockwise and clockwise traveling waves
were orthogonal, as must be the case for distinguishable states. Sup-
pose we want to define a normalized version of the counterclockwise
wave,|ccw〉=Aeiθ. Use an inner product to determine|A|^2 , and
show that normalization doesn’t depend on the phase ofA. (Do not
assume thatAis real.)√17 In example 12 on p. 984, we defined wavefunctions for the
traveling-wave solutions to the Schr ̈odinger equation in a “quantum
moat,” and calculated the inner product〈ccw|cw〉= 0 to verify that
the counterclockwise and clockwise traveling waves were orthogonal,
as must be the case for distinguishable states. Let’s now define
standing-wave versions|c〉= cosθand|s〉= sinθ. Verify by direct
calculation that〈c|s〉= 0.
Remark:Note that, as discussed in the sidebar on p. 966, this does not contradict
the principle that a quantum-mechanical phase is undetectable.18 Consider the wavefunctionsΨ 1 =eikx,
Ψ 2 =e−ikx,
Φ 1 = coskx, and
Φ 2 =isinkx.Show that Ψ 1 = Φ 1 + Φ 2. Similarly, express Ψ 2 in terms of the
Φ’s, and express each of the Φ’s in terms of the Ψ’s. Relate this to
the principle that there is no preferred basis in quantum mechanics
(p. 987).√1012 Chapter 14 Additional Topics in Quantum Physics