Hints and answers

QUANTUM OPERATORS
19.4 Hints and answers
19.1
19.3
19.5
19.7
19.9
Show that the commutator is anti-Hermitian.
Use the Hermitian conjugate of the given equation to obtain the time dependence
of ψ|. The rate of change
of ψ|A|ψ is iψ| [ H, A ] |ψ. Note that [ H, px ] =
[ V , px ] and [ H, x ] = p2x , x /2m.
Show that C2 = 2I.
√ √
sin 2
1 0
1
1
, sin C = √
cos C = cos 2
.
0 1
1 −1
2
Express
the
total Hamiltonian in terms of B = A + gI and determine the value
of B, B † .
Show that, if F (n) is the nth derivative of F(λ), then F (n+1) = A, F (n) . Use a Taylor
series in λ to evaluate F(1), using derivatives evaluated at λ = 0. Successively
reduce the level of nesting of each multiple commutator by using the result of
evaluating the previous term. The given expression reduces to cos θ Ly − sin θ Lz .
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