QUANTUM OPERATORS
19.4 Hints and answers19.1 Show that the commutator is anti-Hermitian.
19.3 Use the Hermitian conjugate of the given equation to obtain the time dependence
of〈ψ|. The rate of change of〈ψ|A|ψ〉isi〈ψ|[H, A]|ψ〉. Note that[H,px]=
[V,px]and[H,x]=
[
p^2 x,x]
/ 2 m.19.5 Show thatC^2 =2I.
cosC=cos√
2
(
10
01
)
, sinC=sin√
2
√
2
(
11
1 − 1
)
.
19.7 Express the total Hamiltonian in terms ofB=A+gIand determine the value
of
[
B, B†
]
.
19.9 Show that, ifF(n)is thenth derivative ofF(λ), thenF(n+1)=
[
A, F(n)]
. Use a Taylor
series inλto evaluateF(1), using derivatives evaluated atλ= 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 cosθLy−sinθLz.