QUANTUM OPERATORS
For a particle of massmmoving in a one-dimensional potentialV(x), prove
Ehrenfest’s theorem:
d〈px〉
dt=−
〈
dV
dx〉
andd〈x〉
dt=
〈px〉
m.
19.4 Show that the Pauli matrices
Sx=^12 (
01
10
)
, Sy=^12 (
0 −i
i 0)
, Sz=^12 (
10
0 − 1
)
,
which are used as the operators corresponding to intrinsic spin of^12 in non-
relativistic quantummechanics, satisfyS^2 x=S^2 y=S^2 z=^14 ^2 I, and have the
same commutation properties as the components of orbital angular momentum.
Deduce that any state|ψ〉represented by the column vector(a, b)Tis an eigenstate
ofS^2 with eigenvalue 3^2 /4.
19.5 Find closed-form expressions for cosCand sinC,whereCis the matrix
C=
(
11
1 − 1
)
.
Demonstrate that the ‘expected’ relationshipscos^2 C+sin^2 C=I and sin 2C=2sinCcosC
are valid.
19.6 OperatorsAandBanticommute. Evaluate (A+B)^2 nfor a few values ofnand
hence propose an expression forcnrin the expansion
(A+B)^2 n=∑nr=0cnrA^2 n−^2 rB^2 r.Prove your proposed formula for general values ofn, using the method of
induction.
Show thatcos(A+B)=∑∞
n=0∑nr=0dnrA^2 n−^2 rB^2 r,where thednrare constants whose values you should determine.By taking asAthe matrixA=(
01
10
)
, confirm that your answer is
consistent with that obtained in exercise 19.5.
19.7 Expressed in terms of the annihilation and creation operatorsAandA†discussed
in the text, a system has an unperturbed HamiltonianH 0 =ωA†A. The system
is disturbed by the addition of a perturbing HamiltonianH 1 =gω(A+A†),
wheregis real. Show that the effect of the perturbation is to move the whole
energy spectrum of the system down byg^2 ω.
19.8 For a system ofNelectrons in their ground state| 0 〉, the Hamiltonian is
H=
∑N
n=1p^2 xn+p^2 yn+p^2 zn
2 m+
∑N
n=1V(xn,yn,zn).Show that[
p^2 xn,xn]
=− 2 ipxn, and hence that the expectation value of the
double commutator[[x, H],x],wherex=∑N
n=1xn,isgivenby〈 0 |[[x, H],x]| 0 〉=N^2
m