PHY 851, HW 5 due Friday 10/10
1. (a) Prove the BakerCampbellHausdorﬀ (BCH) formula where
A
and
B
arelinear operators
e
A
Be
−
A
=
B
+ [
A,B
] + 12! [
A,
[
A,B
]] + 13! [
A,
[
A,
[
A,B
]]] +
···
(1)Hint: Consider the quantity
F
(
λ
) =
e
λA
Be
−
λA
and expand it as a Taylorseries in the parameter
λ
.2. In class, we used the BCH formula to evaluate the Heisenberg picture spinoperators given the Hamiltonian
H
=
ωS
z
. Another method, which is oftentimes more straightforward, is to evaluate the commutators in the HeisenbergEquations of Motion (EOM) and solve the resulting diﬀerential equations.
ddtS
i
(
t
) = 1
i
[
S
i
(
t
)
,H
] = 1
i
[
S
i
(
t
)
,ωS
z
]
.
(2)Do this for
i
=
x,y
and obtain a set of 2 coupled ﬁrstorder diﬀerentialequations. Uncouple them and integrate by forming appropriate linear combinations. Verify that your results for
S
x,y
(
t
) agree with the results obtainedin class (or in the book).3. (a) Solve the Schr¨odinger equation for the timeevolution operator in thecase of a timedependent Hamiltonian
i
ddt
U
(
t,
0) =
H
(
t
)
U
(
t,
0)
,
(3)where [
H
(
t
)
,H
(
t
)] = 0. Hint: Divide the ﬁnite interval into inﬁnitesimalslices
δt
=
tN
where
N
→ ∞
. With the notation
t
n
=
nδt
, use the group composition property to write
U
(
t,
0) =
U
(
t
N
,t
N
−
1
)
···U
(
t
n
,t
n
−
1
)
···U
(
t
1
,
0).Note the Hamiltonian is approximately timeindependent on each inﬁnitesimal slice, so that the “usual” expression
U
(
t
n
,t
n
−
1
)
≈
exp[
−
(
i/
)
δtH
(
t
n
)]can be used.1
(b) Now consider a spin1/2 particle (say, an electron) in a spatially uniform but timedependent magnetic ﬁeld,
B
=
B
(
t
)
z
. Solve the Heisenbergequations of motion to ﬁnd the timedependent spin operators
S
x
(
t
) and
S
y
(
t
).4. In quantum information theory, the work of a quantum computer consistsof transformations through mutually orthogonal quantum states. Then, theminimal time necessary for such a transformation determines the processingrate of the computer. It is clear that a single stationary energy eigenstate

ψ
n
(
t
)
will never become orthogonal to

ψ
n
(0)
since they diﬀer only by thephase exp[
−
i/
E
n
t
].(a) Take the initial state as a superposition of two normalized energy eigenkets with energies
E
0
= 0 and
E
1
>
0. Find the initial superpositionthat will evolve into an orthogonal state where
C
(
t
)
≡
ψ
(0)

ψ
(
t
)
= 0,and determine the minimal time
τ
required to do that. Express
τ
interms of fundamental constants and the expectation value of
H
in theinitial state.(b) Take the initial state as a superposition with equal weights of threenormalized energy eigenkets with energies
E
0
= 0,
E
2
> E
1
>
0. Findthe sets of energies which will guarantee
C
(
t
) = 0 and determine theminimal time
τ
to do that.5. Sakurai 2.146. Sakurai 2.192