Download Physics 234: Quantum Mechanics Homework 4. 1. Consider two

Physics 234: Quantum Mechanics
Homework 4.
1. Consider two operators G and A. We define
[G, A] = C1
[G, [G, A]] = C2
[G, [G, [G, A]]] = C3
...
The Baker-Hausdorff formula is
eiGα Ae−iGα
(iα)2
C2 + ...
2!
∞
X
(iα)n
Cn ,
= A+
n!
n=1
= A + iαC1 +
where α is a number. Use this formula to evaluate
exp
iSz φ
h̄
Sx exp
−iSz φ
h̄
2. Spin-1/2 charged particle in constant magnetic field. The Hamilitonian is given by
~ · B,
~
H = −λS
~ = B k̂. λ is a parameter depending on the charge and
where the magnetic field is along the z-direction, B
mass of the particle. At t = 0, the spin is polarized in the x-direction. Compute the expectation value of
Sy as a function of time.
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