d
dt(beiω^0 t) =iω 1
2
beiω^0 t =
iω 1
2tIt appears that the amplitude grows linearly with time and hence the probability would grow like
t^2. Actually, once we do the calculation (only a bit) more carefully, we willsee that the probability
increases linearly with time and there is a delta function of energy conservation. We will do this
more generally in the section on time dependent perturbation theory.
In any case, we can only cause transitions if the EM field is tuned so thatω≈ 2 ω 0 which means
the photons in the EM wave have an energy equal to the difference inenergy between the spin
down state and the spin up state. The transition rate increases aswe increase the strength of the
oscillating B field.
18.12Homework Problems
- An angular momentum 1 system is in the stateχ=√^126
1
3
4
. What is the probability thata measurement ofLxyields a value of 0?- A spin^12 particle is in an eigenstate ofSywith eigenvalue + ̄h 2 at timet= 0. At that time it is
placed in a constant magnetic fieldBin thezdirection. The spin is allowed to precess for a
timeT. At that instant, the magnetic field is very quickly switched to thexdirection. After
another time intervalT, a measurement of theycomponent of the spin is made. What is the
probability that the value− ̄h 2 will be found? - Consider a system of spin^12. What are the eigenstates and eigenvalues of the operatorSx+Sy?
Suppose a measurement of this quantity is made, and the system is found to be in the eigenstate
with the larger eigenvalue. What is the probability that a subsequentmeasurement ofSyyields
̄h
2? - The Hamiltonian matrix is given to be
H= ̄hω
8 4 6
4 10 4
6 4 8
.
What are the eigen-energies and corresponding eigenstates of the system? (This isn’t too
messy.)- What are the eigenfunctions and eigenvalues of the operatorLxLy+LyLxfor a spin 1 system?
- Calculate theℓ = 1 operator for arbitrary rotations about the x-axis. Use the usual Lz
eigenstates as a basis. - An electron is in an eigenstate ofSxwith eigenvalue ̄h 2. What are the amplitudes to find the
electron with a)Sz= + 2 ̄h, b)Sz=− ̄h 2 ,Sy= +h ̄ 2 ,Su= + ̄h 2 , where theu-axis is assumed to
be in thex−yplane rotated by and angleθfrom thex-axis.