130_notes.dvi

m=-1/2

m=+1/2

Ε=μΒ

Ε=−μΒ

B field excites spin

emittted

Pulse of oscillating

state if hν=2μΒ

As excited state

decays back to

ground state,

EM radiation is

As we derived, the Hamiltonian is

H=−~μ·B~=−

gpe

2 mpc

S~·B~=gpe ̄h

4 mpc

~σ·B~=−

gp

2

μN~σ·B~

Note that the gyromagnetic ratio of the proton is about +5.6. The magnetic moment is 2.79μN
(Nuclear Magnetons). Different nuclei will have different gyromagnetic ratios, giving us more tools
to work with. Let’s choose our strong static B field to be in the z direction and the polarization on
our oscillating EM wave so that the B field points in the x direction. The EM wave has (angular)
frequencyω.

H=−

gp

2

μN(Bzσz+Bxcos(ωt)σx) =−

gp

2

μN

(

Bz Bxcosωt

Bxcosωt −Bz

)

Now we apply the time dependent Schr ̈odinger equation.

i ̄h

dχ

dt

= Hχ

i ̄h

(

a ̇

b ̇

)

= −

gp

2

μN

(

Bz Bxcosωt

Bxcosωt −Bz

)(

a

b

)

(

a ̇

b ̇

)

= i

gpμN

2 ̄h

(

Bz Bxcosωt

Bxcosωt −Bz

)(

a

b

)

= i

(

ω 0 ω 1 cosωt

ω 1 cosωt −ω 0

)(

a

b

)

Thesolution (see section 18.11.8)of these equations represents and early example of time
dependent perturbation theory.

d

dt

(beiω^0 t) =

iω 1

2

(ei(ω+2ω^0 )t+e−i(ω−^2 ω^0 t))

Terms that oscillate rapidly will average to zero. The first term oscillates very rapidly. The second
term will only cause significant transitions ifω≈ 2 ω 0. Note that this is exactly the condition that