QuantumPhysics.dvi

The time-dependent Schr ̈odinger equation for|ψ(t)〉then reduces to the following set of equations
fora±,

ia ̇+(t) =

ω 0

2

a+(t) +

ω 1

2

e−iωta−(t)

ia ̇−(t) =

ω 1

2

e+iωta+(t)−

ω 0

2

a−(t) (13.6)

In a comoving frame of states, defined by

b+(t) = a+(t)e+iωt/^2

b−(t) = a−(t)e−iωt/^2 (13.7)

the equations simplify in that they no longer involve any explicit time dependence,

i ̇b+(t) =

ω 0 −ω

2

b+(t) +

ω 1

2

b−(t)

i ̇b−(t) =

ω 1

2

b+(t)−

ω 0 −ω

2

b−(t) (13.8)

The time evolution of theb±variables is thus effectively given by a reduced Hamiltonian,

H ̃= ̄h

2

(

ω 0 −ω ω 1

ω 1 −ω 0 +ω

)

(13.9)

The eigenvalues ofH ̃ are± ̄hΩ with

Ω =

√

(ω 0 −ω)^2 +ω 12 (13.10)

Thus, we get the following general solution,
(
b+(t)
b−(t)

)

=γ+

(

ω 1

Ω +ω−ω 0

)

e−iΩt/^2 +γ−

(

ω 1

−Ω +ω−ω 0

)

e+iΩt/^2 (13.11)

whereγ±are constants. This gives a complete solution of the problem.

We now introduce also physical initial conditions and a physical problem : suppose the system
at timet= 0 is in the state|+〉, what is the probability of finding the system in the state|−〉after
timet? To solve this problem, we first enforce the initial condition on the general solution. This
means that at timet= 0, we havea+(0) =b+(0) = 1 anda−(0) =b−(0) = 0, so that we must have

γ±=

1

2 ω 1

(

1 ±

ω 0 −ω

Ω

)

(13.12)

The desired probability is then given by

P+−(t) =

ω^21

(ω−ω 0 )^2 +ω^21

sin^2

(√

(ω−ω 0 )^2 +ω^21

t

2

)

(13.13)

This is the Rabi formula for magnetic resonance. The resonance effect occurs whenω∼ω 0 , in
which case the probability for the spin flip is maximal. In fact, whenω=ω 0 , then this formula
tells us that after a timet=π/ω 1 , the spin is flipped with probability 1.