Quantum Mechanics for Mathematicians

3.3 Dynamics of a two-state system

Recall that the time dependence of states in quantum mechanics is given by the
Schr ̈odinger equation
d
dt

|ψ(t)〉=−iH|ψ(t)〉

whereHis a particular self-adjoint linear operator onH, the Hamiltonian op-
erator. Considering the case ofH time-independent, the most general such
operatorHonC^2 will be given by

H=h 01 +h 1 σ 1 +h 2 σ 2 +h 3 σ 3

for four real parametersh 0 ,h 1 ,h 2 ,h 3. The solution to the Schr ̈odinger equation
is then given by exponentiation:

|ψ(t)〉=U(t)|ψ(0)〉

where
U(t) =e−itH

Theh 01 term inHcontributes an overall phase factore−ih^0 t, with the remaining
factor ofU(t) an element of the groupSU(2) rather than the larger groupU(2)
of all 2 by 2 unitaries.
Using our equation 3.3, valid for a unit vectorv, ourU(t) is given by taking
h= (h 1 ,h 2 ,h 3 ),v=|hh|andθ=−t|h|, so we find

U(t) =e−ih^0 t

(

cos(−t|h|) 1 +isin(−t|h|)

h 1 σ 1 +h 2 σ 2 +h 3 σ 3

|h|

)

=e−ih^0 t

(

cos(t|h|) 1 −isin(t|h|)

h 1 σ 1 +h 2 σ 2 +h 3 σ 3

|h|

)

=e−ih^0 t

(

cos(t|h|)−i|hh^3 |sin(t|h|) −isin(t|h|)h^1 |−hih|^2

−isin(t|h|)h^1 +|hih|^2 cos(t|h|) +ih|h^3 |sin(t|h|)

)

In the special caseh= (0, 0 ,h 3 ) we have

U(t) =

(

e−it(h^0 +h^3 ) 0

0 e−it(h^0 −h^3 )

)

so if our initial state is

|ψ(0)〉=α|+

1

2

〉+β|−

1

2

〉

forα,β∈C, at later times the state will be

|ψ(t)〉=αe−it(h^0 +h^3 )|+

1

2

〉+βe−it(h^0 −h^3 )|−

1

2

〉