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 )|−