Chapter 24 Lecture 23 - Temporal Evolution in a Two Level System

24.1 Transitions Between Eigenstates

We consider

H(t)=H₀+V(t) (24.1)

where V(t) changes rapidly, so the time scale for this operator is

The outcomes of this are

in this case we cannot define new instantaneous energy eigenstates/eigenvalues because
D ED t (24.2)
but we can describe the state of the system as a superposition of eigenstates of H₀ (here we are using the Completeness Theorem)

Y (r ,t)=

å

n
c_n(t)U_n(r )e

-

E_n

t

(24.3)
where c_n is the time dependent evolution and e^-^E_ⁿ^/^t is the usual evolution of an eigenstate.
c_n(t) are time varying coefficients
P(U_n)=|c_n(t)|² (24.4)
this produces time dependent occupation probabilities. Therefore there are transfers of probability (transitions) between the eigenstates of the system exposed to a rapidly varying interaction.

24.2 Two Level Systems

The resonance condition is

w₀=E₂-E₁ (24.5)

In practice many systems simplify to two levels. eg. the spin of a half by particles in a magnetic field, or atoms in resonant laser fields.

For a two level system equation V.14 reduces to

Y (r,t)=c₁(t)U₁(r

E₁

+c₂(t)U₂(r

E₂

(24.6)

The time development is given by the Schrödinger equation

¶ t

Y (r,t)=(H₀+V(t))Y (r,t)

This gives

é
ê
ê
ë

E₁c₁(t)+

¶ t

c₁(t)

ù
ú
ú
û

U₁(r

E₁

é
ê
ê
ë

E₂c₂(t)+

¶ t

c₂(t)

ù
ú
ú
û

U₂(r

E₂

=c₁(t)

[

E₁+V(t)

]

U₁(r

E₁

+c₂(t)

[

E₂+V(t)

]

U₂(r

E₂

(24.7)

We recall the orthonormality condition

ó
õ

all space

U_m^*U_ndV=d_mn

ó
õ

all space

U_n^*dV (24.8)

this can be used to sieve out the terms of U_n that are m¹ n where

U_n^*=U_n^*(r

E_n

(24.9)

We now operate on equation V.18 with ò_all spaceU₁^*dV and ò_all spaceU₂^*dV and obtain

c₁(t)=c₁(t)

ó
õ

all space

U₁^*V

(t)U₁dV+c₂(t)

ó
õ

all space

U₁^*V(t)U₂dV

c₂(t)=c₂(t)

ó
õ

all space

U₂^*V

(t)U₂dV+c₁(t)

ó
õ

all space

U₂^*V(t)U₁dV (24.10)

An object like

ó
õ

all space

U_n^*VU_mdV (24.11)

is called a matrix element. Dirac invented a succinct notation for this.

<n|V|m> (24.12)

Equation V.21 is a master equation that discribes the evolution of c₁(t) and c₂(t)

24.3 Resonant Interaction

To solve equation V.21 we will assume the following

the boundary conditions are
c₁(0) and c₂(0)=0 (24.13)
<1|V|1>=<2|V|2>=0 (24.14)
from simmetry if we have an electric dipole interaction
the interaction is oscillatory
V(t)=Ucos(w₀t) (24.15)
and resonant

w₀=

E₂-E₁

the solutions become

c₁(t)=cos

æ
ç
ç
è

<2|U|1>

ö
÷
÷
ø

c₂(t)=-sin

æ
ç
ç
è

<1|U|2>

ö
÷
÷
ø

(24.16)

therefore the amplitudes are the probabilities which are proportional to

cos²

é
ê
ê
ë

<2|U|1>

ù
ú
ú
û

(24.17)

Figure 23.1 - The Nutation or Rabi Plopping

If the interaction is weak ( P₂<<1 ) and we have a frequency spread in Dw for V(t) we obtain a slightly different form of the solution

P(2)=

g(E)|<2|U|1>|²t (24.18)

this is known as the Fermi-Golden rule.