Observales and time
Time evolution pictures

In one dimension, the state of a quantum system is 
represented by:

ψ(x,t) = ψ(x) ψ(t)

ψ(x) is the spatial part of the wave function, called 
the stationary state, and ψ(t) is the temporal part.

ψ(x,t)  is  a solution of the Schrodinger 
time-dependent equation:

i ℏ ∂ ψ(x,t) /∂t = H ψ(x,t) 

In quantum mechanics, the time evolution of operators 
and state vectors can be expressed in three different representations. 
The Schrodinger, the Interaction, and the Heisenberg representations. 

The Hamiltonian of a perturbed system is expressed in two parts as:
H = H0 + Hint
Where:
H0 is the exactly solvable part without any interactions, and
Hint that contains all the interactions.

1 Schrodinger Picture

In the Schrodinger picture the operators are 
time-independent, but the states evolve

At the initial time or reference time t0 or 
later at the time "t", the operator remains the same.

As(t) = As(t0) = As

But the states of the system evolve. The time dependence 
of the state vector |ψs(t)> is obtained from the 
Schrodinger equation iℏ∂/∂t (|ψs(t)>) = H|ψs(t)> 
which has the solution:

|ψs(t)> = exp{- i H(t - t0)/ℏ}ψs(0)>


Schrodinger Picture
As(t) = As(t0) = As
|ψs(t)> = exp{- i H(t - t0)/ℏ}ψs(0)>

2. Interaction Picture

  
In the interaction representation both the state 
vectors and the operators are time-dependent. 

In this representation, the state vector is written as:
|ψi(t)> = exp{+ i H0 t/ℏ} |ψs(t)>

Its related Schrodinger equation is:
iℏ∂/∂t (|ψi(t)>) = - H0exp{+ i H0 t/ℏ} |ψs(t)>
+ exp{+ i H0 t/ℏ} i ℏ  &part/∂ [ψs(t)>]

= exp{+ i H0 t/ℏ}[ - H0 + H]ψs(t)> 
= exp{+ i H0 t/ℏ}  [ - H0
+ H0 + Hint]  exp{- i H0 t/ℏ}ψi(t)> 

We have then:
iℏ∂/∂t (|ψi(t)>) =  Hint(t) ψi(t)>
With:
Hint(t) = exp{+ i H0 t/ℏ} Hintexp{- i H0 t/ℏ

In the Schrodinger picture, we can write for 
any operator: 
<ψ's(t)>|As| ψs(t)> =  <ψ's(t)>|As| ψs(t)> =
<ψ'i(t)>|exp{+ i H0 t/ℏ}As| exp{- i H0 t/ℏ}ψi(t)>

Hence:
Ai(t) = exp{+ i H0 t/ℏ}Asexp{- i H0 t/ℏ}

Interaction picture:
|ψi(t)> = exp{+ i H0 t/ℏ} |ψs(t)>
Ai(t) = exp{+ i H0 t/ℏ}Asexp{- i H0 t/ℏ}

3. Heisenberg Picture

  
In the Heisenberg  picture the states are time-independent, 
but the operators evolve.

If the ket of the Schrodinger evolves, it is writen in the 
Heisenberg picture in the manner that it is time-independent:

|ψh(t)> = exp{+ i H t/ℏ} |ψs(t)> = exp{+ i H t0/ℏ} |ψs(0)>

|ψh(t)> is then time-independent.

If A is an operator that It doesn't evolve in time in 
the Schrodinger picture, then, in the Heisenberg picture:

<ψ's(t)|As|ψs(t)> = 
<ψ'h(t)| exp{+ i H t/ℏ}Asexp{- i H t/ℏ}|ψh(t)>

And we write:

Ah(t) = exp{+ i H t/ℏ} As exp{- i H t/ℏ}

This expression can be expressed in terms of an operator 
in the interaction picture:

Ah(t) = exp{+ i H t/ℏ}exp{- i H0 t/ℏ} Ai(t) exp{+ i H0 t/ℏ}exp{- i H t/ℏ}

We have also:
Ah(t) = U(0,t) Ai(t) U(t,0) Where U(t,t0)  is the unitary operator that 
determines the evolution of the state vector in the interaction 
picture.



Heisenberg picture:
|ψh(t)> = exp{+ i H t/ℏ} |ψs(t)> = 
exp{+ i H t0/ℏ} |ψs(0)>

Is constant since: ∂ |ψh(t)>/∂ = 0

Ah(t) = exp{+ i H t/ℏ}exp{- i H0 t/ℏ} Ai(t) 
exp{+ i H0 t/ℏ}exp{- i H t/ℏ} 
or:
Ah(t) = Ah(t) = U(0,t) Ai(t) U(t,0) 
with: 
U(t,t0) = exp {- (i/ℏ) ∫ Hint(τ) dτ}
from  t0 to t.

4. The Evolution Operator U(t,t₀)

  
In the interaction picture:

|ψi(t) = U(t,t0)|ψi(t0)
The initial condition is: U(t0,t0) = 1

In the Schrodinger picture:
|ψi(t) = exp{+ i H0 t/ℏ}}|ψs(t) = 
exp{+ i H0 t/ℏ} exp{- i H (t - t0)/ℏ}|ψi(t0) = 
exp{+ i H0 t/ℏ} exp{- i H (t - t0)/ℏ}|ψi(t0) exp{- i H0 t0/ℏ} |ψi(t0)

Therefore:
U(t,t0) = exp{+ i H0 t/ℏ} exp{- i H (t - t0)/ℏ}|ψi(t0) exp{- i H0 t0/ℏ}

We have from the interaction picture:
iℏ∂/∂t (|ψi(t)>) 
=  Hint(t) ψi(t)>
With:
Hint(t) = exp{+ i H0 t/ℏ} Hint exp{- i H0 t/ℏ
Then, from  |ψi(t) = U(t,t0)|ψi(t0), we have:
iℏ∂/∂t (|ψi(t)>) = 
iℏ∂/∂t [U(t,t0)] |ψi(0)> = Hint(t) |ψi(t)> = Hint(t) U(t,t0)ψi(0)>

Then:
iℏ∂/∂t [U(t,t0)] = Hint(t) U(t,t0)

Integrating gives:

U(t,t0) = U(t0,t0) exp {- (i/ℏ) ∫ Hint(τ) dτ}

With the initial condition:
Ui(t,t0) = exp {- (i/ℏ) ∫ Hint(τ) dτ}
from t0 to t.

5. Time evolution of operators in
an unperturbed system

In the Heisenberg representation, the state vectors does 
not evolve in time, but operators do.

H = H0

ψ(t) has the following wxpression:
ψ(t) = U(t) ψ(0), where U(t) is the time evolution 
operator. U(t) = exp{-i Ht/ℏ}, 
where H is the Hamiltonian of the system.

An observable of the system A has the following 
expectation:

<|A|>t = <ψ(t)|A|ψ(t)> = <U*(t) ψ*(0)|A|U(t) ψ(0)> =
<exp{+i Ht/ℏ} ψ*(0)|A|exp{-i Ht/ℏ} ψ(0)> = 
<ψ*(0) exp{+i Ht/ℏ} |A|exp{-i Ht/ℏ} ψ(0)> 

We define:
A(t) = exp{+i Ht/ℏ} |A|exp{-i Ht/ℏ} 

So
<|A|>t =  <ψ*(0) | A(t)| ψ(0)> 

Deriving gives:
dA(t)/dt = i H/ℏ exp{-i Ht/ℏ}|A|exp{-i Ht/ℏ + 
exp{+i Ht/ℏ} (∂A/∂)exp{-i Ht/ℏ} 
- i H/ℏ exp{+i Ht/ℏ} |A|exp{-i Ht/ℏ} =
i /ℏ exp{-i Ht/ℏ}(HA - AH) exp{-i Ht/ℏ + 
exp{+i Ht/ℏ} (∂A/∂)exp{-i Ht/ℏ} 
= i /ℏ (HA(t) - A(t)H)  + exp{+i Ht/ℏ} (∂A/∂)exp{-i Ht/ℏ} 
= (i/ℏ) [H,A(t)]  + (∂A/∂t)

dA(t)/dt = (i/ℏ) [H,A(t)]  + (∂A/∂t)