Quantum Mechanics Operators

Operators in Quantum Mechanics are useful tools to do 
the related mathematics calculations more rapidly. Quantum 
Mechanics is just Quantum Mathematics operating all the time 
on the wave function ψ(r,t).

1. operation
an operation is an action that produces a new value 
from one or more input values.

2. operator
an operator is a symbol or function that represents a 
mathematical operation.

3. Self-adjoint operator
a self-adjoint operator is an operator that is its own adjoint

4. Adjoint operator
adjoint = Conjugate transpose

5. Conjugate Operator
The conjugate  of the complex matrix A, written as A*, is 
obtained by taking the complex conjugate of each entry.

6. Transpose operator
If A is an ij matrix, then the (i,j) element of itf 
transpose AT is the (j,i) element of A.

[AT]ij = [A]ji 

7. Conjugate transpose operator
The conjugate transpose of the complex matrix A, written as A*, is 
obtained by taking the transpose of A and the complex conjugate of 
each entry.

	
8. Hermitian operator
If A = its adjoint, then A is called Hermitian
In quantum mechanics an observable is a Hermitian operator on 
the physical Hilbert space.
	
9. Linear operator
		
A linear map, or linear operator is a function between two 
vector spaces that obeys the operations of vector addition 
and scalar multiplication; that is:
If A and B are vector spaces over the same field C, and a function 
f (or oerator f) : A → B is linear if for any two 
vectors x and y in A and any scalar a in C, we have:
f(x + y) = f(x) + f(y),a nd 
f(α x) = α f(s)

Or, in general manner:f(Σ αn xn) = Σ αn f(xn )

	
10. Observable
Any measurable quantity for which we can calculate the 
expectation value is called a physical observable.
	
11. Observable operators
	
We take the simplest case where the particle is free, that is 
V(r) = 0. Thereore: ψ(x,t) = exp{i(kx - ωt)}

 1. Linear momenum Operator
∂ ψ(x,t) /∂x = ik ψ(x,t) 
With p = ℏ k, we have:
∂ ψ(x,t) /∂x = (ip/ℏ)  ψ(x,t) 
then:
p = - i ℏ) ∂ /∂x 
This is the operator associated to the physical observable 
linear momentum P. 

Linear momenum Operator:
p = - i ℏ ∂ /∂x 

 2. Energy Operator
Now, the time derivative of the wave function is:

∂ ψ(x,t) /∂t = - i ω ψ(x,t) 
With E = ℏ ω, we have:
∂ ψ(x,t) /∂t = (- iE/ℏ)  ψ(x,t) 
then:
E = i ℏ ∂ /∂t 
This is the operator associated to the physical observable 
energy. 


Energy Operator:
E = i ℏ ∂ /∂t 

 3. Time-dependent Schrodinger equation
Using operators, we can verify the time-dependent 
Schrodinger equation:

p = - i ℏ ∂ /∂x 
p2 = - ℏ2 ∂2 /∂x2 
KE = p2/2m = - (ℏ2/2m) ∂2 /∂x2 

If we use the conservation of energy, that is KE + PE = E, 
and take the simplest case where the particle is free (PE = 0), 
we have KE = E, and then : 
- (ℏ2/2m) ∂2 /∂x2  = i ℏ ∂ /∂t . That is the time-dependent 
Schrodinger equation.

 4. General case
 The Schrödinger time-independant equation: 
- (ℏ 2/2m)∂2ψ(r)/∂r2 + V(r) ψ(r) = E ψ(r) 
It gives the total energy operator Hamiltonian E:
  
 E = - (ℏ 2/2m)∂2/∂r2 + V(r)

 The Schrödinger time-dependant equation:
iℏ ∂ψ(r,t)/∂t = - (ℏ 2/2m)∂2ψ(r,t)/∂r2 + V(r) ψ(r,t) 
It gives the Hamiltonian E: 
E = iℏ ∂/∂t   

The kinetic energy operator KE is:
 KE = - (ℏ 2/2m)∂2/∂r2 

If KE = P2 /2m = - (ℏ 2/2m)∂2/∂r2, then:  
P =  - i ℏ∂/∂r 

Remark  ∂/∂r (∂/∂r) = - ∂2/∂r2 when 
acting on ψ(r), because ψ(r) = ψ( ω t - kr) .



12. Commutator
	
By definition, the commutator of two operators 
A and B is : [A,B] = AB - BA

The two operators A and B  commute if [A,B] = 0.

Examples:

1. The two integers 1 and 3 commute because [1,3] = 1x3 - 3x1 = 0.

2. The momentum and position do not.

Proof:
p = - i ℏ ∂ /∂x 
[p, x] = px - xp
px ψ = p (x ψ) =  - i ℏ ∂(x ψ) /∂x = - i ℏ (ψ + x ∂ψ/∂x )
And:
xp ψ = x p(ψ) = x (- i ℏ ∂ ψ/∂x )
Thus:
[p, x] ψ = - i ℏ (ψ + x ∂x (ψ) /∂x ) - x (- i ℏ ∂ (ψ)/∂x ) = 
iℏ (- ψ - x ∂x (ψ) /∂x + x ∂ (ψ)/∂x )  = - i  ℏ ψ

Therefore:  [P,X] = - i ℏ 

A manner to  show that we can not know two physical values 
(here position and momentum) at the same time if they do not 
commute.