Nuclear Fusion Physics and Technology/Atomic Physics summary

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Definition: Linear Harmonic Oscillator HamiltonianEdit

Linear Harmonic Oscillator Hamiltonian   of a particle  is defined as
 
where operators  

Definition: Anihilation OperatorEdit

Anihilation Operator   is defined as
 

Definition: Creation OperatorEdit

Creation Operator   is defined as
 

Theorem: a, a+ compound expressionsEdit

Lets assume Linear Harmonic Oscillator Hamiltonian. Then

 

Proof:
Directly form a, a+ definitions with respect to LHO Hamiltonian definition and [x,p] commutator

 
 
 
 

Theorem: H,a,a+ commutatorsEdit

Lets assume Linear Harmonic Oscillator Hamiltonian. Then

 

Proof:
The first expression may be evaluated from commutator definition directly from a, a+ compound expression theorem

 

The second and the third expressions needs   first, which can be obtained from a,a+ compound expression theorem again: