## Contents

#### 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: