Definition: Linear Harmonic Oscillator Hamiltonian[edit | edit source]
Linear Harmonic Oscillator Hamiltonian of a particle is defined as
where operators
Definition: Anihilation Operator[edit | edit source]
Anihilation Operator is defined as
Definition: Creation Operator[edit | edit source]
Creation Operator is defined as
Theorem: a, a+ compound expressions[edit | edit source]
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+ commutators[edit | edit source]
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: