Definition: Linear Harmonic Oscillator Hamiltonian
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Definition: Anihilation Operator
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Anihilation Operator is defined as
Definition: Creation Operator
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Creation Operator is defined as
Theorem: a, a+ compound expressions
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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
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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: