Nuclear Fusion Physics and Technology/Atomic Physics summary

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Definition: Linear Harmonic Oscillator Hamiltonian[edit]

Linear Harmonic Oscillator Hamiltonian of a particle is defined as

where operators

Definition: Anihilation Operator[edit]

Anihilation Operator is defined as

Definition: Creation Operator[edit]

Creation Operator is defined as

Theorem: a, a+ compound expressions[edit]

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]

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: