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: