Nuclear Fusion Physics and Technology/Atomic Physics summary

Definition: Linear Harmonic Oscillator Hamiltonian

Linear Harmonic Oscillator Hamiltonian ${\hat{H}}_{LHO}:𝕍\to 𝕍$ of a particle $p\in 𝔓𝔞𝔯𝔱𝔦𝔠𝔩𝔢𝔖𝔢𝔱$is defined as
$\hat{H}(|n>)=\frac{{\hat{p}}^2(|n>)}{2m}+\frac{1}{2}m\omega {\hat{x}}^2(|n>)$
where operators $\hat{x}(|n>),\hat{p}(|n>):𝕍\to 𝕍$

Definition: Anihilation Operator

Anihilation Operator $\hat{a}:𝕍\to 𝕍$ is defined as
$\hat{a}(|n>)=\sqrt{\frac{m\omega }{2\hslash }}\hat{x}+i\sqrt{\frac{1}{2\hslash m\omega }}\hat{p}$

Definition: Creation Operator

Creation Operator ${\hat{a}}^{+}:𝕍\to 𝕍$ is defined as
${\hat{a}}^{+}(|n>)=\sqrt{\frac{m\omega }{2\hslash }}\hat{x}-i\sqrt{\frac{1}{2\hslash m\omega }}\hat{p}$

Theorem: a, a+ compound expressions

Lets assume Linear Harmonic Oscillator Hamiltonian. Then

$\hat{a}[{\hat{a}}^{+}(|n>)]=\frac{\hat{H}(|n>)}{\hslash \omega }+\frac{\hat{1}(|n>)}{2},{\hat{a}}^{+}[\hat{a}(|n>)]=\frac{\hat{H}(|n>)}{\hslash \omega }-\frac{\hat{1}(|n>)}{2},$

Proof:
Directly form a, a+ definitions with respect to LHO Hamiltonian definition and [x,p] commutator

$a[a^{+}(|n>)]\stackrel{defa,a^{+}}{=}(\sqrt{\frac{m\omega }{2\hslash }}\hat{x}+i\sqrt{\frac{1}{2\hslash m\omega }}\hat{p})(\sqrt{\frac{m\omega }{2\hslash }}\hat{x}-i\sqrt{\frac{1}{2\hslash m\omega }}\hat{p})=$
$=\frac{m\omega }{2\hslash }{\hat{x}}^2(|n>)-i^2\frac{1}{2\hslash m\omega }{\hat{p}}^2(|n>)-i\sqrt{\frac{m\omega }{2\hslash }}\sqrt{\frac{1}{2\hslash m\omega }}\hat{x}[\hat{p}(|n>)]+i\sqrt{\frac{m\omega }{2\hslash }}\sqrt{\frac{1}{2\hslash m\omega }}\hat{p}[\hat{x}(|n>)]=$
$=|\hat{H}=\frac{{\hat{p}}^2|n>}{m}+\frac{1}{2}m\omega {\hat{x}}^2|n>|=\frac{\hat{H}}{2\hslash \omega }-i\sqrt{\frac{m\omega }{2\hslash }}\sqrt{\frac{1}{2\hslash m\omega }}(\hat{x}\hat{p}|n>-\hat{p}\hat{x}|n>)=$
$=\frac{\hat{H}}{2\hslash \omega }-\frac{i}{2\hslash }[\hat{x},\hat{p}]|n>=\frac{\hat{H}}{2\hslash \omega }-\frac{i}{2\hslash }(i\hslash \hat{1})|n>=\frac{\hat{H}|n>}{2\hslash \omega }+\frac{\hat{1}|n>}{2}$

Theorem: H,a,a+ commutators

Lets assume Linear Harmonic Oscillator Hamiltonian. Then

$[\hat{a},{\hat{a}}^{+}](|n>)=\hat{1}(|n>),[\hat{H},{\hat{a}}^{+}](|n>)=\hslash \omega {\hat{a}}^{+}(|n>)[\hat{H},\hat{a}](|n>)=-\hslash \omega \hat{a}(|n>)$

Proof:
The first expression may be evaluated from commutator definition directly from a, a+ compound expression theorem

$[\hat{a},{\hat{a}}^{+}]=\hat{a}{\hat{a}}^{+}-{\hat{a}}^{+}\hat{a}\stackrel{theorem}{=}\frac{\hat{H}|n>}{2\hslash \omega }+\frac{\hat{1}|n>}{2}-\frac{\hat{H}|n>}{2\hslash \omega }-\frac{\hat{1}|n>}{2}=\hat{1}|n>$

The second and the third expressions needs $\hat{H}=\hat{H}(\hat{a},{\hat{a}}^{+})$ first, which can be obtained from a,a+ compound expression theorem again: Template:BookCat