harmonic-oscillator-ladder-operators

Ladder operators are operators that raise or lower the eigenvalues of an observable. They are used to transform the Harmonic oscillator state wave function into a lower or higher energy state

• $a_{+}$ = raising operator
• $a_{-}$ = lowering operator

$$a_{\pm }=\frac{1}{\sqrt{2\hslash m\omega }}(\mp i\hat{p}+m\omega \hat{x})$$ $$a_{+}{a}_{-}=\frac{1}{\hslash \omega }\mathbb{H}-\frac{1}{2},\mathbb{H}=\hslash \omega \left(a_{+}{a}_{-}+\frac{1}{2}\right)$$

The ladder operators do not commute =

$$[\widehat{a_{-}},\widehat{a_{+}}]=a_{-}{a}_{+}-a_{+}{a}_{-}=1$$

Note that the position operator and momentum operator can be rewritten in terms of the ladder operators =

$$\hat{X}=\sqrt{\frac{\hslash }{2m\omega }}(a_{+}+a_{-})$$ $$\hat{P}=i\sqrt{\frac{\hslash m\omega }{2}}(a_{+}-a_{-})$$

Number operator

$$N=a_{+}{a}_{-}$$

such that

$$N{\psi }_n(x)=a_{+}{a}_{-}{\psi }_n(x)=n{\psi }_n(x)$$

Action on number states

$a_{-}|n⟩=\sqrt{n}|n-1⟩$ $a_{+}|n⟩=\sqrt{n+1}|n+1⟩$

expectation values

$⟨n|a_{-}|n⟩=0\text{ , }⟨n|a_{+}|n⟩=0$ This is because for expectation value of the lowering operator the two states ($|n⟩$ and $|n-1⟩$) are orthogonal since they have different eigenvalues ($E_n\ne E_{n-1}$)

expectation value of the number operator $⟨n|a_{+}{a}_{-}|n⟩=n$