QHO

Professor Dave Explains Video

Quantum Physics

Topics

• Classical Harmonic Oscillator

$U(x)=\frac{1}{2}k{x}^2$

• This is the potential of a quantum harmonic oscillator, and it applies to all the equations below

${\psi }_n(x)={\left(\frac{m\omega }{\pi \hslash }\right)}^{\frac{1}{4}}\frac{1}{\sqrt{2^{n}n!}}{H}_n(\xi )e^{-\frac{1}{2}{\xi }^2}$

• When the potential is that of a harmonic oscillator, the wave eigenfunctions of energy $E_n$ takes the above form
• You may also substitute $k=m{\omega }^2$ to get $U(x)=\frac{1}{2}m{\omega }^2{x}^2$
• $\xi \equiv \sqrt{\frac{m\omega }{\hslash }}x$ for substitution
• ${\left(\frac{m\omega }{\pi \hslash }\right)}^{\frac{1}{4}}\frac{1}{\sqrt{2^{n}n!}}$ is the normalization factor
• $H_n(\xi )$ is a Hermite polynomial
• Desmos Demo of Different Standing Waves

${\psi }_n(x)=\frac{1}{\sqrt{n}}({\hat{a}}_{+}{)}^n{\psi }_0(x)$

• ${\hat{a}}_{+}$ is the raising operator
• ${\psi }_0(x)$ is the ground state and is equal to ${\left(\frac{m\omega }{\pi \hslash }\right)}^{1/4}{e}^{-\frac{m\omega }{2\hslash }{x}^2}$

$\psi (\xi )=h(\xi )e^{-{\xi }^2/2}$

• General wave function of a harmonic oscillator
• $h(\xi )=a_0+a_1\xi +a_2{\xi }^2+\dots =\sum _{j=0}^{\infty }{a}_n{\xi }^n,a_{n+2}=\frac{2n-k+1}{(n+1)(n+2)}{a}_n$

$E_n=\left(n+\frac{1}{2}\right)\hslash \omega =\frac{1}{2}⟨K⟩=\frac{1}{2}⟨U⟩$

• Shows how energy is quantized in a harmonic oscillator
• $E_n$ is the energy of $n$th energy state
• $\hslash$ is the reduced planck constant
• $\omega$ is the angular frequency of oscillation
• $⟨K⟩$ is the average kinetic energy of the particle
• $⟨U⟩$ is the average potential energy of the particle

Statistical Mechanics

Topics

• Planck Distribution Function
• Stefan-Boltzmann Law
• Planck’s Law

$H=\frac{1}{8\pi }\int \left(ϵE^2+\frac{B^2}{\mu }\right)\mathrm{d}V\to \epsilon =\hslash {\omega }_0\left(n+\frac{1}{2}\right)$

• Delocalized harmonic oscillator

$\frac{{\mathrm{d}}^2\psi }{\mathrm{d}{x}^2}=\frac{2m}{\hslash }[V(x)-E]\psi$