physref

Solutions of the Schrödinger equation (WIP)

  1. Harmonic oscillator
  2. Infinite square well (particle in a box)

Harmonic oscillator

Potential
V(x)=12mω2x2V(x) = \tfrac{1}{2}m{\omega ^2}{x^2}
(1)
Energy levels
En=(n+12)ℏωE_n = (n+\tfrac{1}{2})\hbar\omega
(2)

Here n=0,1,2,...n=0,1,2,...

Stationary states
∣n⟩=(a+)nn!∣0⟩=Hn(ξ)2nn!∣0⟩|n\rangle = \frac{{{{({a_ + })}^n}}}{{\sqrt {n!} }}|0\rangle = \frac{{{H_n}(\xi )}}{{\sqrt {{2^n}n!} }}|0\rangle
(3)

These are eigenstates of the time-independent Schrödinger equation H^ψ=Eψ\hat{H}\psi = E\psi. Here ξ≡mωℏx\xi \equiv \sqrt {\frac{{m\omega }}{\hbar }} x, and HnH_n are Hermite polynomials.

Ground state wave function
ψ0(x)=(mωπℏ)1/4e−ξ2/2{\psi _0}(x) = {{( {\frac{{m\omega }}{{\pi \hbar }}} )}^{1/4}}{e^{ - {\xi ^2}/2}}
(4)
Raising/lowering operators
a^±≡(2ℏmω)−1/2(mωx^∓ip^){{\hat a}_ \pm } \equiv {{(2\hbar m\omega )}^{ - 1/2}}(m\omega \hat x \mp i\hat p)
(5)
Position operator
x^=ℏ2mω(a^++a^−)\hat x = \sqrt {\frac{\hbar }{{2m\omega }}} ( {{\hat a_ + } + {\hat a_ - }} )
(6)
Momentum operator
p^=iℏmω2(a^+−a^−)\hat p = i\sqrt {\frac{{\hbar m\omega }}{2}} ( {{\hat a_ + } - {\hat a_ - }} )
(7)
Action of raising/lowering operators on stationary states
  • a^+∣n⟩=n+1 ∣n+1⟩{{\hat a}_ + }|n\rangle = \sqrt {n + 1} \,|n+1\rangle
  • ⟨n∣a^−=n+1 ⟨n+1∣\langle n|{{\hat a}_ - } = \sqrt{n+1} \,\langle n+1|
  • a^−∣n⟩=n ∣n−1⟩{{\hat a}_ - }|n\rangle = \sqrt n \,|n-1\rangle
  • ⟨n∣a^+=n ⟨n−1∣\langle n|{{\hat a}_ + } = \sqrt{n} \,\langle n-1|
Occupation number operator
n^=a^+a^−\hat{n} = \hat{a}_+ \hat{a}_-
(8)
Hamiltonian
H=(n^+12)ℏωH = (\hat{n} + \tfrac{1}{2})\hbar\omega
(9)

Infinite square well (particle in a box)

Potential
V(x)={0if 0⩽x⩽a∞otherwise V(x) = \begin{cases} 0 &\text{if } 0 \leqslant x \leqslant a \\ \infty &\text{otherwise } \end{cases}
(10)
Energy levels
En=n2π2ℏ22ma2E_n = \dfrac{n^2 \pi^2 \hbar^2}{2ma^2}
(11)

Here n=1,2,3,...n=1,2,3,...

Stationary states
ψn(x)=2asin⁡(nπxa)\psi _n(x) = \sqrt{\dfrac{2}{a}} \sin(\frac{n \pi x}{a})
(12)

These are eigenstates of the time-independent Schrödinger equation H^ψ=Eψ\hat{H}\psi = E\psi

Orthonormality of stationary states
∫ψm∗ψndx=δmn\int{\psi_m^* \psi_n dx = \delta_{mn}}
(13)

Here δmn\delta_{mn} is the Kronecker delta.

More to add...

Resources