9.3 Quantum harmonic oscillator and ladder operators

The quantum harmonic oscillator is a fundamental model in quantum mechanics. It describes a particle in a parabolic potential well, showcasing key quantum phenomena like discrete energy levels and wave-like behavior.

This model introduces crucial concepts like the Schrödinger equation, ladder operators, and energy eigenstates. Understanding these ideas is essential for grasping more complex quantum systems and their applications in physics and chemistry.

Quantum Harmonic Oscillator Fundamentals

Schrödinger equation for harmonic oscillator

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• Hamiltonian combines potential energy $V(x) = \frac{1}{2}kx^2$ and kinetic energy $T = \frac{p^2}{2m}$
• Time-independent Schrödinger equation $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2}kx^2\psi = E\psi$ describes quantum state
• Dimensionless variables $\xi = \sqrt{\frac{m\omega}{\hbar}}x$ and $\omega = \sqrt{\frac{k}{m}}$ simplify equation
• Dimensionless form $\frac{d^2\psi}{d\xi^2} + (2\epsilon - \xi^2)\psi = 0$ solved using power series or Hermite polynomials
• General wavefunction solution $\psi_n(\xi) = N_n H_n(\xi)e^{-\xi^2/2}$ with Hermite polynomials $H_n(\xi)$

Creation and annihilation operators

• Ladder operators manipulate quantum states
  • Annihilation operator $a = \frac{1}{\sqrt{2}}(\xi + \frac{d}{d\xi})$ lowers energy state
  • Creation operator $a^\dagger = \frac{1}{\sqrt{2}}(\xi - \frac{d}{d\xi})$ raises energy state
• Ladder operators act on energy eigenstates
  • $a|n\rangle = \sqrt{n}|n-1\rangle$ removes one quantum of energy
  • $a^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle$ adds one quantum of energy
• Hamiltonian expressed as $H = \hbar\omega(a^\dagger a + \frac{1}{2})$ using ladder operators
• Commutation relations govern operator behavior
  • $[a, a^\dagger] = 1$ fundamental commutation relation
  • $[a, H] = \hbar\omega a$ and $[a^\dagger, H] = -\hbar\omega a^\dagger$ relate to energy changes

Eigenvalues and eigenfunctions via ladder operators

• Energy eigenvalues $E_n = \hbar\omega(n + \frac{1}{2})$ show quantized energy levels
• Energy eigenfunctions generated using $|n\rangle = \frac{(a^\dagger)^n}{\sqrt{n!}}|0\rangle$
• Normalized wavefunctions $\psi_n(x) = \frac{1}{\sqrt{2^n n!}}\left(\frac{m\omega}{\pi\hbar}\right)^{1/4} H_n(\sqrt{\frac{m\omega}{\hbar}}x) e^{-\frac{m\omega x^2}{2\hbar}}$ describe spatial distribution
• Matrix elements of position and momentum operators
  • $\langle n|x|n+1\rangle = \sqrt{\frac{\hbar}{2m\omega}}\sqrt{n+1}$ for position
  • $\langle n|p|n+1\rangle = i\sqrt{\frac{m\hbar\omega}{2}}\sqrt{n+1}$ for momentum

Properties of quantum harmonic oscillator

• Ground state (n = 0) characteristics
  • Energy $E_0 = \frac{1}{2}\hbar\omega$ represents zero-point energy
  • Wavefunction $\psi_0(x) = (\frac{m\omega}{\pi\hbar})^{1/4} e^{-\frac{m\omega x^2}{2\hbar}}$ Gaussian distribution
• Excited states (n > 0) features
  • Equally spaced energy levels $\Delta E = \hbar\omega$
  • Wavefunctions have n nodes for nth excited state
• Probability density shows classical turning points and quantum tunneling
• Correspondence principle links quantum and classical behavior for large n
• Parity of eigenfunctions alternates (even n: even parity, odd n: odd parity)
• Uncertainty principle $\Delta x \Delta p \geq \frac{\hbar}{2}$ limits simultaneous measurement
• Virial theorem $\langle T \rangle = \langle V \rangle = \frac{1}{2}\langle E \rangle$ relates kinetic and potential energies