The Quantum Harmonic Oscillator

Chem 3240 · Lecture 4.3

Davit Potoyan

Why the Harmonic Oscillator?

  • Near equilibrium, any bond is a spring.
  • Molecular vibrations are quantized.
  • Foundation for IR and Raman spectroscopy.
  • First place we meet zero-point energy and tunneling.

The Schrodinger Equation

  • Classical Hamiltonian: kinetic plus a quadratic well.

\[\hat{H} = -\frac{\hbar^2}{2\mu}\frac{d^2}{dx^2} + \frac{1}{2}kx^2\]

\[\hat{H}\psi_v(x) = E_v\,\psi_v(x)\]

  • Spring constant \(k\), angular frequency \(\omega = \sqrt{k/\mu}\).

Quantized Energies

\[E_v = \left(v + \tfrac{1}{2}\right)\hbar\omega, \qquad v = 0,1,2,\dots\]

  • Levels are evenly spaced by \(\hbar\omega\).
  • The integer \(v\) is the vibrational quantum number.
  • Energy never goes to zero, even in the lowest state.

Zero-Point Energy

\[E_0 = \tfrac{1}{2}\hbar\omega \neq 0\]

  • The oscillator cannot sit still at the bottom of the well.
  • A consequence of the uncertainty principle: pinning \(x\) would blow up \(p\).
  • Atoms keep moving even at absolute zero.

The Wavefunctions

\[\psi_v(x) = N_v\,H_v\!\left(\sqrt{\alpha}\,x\right)e^{-\alpha x^2/2}\]

  • Hermite polynomial \(H_v\) times a Gaussian envelope.
  • Scaling \(\alpha = \sqrt{k\mu/\hbar^2}\), normalization \(N_v = \dfrac{1}{\sqrt{2^v v!}}\left(\dfrac{\alpha}{\pi}\right)^{1/4}\).

The Gaussian Ground State

\[\psi_0(x) = \left(\frac{\alpha}{\pi}\right)^{1/4} e^{-\alpha x^2/2}\]

  • No nodes, a single peak at the bottom of the well.
  • A pure Gaussian: the minimum-uncertainty state.
  • First excited state gains one node: \(\psi_1(x) \propto x\,e^{-\alpha x^2/2}\).

Hermite Polynomials

\(v\) \(H_v(y)\)
0 \(1\)
1 \(2y\)
2 \(4y^2 - 2\)
3 \(8y^3 - 12y\)
  • Each added quantum number adds one node.
  • They are orthogonal with weight \(e^{-y^2}\).

Even / Odd Parity

  • Even \(v = 0,2,4,\dots\): \(\psi_v(-x) = \psi_v(x)\).
  • Odd \(v = 1,3,5,\dots\): \(\psi_v(-x) = -\psi_v(x)\).

\[\langle x \rangle = 0, \qquad \langle p \rangle = 0\]

  • Symmetry kills odd integrals, so spreads survive: \(\langle x^2\rangle, \langle p^2\rangle \neq 0\).

Tunneling

  • Wavefunction tails leak past the classical turning points.
  • Finite probability in the classically forbidden region where \(E < V\).
  • Strictly impossible in classical mechanics.
  • At high \(v\), probability piles up near the edges, approaching the classical result.

Beyond Harmonic: Anharmonicity

  • Real bonds dissociate, the well is not a perfect parabola.
  • Levels bunch up toward \(D_e\).
  • Harmonic model is only the near-equilibrium limit.

Takeaway

The quantum oscillator has evenly spaced levels \(E_v = \hbar\omega(v+\tfrac{1}{2})\) with an irreducible zero-point energy \(\tfrac{1}{2}\hbar\omega\). Its states are Hermite polynomials times a Gaussian, with definite parity and tails that tunnel into the forbidden region.