The Classical Harmonic Oscillator

Chem 3240 · Lecture 4.2

Davit Potoyan

Bead, spring, and a wall

  • A mass tied to a wall by a spring
  • Displace it by \(x\) from equilibrium
  • A restoring force pulls it back

\[F = -kx\]

The minus sign always points toward equilibrium

Hooke’s law

\[F = -kx\]

  • \(k\) is the spring constant: the stiffness
  • Stiffer spring means a stronger pull back
  • The only ingredient we need to build the whole model

Equation of motion

  • Newton’s second law with the Hooke force:

\[m\ddot{x} = -kx\]

\[\ddot{x} + \omega^2 x = 0\]

  • A linear second-order ODE
  • One constant \(\omega\) governs everything

The solution

\[x(t) = A\cos(\omega t + \phi)\]

  • Amplitude \(A\) and phase \(\phi\) set by initial conditions
  • Motion is a pure cosine: simple harmonic motion

Frequency

\[\omega = \sqrt{\frac{k}{\mu}}\]

  • Stiffer spring (larger \(k\)): faster oscillation
  • Heavier mass (larger \(\mu\)): slower oscillation
  • Period \(T = 2\pi/\omega\), frequency \(f = \omega/2\pi\)

Energy of the oscillator

  • Force is the slope of a potential: \(F = -\partial V/\partial x\)
  • Integrating gives a parabolic well: \(V = \tfrac{1}{2}kx^2\)

\[E = \frac{p^2}{2m} + \frac{kx^2}{2}\]

  • Kinetic and potential energy trade off; total energy is constant

Diatomic molecule: a two-body problem

  • Two atoms bound by a spring, masses \(m_1\) and \(m_2\)
  • Center of mass drifts freely; only the relative coordinate vibrates

\[\ddot{x} = -\left(\frac{1}{m_1} + \frac{1}{m_2}\right)kx = -\frac{k}{\mu}x\]

Reduced mass

  • The two-body problem becomes one body with an effective mass:

\[\mu = \frac{m_1 m_2}{m_1 + m_2}\]

  • Same equation as the bead on a wall, with \(m \to \mu\)
  • One particle, one spring, one frequency \(\omega = \sqrt{k/\mu}\)

Real bonds are not parabolas

  • Real bonds dissociate; a parabola never does
  • Taylor expand any well about its minimum \(x_0\):

\[U(x) = \frac{1}{2}k(x-x_0)^2 + \frac{1}{6}\gamma(x-x_0)^3 + \cdots\]

The harmonic approximation

  • First derivative vanishes at the minimum
  • Keep only the first non-vanishing term: the quadratic
  • \(k\) is the curvature \(U''(x_0)\) at the bottom of the well
  • Good near equilibrium; anharmonic terms grow far out

Takeaway

A restoring force \(F=-kx\) gives sinusoidal motion \(x(t)=A\cos(\omega t+\phi)\) at \(\omega=\sqrt{k/\mu}\). Any bond near its minimum is harmonic, and the parabola is the launching point for the quantum oscillator.