Harmonic Oscillator

Quantum Mechanical Harmonic Oscillator

The quantum harmonic oscillator is a fundamental model in quantum mechanics describing a particle in a potential well, commonly used to approximate molecular vibrations. Its potential energy function is the same as the classical case:

\[ V(x) = \frac{1}{2} k x^2 \]

Where:

• \( V(x) \): Potential energy as a function of displacement \(x\).
• \( k \): Force constant (bond stiffness).
• \( x \): Displacement from equilibrium (assumed equilibrium at \(x = 0\)).

The quantum mechanical solution involves solving the Schrödinger equation:

\[ \hat{H} \psi_n(x) = E_n \psi_n(x) \]

Where \( \hat{H} \) is the Hamiltonian operator, \( \psi_n(x) \) are the wavefunctions, and \( E_n \) are the quantized energy levels.

Quantized Energy Levels

The allowed energy levels are given by:

\[ E_n = \hbar \omega \left( n + \frac{1}{2} \right) \]

Where:

• \( n \): Quantum number (\(n = 0, 1, 2, \dots\)).
• \( \hbar \): Reduced Planck's constant.
• \( \omega = \sqrt{k/m} \): Angular frequency of the oscillator.
• \( m \): Mass of the particle.

Key features:

• The lowest energy level (\(n = 0\)) is not zero; instead, it is the zero-point energy (\(\frac{1}{2} \hbar \omega\)).
• Energy levels are evenly spaced by \(\hbar \omega\).

Wavefunctions (\( \psi_n(x) \))

The wavefunctions \( \psi_n(x) \) are solutions to the Schrödinger equation, and they correspond to the probability distributions of the particle at each energy level. They are expressed in terms of Hermite polynomials \( H_n(x) \):

\[ \psi_n(x) = N_n e^{-\frac{\alpha x^2}{2}} H_n(\sqrt{\alpha} x) \]

Where:

• \( \alpha = \frac{m \omega}{\hbar} \): A scaling factor based on mass and angular frequency.
• \( N_n \): Normalization constant ensuring total probability equals 1.
• \( H_n \): Hermite polynomials.

Key Differences From Classical Behavior:

• In classical mechanics, the particle oscillates continuously within the potential well, with all energy levels being possible.
• In quantum mechanics:
  • The energy is quantized.
  • The particle has a nonzero probability of being found outside the classical turning points due to quantum tunneling.
  • Zero-point energy prevents the particle from being at rest at \(x = 0\).