Oscillator Eigenfunctions and Hermite Polynomials

The Shape of the Wavefunctions

Imagine the ground state of the harmonic oscillator. It is a simple, symmetric bell curve centered at the origin, representing the particle's most probable position. When we apply the raising operator to create the first excited state, the wavefunction does not simply get wider; it develops a node at the origin. The function becomes positive on one side and negative on the other, creating a "dumbbell" shape. This sign change is a direct consequence of the parity symmetry of the potential. As we climb higher in energy, adding more nodes, the wavefunctions oscillate more rapidly within the classical turning points. The overall envelope of these oscillations remains Gaussian, decaying exponentially far from the origin, but the internal structure is defined by a polynomial. These polynomials are not arbitrary; they are the Hermite polynomials, which naturally arise when solving the differential equation for the oscillator in position space. The interplay between the Gaussian decay and the polynomial oscillation defines the unique character of each quantum state.

Generating the Eigenfunctions

We begin with the normalized ground state wavefunction, which is a Gaussian:

The excited states are generated by applying the raising operator \( \hat{a}^\dagger \) repeatedly. In position space, the raising operator acts as a differential operator. Specifically, \( \hat{a}^\dagger \) corresponds to \( \frac{1}{\sqrt{2}} \left( \xi - \frac{d}{d\xi} \right) \), where \( \xi = \sqrt{\frac{m\omega}{\hbar}} x \) is the dimensionless coordinate. Applying this operator to the Gaussian ground state produces a function that is the product of a linear polynomial and the original Gaussian. Repeated application generates higher-degree polynomials multiplied by the same Gaussian envelope.

The general form of the \( n \)-th eigenfunction is therefore:

where \( H_n(\xi) \) is the physicists' Hermite polynomial of degree \( n \), and \( N_n \) is a normalization constant. The Hermite polynomials are defined by the Rodrigues formula or by the recurrence relation, and they possess the crucial property of orthogonality with respect to the Gaussian weight function.

Normalization and the Hermite Polynomials

To determine the constant \( N_n \), we impose the condition that the probability of finding the particle anywhere in space is unity. This requires evaluating the integral of the squared wavefunction over all space. The orthogonality relation for Hermite polynomials is:

Using this result, the normalization integral for \( \psi_n(x) \) becomes straightforward. The square of the wavefunction involves \( H_n^2(\xi) e^{-\xi^2} \). Substituting the orthogonality integral, we find that the normalization constant must be:

Thus, the fully normalized eigenfunctions are:

This expression is the central result of this lecture. It explicitly links the quantum state \( n \) to the polynomial degree \( n \) and provides the precise amplitude for probability calculations.

Parity and Nodes

The Hermite polynomials have definite parity. \( H_n(\xi) \) is an even function if \( n \) is even, and an odd function if \( n \) is odd. Since the Gaussian \( e^{-\xi^2/2} \) is always even, the parity of the total wavefunction \( \psi_n(x) \) is determined solely by \( n \). This means \( \psi_n(-x) = (-1)^n \psi_n(x) \). Consequently, the probability density \( |\psi_n(x)|^2 \) is always an even function, symmetric about the origin.

The number of nodes (zeros) in the wavefunction is exactly \( n \). For \( n=0 \), there are no nodes. For \( n=1 \), there is one node at \( x=0 \). For \( n=2 \), \( H_2(\xi) = 4\xi^2 - 2 \), so there are two nodes, at \( \xi = \pm 1/\sqrt{2} \). This node structure is a direct manifestation of the quantum number \( n \) and is a key feature distinguishing quantum states from classical trajectories. The nodes represent positions where the probability of finding the particle is exactly zero, a phenomenon with no classical analogue.