Hermite Functions

Hermite functions are the normalized shape functions of the one-dimensional quantum harmonic oscillator. They are Hermite polynomials multiplied by a Gaussian and normalized in $L^2(\mathbb R)$.

This page is a focused reference for the eigenfunctions. The differential-equation derivation belongs to Differential-Equation Solution, and the polynomial identities belong to Hermite Polynomials.

Dimensionless Functions

In the dimensionless coordinate $\xi$, define

$$\varphi_n(\xi) = \frac{1}{ \pi^{1/4}\sqrt{2^n n!} } H_n(\xi)e^{-\xi^2/2}, \qquad n=0,1,2,\ldots$$

Here $H_n$ is the physicists’ Hermite polynomial. The first few are

$$H_0(\xi)=1, \qquad H_1(\xi)=2\xi, \qquad H_2(\xi)=4\xi^2-2.$$

Thus the first Hermite functions are a Gaussian, a Gaussian times a linear function, and a Gaussian times a quadratic function.

The first four normalized Hermite functions $\varphi_n(\xi)$. The parity alternates with $n$, and the number of nodes is exactly $n$.

Oscillator Scaling

For the physical oscillator with mass $m$ and angular frequency $\omega$, the oscillator length is

$$\ell=\sqrt{\frac{\hbar}{m\omega}}.$$

The dimensionless coordinate is

$$\xi=\frac{x}{\ell}.$$

The normalized position-space energy eigenfunctions are scaled Hermite functions:

$$\psi_n(x) = \frac{1}{\sqrt{\ell}}\, \varphi_n\left(\frac{x}{\ell}\right).$$

Equivalently,

$$\psi_n(x) = \frac{1}{\sqrt{2^n n!}} \left(\frac{1}{\pi\ell^2}\right)^{1/4} H_n\left(\frac{x}{\ell}\right) e^{-x^2/(2\ell^2)}.$$

The factor $\ell^{-1/2}$ is required because $dx=\ell\,d\xi$. Without it, the physical wavefunction would have the wrong units and the wrong normalization.

Orthonormality

The dimensionless Hermite functions obey

$$\int_{-\infty}^{\infty} \varphi_m(\xi)^*\varphi_n(\xi)\,d\xi = \delta_{mn}.$$

Therefore the physical oscillator wavefunctions obey

$$\int_{-\infty}^{\infty} \psi_m(x)^*\psi_n(x)\,dx = \delta_{mn}.$$

The normalization follows from the weighted Hermite-polynomial identity

$$\int_{-\infty}^{\infty} e^{-\xi^2}H_m(\xi)H_n(\xi)\,d\xi = \sqrt\pi\,2^n n!\,\delta_{mn}.$$

The Gaussian weight is essential. The polynomials $H_n$ are not orthogonal with respect to the ordinary unweighted integral on the real line.

Parity and Nodes

Hermite polynomials satisfy

$$H_n(-\xi)=(-1)^nH_n(\xi).$$

Because the Gaussian envelope is even,

$$\varphi_n(-\xi)=(-1)^n\varphi_n(\xi),$$

and hence

$$\psi_n(-x)=(-1)^n\psi_n(x).$$

Even $n$ gives even functions, and odd $n$ gives odd functions. The $n$th Hermite function has exactly $n$ real nodes. This agrees with the general one-dimensional bound-state rule that higher energy states have more nodes.

Energy Labels

The Hermite function label $n$ is also the oscillator energy label:

$$E_n=\hbar\omega\left(n+\frac12\right).$$

The wavefunction $\psi_n$ is the coordinate representation of the number state $\lvert n\rangle$:

$$\psi_n(x)=\langle x\vert n\rangle.$$

Number-state notation and ladder actions are collected in Number States. This page focuses on the functions themselves.

Recurrence and Matrix Elements

The dimensionless Hermite functions satisfy useful ladder identities:

$$\xi\varphi_n(\xi) = \sqrt{\frac{n+1}{2}}\,\varphi_{n+1}(\xi) + \sqrt{\frac{n}{2}}\,\varphi_{n-1}(\xi),$$

and

$$\frac{d\varphi_n}{d\xi} = \sqrt{\frac{n}{2}}\,\varphi_{n-1}(\xi) - \sqrt{\frac{n+1}{2}}\,\varphi_{n+1}(\xi).$$

After restoring $x=\ell\xi$, the position operator has matrix elements

$$\langle m\vert \hat x\vert n\rangle = \ell \left[ \sqrt{\frac{n+1}{2}}\,\delta_{m,n+1} + \sqrt{\frac{n}{2}}\,\delta_{m,n-1} \right].$$

These formulas are often faster than doing integrals directly. They are the wavefunction version of the ladder-operator algebra.

Fourier-Transform Property

With the unitary dimensionless Fourier convention

$$(\mathcal F f)(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ik\xi}f(\xi)\,d\xi,$$

the Hermite functions are Fourier-transform eigenfunctions:

$$\mathcal F\varphi_n = (-i)^n\varphi_n.$$

This property is one reason Hermite functions are useful in spectral methods and phase-space calculations. The phase depends on the Fourier convention; the eigenfunction structure does not.

Completeness

The set

$$\{\varphi_n:n=0,1,2,\ldots\}$$

is a complete orthonormal basis of $L^2(\mathbb R)$. A square-integrable function can be expanded as

$$f(\xi) = \sum_{n=0}^{\infty} c_n\varphi_n(\xi),$$

with coefficients

$$c_n = \int_{-\infty}^{\infty} \varphi_n(\xi)^*f(\xi)\,d\xi.$$

Completeness is a Hilbert-space statement about convergence in norm. It does not mean every function is well approximated pointwise by a short Hermite expansion.

Common Mistakes

• Confusing the Hermite polynomial $H_n$ with the normalized Hermite function $\varphi_n$.
• Dropping the factor $\ell^{-1/2}$ when converting to $\psi_n(x)$.
• Writing the Gaussian envelope as $e^{-x^2/\ell^2}$ instead of $e^{-x^2/(2\ell^2)}$.
• Forgetting that $n$ starts at $0$.
• Mixing physicists’ Hermite polynomials with probabilists’ $\operatorname{He}_n$.
• Treating Fourier-transform phases of Hermite functions as convention-independent.
• Assuming a short Hermite-function expansion is automatically good for discontinuous or sharply localized functions.

Where This Is Used

• Quantum Harmonic Oscillator summarizes the oscillator spectrum and scales.
• Differential-Equation Solution derives why Hermite functions appear.
• Hermite Polynomials gives the special-function identities used here.
• Ladder-Operator Solution: First Encounter gives the operator origin of the recurrence relations.
• Number States interprets $\psi_n(x)$ as $\langle x\vert n\rangle$.
• Harmonic Oscillator Spectrum gives a quick formula reference.
• Fourier Transform supplies the convention-sensitive transform background.
• Harmonic-Oscillator Propagator Notebook tests stable Hermite recurrences, spectral completeness, and kernel convergence numerically.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds., NIST Handbook of Mathematical Functions, Cambridge University Press, 2010.
• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.

Exercises

1. Show that $\psi_n(x)=\ell^{-1/2}\varphi_n(x/\ell)$ is normalized if $\varphi_n$ is normalized.

Solution

Compute

$$\int_{-\infty}^{\infty} \lvert\psi_n(x)\rvert^2\,dx = \int_{-\infty}^{\infty} \frac{1}{\ell} \left\lvert \varphi_n\left(\frac{x}{\ell}\right) \right\rvert^2 dx.$$

Set $\xi=x/\ell$, so $dx=\ell\,d\xi$. Then

$$\int_{-\infty}^{\infty} \lvert\psi_n(x)\rvert^2\,dx = \int_{-\infty}^{\infty} \lvert\varphi_n(\xi)\rvert^2\,d\xi =1.$$

2. Use the parity of $H_n$ to prove the parity of $\psi_n$.

Solution

The Gaussian factor is even:

$$e^{-(-x)^2/(2\ell^2)} = e^{-x^2/(2\ell^2)}.$$

The Hermite polynomial satisfies

$$H_n\left(-\frac{x}{\ell}\right) = (-1)^nH_n\left(\frac{x}{\ell}\right).$$

Therefore

$$\psi_n(-x)=(-1)^n\psi_n(x).$$

3. Derive $\langle m\vert \hat x\vert n\rangle$ from the recurrence relation for $\xi\varphi_n$.

Solution

Since $\hat x=\ell\xi$,

$$\langle m\vert \hat x\vert n\rangle = \ell \int_{-\infty}^{\infty} \varphi_m(\xi)^* \xi\varphi_n(\xi)\,d\xi.$$

Use

$$\xi\varphi_n = \sqrt{\frac{n+1}{2}}\,\varphi_{n+1} + \sqrt{\frac{n}{2}}\,\varphi_{n-1}.$$

Orthonormality gives

$$\langle m\vert \hat x\vert n\rangle = \ell \left[ \sqrt{\frac{n+1}{2}}\,\delta_{m,n+1} + \sqrt{\frac{n}{2}}\,\delta_{m,n-1} \right].$$

4. Why are Hermite polynomials alone not acceptable oscillator wavefunctions?

Solution

The polynomial $H_n(\xi)$ grows like $\xi^n$ at large $\lvert\xi\rvert$. It is not square-integrable on the real line. The Gaussian factor $e^{-\xi^2/2}$ supplies the decay needed for normalizability. The normalized Hermite function is the product of the polynomial, the Gaussian envelope, and the normalization constant.