Charles Hermite (1822-1901) gave us these polynomials, famous in the quantum mechanics of the harmonic oscillator
A polynomial is a finite sum of terms like akxk, where k is a positive integer or zero. There are sets of polynomials such that the product of any two different ones, multiplied by a function w(x) called a weight function and integrated over a certain interval, vanishes. Such a set is called a set of orthogonal polynomials. Among other things, this makes it possible to expand an arbitrary function f(x) as a sum of the polynomials, each multiplied by a coefficient c(k), which is easily and uniquely determined by integration. A Fourier series is similar, but the orthogonal functions are not polynomials. These functions can also be used to specify basis states in quantum mechanics, which must be orthogonal.
The Hermite polynomials Hn(x) are orthogonal on the interval from -∞ to +∞ with respect to the weight function w(x) = exp(x2). [exp (x) = ex] Surprisingly, this is sufficient to determine the polynomials up to a multiplicative factor. Let's start with the expression Hn = exp(x2)(dn/dxn)exp(-x2). We notice that each differentiation will bring down a factor -2x, and that the exponential will survive in each term. Each term will contain the factor exp(-x2) that will be cancelled by the exponential in front. Therefore, the result will be a polynomial of degree n, and the leading term will be (-2x)n. The powers of x will either be all even or all odd, as well. Moreover, the form of this definition will guarantee that the polynomials belonging to different values of n are orthogonal. It is, in fact, a Rodrigues' formula, like those used to define other kinds of orthogonal polynomials.
An alternative definition uses the weight function w(x) = exp(x2/2) instead. Then, Hen = exp(x2/2)(dn/dxn)exp(-x2/2), where the notation He is used instead of H to make the different weight function clear. Of course, the corresponding polynomials will be very similar, and one could be used as well as the other, with appropriate changes of variable. In this case, each differentiation brings down the simpler factor (-x), so that the coefficient of xn is (-1)n. In fact, the usual definitions include the (-1)n factor, so that the coefficient of xn is always positive. Of three classic references (see References), Jackson uses He, Pauling and Wilson H, and both are given in Abramowitz and Stegun. The possibilities of confusion are not as great as in the case of Laguerre polynomials.
The H polynomials are H0 = 1, H1 = 2x, with the recursion relation Hn+1(x) = 2xHn(x) - 2nHn-1(x), and they satisfy the differential equation H"n -2xH'n + 2nHn = 0.
The He polynomials are He0 = 1, He1 = x, with the recursion relation Hen+1 = xHen - nHen-1, and they satisfy the differential equation He"n - xHe'n + nHen = 0.
It is of interest that y(x) = exp(-x2/2)Hn(x) is a solution of the differential equation y" + (2n + 1 - x2)y = 0. This equation arises in the quantum mechanics of the harmonic oscillator. These solutions are called Hermite functions, and each includes the square root of the weight function w(x), so that the wave functions y(x) are orthogonal when integrated from -∞ to +∞, which is required by the theory.
The orthogonality of the Hermite polynomials is expressed by , where the Kronecker delta is zero if m is not equal to n, and unity if m equals n. To prove this, simply express the exponential times the Hermite polynomial of larger order as an nth derivative using the Rodrigues formula, then integrate by parts until the polynomial of smaller order is differentiated to zero. If the orders are equal, the final integral is simply the integral of exp(-x2) times a constant, and the result is established. The normalization constant becomes sqrt(2π)n! for He.
The orthogonality can be used to expand an arbitrary function f(x) in a series of Hermite polynomials, in exactly the same way that a Fourier series is formed. If the expansion functions are exp(-x2/2)Hn(x), the series is called a Gram-Charlier series, and is useful in mathematical statistics.
Surely the most famous use of Hermite polynomials is in the Schrödinger theory of the harmonic oscillator. I will sketch the development here. For more information, see Pauling and Wilson or Dirac, or indeed any introductory quantum mechanics text. A model of a harmonic oscillator is a point mass m moving along the x-axis under a restoring force -kx. The natural frequency ν is given by 2πν = ω = √(k/m). In terms of the momentum p = mv the kinetic energy T = p2/2m, and the potential energy V = mω2x2/2. The Hamiltonian H(p,q) = T + V.
In quantum mechanics, the state of the oscillator is described by an amplitude Ψ that obeys the equation of motion (ih/2π)dΨ/dt = HΨ, where H is an operator formed from the operators p and x. In the Schrödinger picture, the state is described by the generalized coordinate x, and the momentum is the operator p = -(ih/2π)d/dx. A state of definite energy E satisfies the relation HΨ = EΨ, and is called an energy eigenstate. Since H does not contain the time, Ψ can be expressed as the product of a time factor and a space factor ψ. The equation of motion shows that the time factor is exp(i2πEt/h) for a state of this kind, which represents a steady change of phase at frequency E/h. It is the complex nature of Ψ that allows it to express the complete behavior of the oscillator with the x coordinate alone, instead of the position and momentum separately. The operators for p and x satisfy xp - px = ih/2π; that is, they do not commute, which is a fundamental property.
Since the time function factors out, ψ satisfies Hψ = Eψ, or -(h2/8π2m)d2ψdx2 + mω2x2ψ/2 = Eψ. This is the Schrödinger Equation, and ψ(x) is the Schrödinger wave function. If we put u = x√α, where α = 2πmω/h, this equation becomes
d2ψ/du2 + (2E/hν - u2)ψ = 0
From what we said above, we see that 2E/hν = 2n + 1, and ψ = Cnexp(-u2/2)Hn(u), where Cn is chosen so that the integral of the square of ψ from -∞ to +∞ is unity; this is said to normalize the wave function. For energy eigenstates such as this, we can always choose ψ real. The energy of the oscillator takes the values E = hν(n + 1/2). In the ground state, n= 0, there remains the zero-point energy hν/2. This unexpected and quite real result was one of the triumphs of quantum mechanics.
The energy eigenstates of the harmonic oscillator are by no means the only or the usually observed states. They are mainly useful in expanding more general states in terms of the energy eigenstates, and other theoretical developments. In particular, it is impossible to realize a macroscopic energy eigenstate, so we never see the nodes or the spreading in daily life. The widest applications of this theory are to optics and radiation, incidentally.
Composed by J. B. Calvert
Created 15 November 2000