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HERMITE FUNCTIONS

Generating Functions — Hermite Polynomials 

The Hermite polynomials (Fig. 13.1), H_n (x ), may be defined by the generating function ²

           (13.1)

Recurrence Relations

Note the absence of a superscript, which distinguishes Hermite polynomials from the unrelated Hankel functions. From the generating function we find that the Hermite polynomials satisfy the recurrence relations

 (13.2)

and

 (13.3)

Equation (13.2) is obtained by differentiating the generating function with respect to t :

which can be rewritten as

Because each coefficient of this power series vanishes, Eq. (13.2) is established. Similarly, differentiation with respect to x leads to

which yields Eq. (13.3) upon shifting the summation index n in the last sum n + 1 → n.

Table 13.1 Hermite Polynomials

The Maclaurin expansion of the generating function

gives H₀ (x ) = 1 and H₁ (x ) = 2x , and then the recursion Eq. (13.2) permits the construction of any H_n (x ) desired (integral n). For convenient reference the first several Hermite polynomials are listed in Table 13.1.
Special values of the Hermite polynomials follow from the generating function for
x = 0:

that is,

We also obtain from the generating function the important parity relation

by noting that Eq. (13.1) yields

Alternate Representations 

The Rodrigues representation of H_n (x ) is

Let us show this using mathematical induction as follows.

Example 13.1.1 RODRIGUES REPRESENTATION 

We rewrite the generating function as  and note that

This yields

to establish the n + 1 case, with the last equality following from Eqs.(13.2) and (13.3).

More directly, differentiation of the generating function n times with respect to t and then setting t equal to zero yields

A second representation may be obtained by using the calculus of residues (Section 7.1).
If we multiply Eq. (13.1) by t ^−m−1 and integrate around the origin in the complex t -plane, only the term with H_m (x ) will survive:

             (13.8)

Also, from the Maclaurin expansion, Eq. (13.4), we can derive our Hermite polynomial H_n (x ) in series form: Using the binomial expansion of (2x − t)^ν and the index N = s + ν ,

where [N/2] is the largest integer less than or equal to N/2. Writing the binomial coefficient in terms of factorials and using Eq. (13.1) we obtain

More explicitly, replacing N → n,wehave

This series terminates for integral n and yields our Hermite polynomial.

Orthogonality

If we substitute the recursion Eq. (13.3) into Eq. (13.2) we can eliminate the index n − 1, obtaining

which was used already in Example 13.1.1. If we differentiate this recursion relation and substitute Eq. (13.3) for the index n + 1 we find

which can be rearranged to the second-order ODE for Hermite polynomials. Thus, the recurrence relations (Eqs. (13.2) and (13.3)) lead to the second-order ODE

which is clearly not self-adjoint.
To put the ODE in self-adjoint form, following Section 10.1, we multiply by exp (−x 2 ),

Exercise 10.1.2. This leads to the orthogonality integral

with the weighting function exp(−x ²), a consequence of putting the ODE into self-adjoint form. The interval (−∞, ∞) is chosen to obtain the Hermitian operator boundary conditions, Section 10.1. It is sometimes convenient to absorb the weighting function into the Hermite polynomials. We may define

with ϕ_n (x) no longer a polynomial.

Substitution into Eq. (13.10) yields the differential equation for ϕ_n (x),

This is the differential equation for a quantum mechanical, simple harmonic oscillator, which is perhaps the most important physics application of the Hermite polynomials.

Equation (13.13) is self-adjoint, and the solutions ϕn (x ) are orthogonal for the interval (−∞ <x < ∞) with a unit weighting function.
The problem of normalizing these functions remains. Proceeding as in Section 12.3, we multiply Eq. (13.1) by itself and then by e^{−x 2} . This yields

When we integrate this relation over x from −∞ to +∞, the cross terms of the double sum drop out because of the orthogonality property:³

using Eqs. (8.6) and (8.8). By equating coefficients of like powers of st , we obtain

Quantum Mechanical Simple Harmonic Oscillator 

The following development of Hermite polynomials via simple harmonic oscillator wave functions φ_n (x ) is analogous to the use of the raising and lowering operators for angular momentum operators presented in Section 4.3. This means that we derive the eigenvalues n + 1/2 and eigenfunctions (the H_n (x )) without assuming the development that led to Eq. (13.13). The key aspect of the eigenvalue Eq. (13.13),  −(2n + 1)ϕ_n (x), is that the Hamiltonian

almost factorizes. Using naively a² − b² = (a − b)(a + b), the basic commutator i of quantum mechanics (with momentum px =  i)d/dx) enters as a correction in Eq. (13.16). (Because p_x is Hermitian, d/dx is anti-Hermitian, (d /dx)^† =−d/dx .) This commutator can be evaluated as follows. Imagine the differential operator d/dx acts on a wave function ϕ(x ) to the right, as in Eq. (13.13), so

by the product rule. Dropping the wave function ϕ from Eq. (13.17), we rewrite Eq. (13.17) as

                      (13.18)

a constant, and then verify Eq. (13.16) directly by expanding the product of operators.
The product form of Eq. (13.16), up to the constant commutator, suggests introducing the non-Hermitian operators

with  which are adjoints of each other. They obey the commutation relations

which are characteristic of these operators and straightforward to derive from Eq. (13.18) and

Returning to Eq. (13.16) and using Eq. (13.19) we rewrite the Hamiltonian as

we see that N has nonnegative igenvalues

We now show that if  is nonzero it is an eigenfunction with eigenvalue λ_n−1 = λ_{n − 1}. After normalizing , this state is designated |n − 1}.Thisisprovedbythe commutation relations

which follow from Eq. (13.20). These commutation relations characterize N as the number operator. To see this, we determine the eigenvalue of N for the states   Using

Applying  repeatedly, we can reach the lowest, or ground, state |0} with eigenvalue λ₀ .We cannot step lower because λ₀ ≥ 1/2. Therefore  |0}≡ 0, suggesting we construct ψ₀ = |0} from the (factored) first-order ODE

Integrating

              (13.28)

we obtain

               (13.29)

where c₀ is an integration constant. The solution,

(13.30)

is a Gaussian that can be normalized, with c₀ = π ^−1/4 using the error integral, Eqs. (8.6) and (8.8). Substituting ψ₀ into Eq. (13.13) we find

(13.31)

so its energy eigenvalue is λ₀ = 1/2 and its number eigenvalue is n = 0, confirming the notation |0}. Applying  repeatedly to ψ₀ =|0}, all other eigenvalues are confirmed to be λ_n = n + 1/2, proving Eq. (13.13). The normalizations needed for Eq. (13.26) follow from Eqs. (13.25) and (13.23) and

       (13.32)

showing

Thus, the excited-state wave functions, ψ₁ ,ψ₂ , and so on, are generated by the raising operator

yielding (and leading to upcoming Eq. (13.38))

As shown, the Hermite polynomials are used in analyzing the quantum mechanical sim

Our oscillating particle has mass m and total energy E . By use of the abbreviations

in which ω is the angular frequency of the corresponding classical oscillator, Eq. (13.36) becomes (with ψ(z) = ψ(x /α) = ψ(x ))

This is Eq. (13.13) with λ = 2n + 1  Hence (fig. 13.2)

 (normalized). (13.39)

Alternatively, the requirement that n be an integer is dictated by the boundary conditions of the quantum mechanical system,

FIGURE 13.2 Quantum mechanical oscillator wave functions: The heavy bar on the x -axis indicates the allowed range of the classical oscillator with the same total energy.

As n ranges over integral values (n ≥ 0), we see that the energy is quantized and that there is a minimum or zero point energy

                 (13.41)

This zero point energy is an aspect of the uncertainty principle, a genuine quantum phenomenon.

In quantum mechanical problems, particularly in molecular spectroscopy, a number of integrals of the form

are needed. Examples for r = 1 and r = 2 (with n = m) are included in the exercises at the end of this section. A large number of other examples are contained in Wilson, Decius, and Cross.

In the dynamics and spectroscopy of molecules in the Born–Oppenheimer approximation, the motion of a molecule is separated into electronic, vibrational and rotational motion. Each vibrating atom contributes to a matrix element two Hermite polynomials, its initial state and another one to its final state. Thus, integrals of products of Hermite polynomials are needed.

Example 13.1.2 THREEFOLD HERMITE FORMULA

We want to calculate the following integral involving m = 3 Hermite polynomials:

where Ni ≥ 0 are integers. The formula (due to E. C. Titchmarsh, J. Lond. Math. Soc. 23: 15 (1948), see Gradshteyn and Ryzhik, p. 838, in the Additional Readings) generalizes the m = 2 case needed for the orthogonality and normalization of Hermite polynomials. To derive it, we start with the product of three generating functions of Hermite polynomials, multiply by e^{−x 2} , and integrate over x in order to generate I₃ :

using the polynomial expansion

The powers of the foregoing t_j t_k become

That is, from

there follows

so we obtain

The n_i are all fixed (making this case special and easy) because the N_i are fixed, and

an integer by parity. Hence, upon comparing the foregoing like t₁ t₂ t₃ powers,

which is the desired formula. If we order N₁ ≥ N₂ ≥ N₃ ≥ 0, then n₁ ≥ n₂ ≥ n₃ ≥ 0 follows, being equivalent to N − N₃ ≥ N − N₂ ≥ N − N₁ ≥ 0, which occur in the denominators of the factorials of I₃

Example : DIRECT EXPANSION OF PRODUCTS OF HERMITE POLYNOMIALS 

In an alternative approach, we now start again from the generating function identity

Using the binomial expansion and then comparing like powers of t₁ t₂ we extract an identity due to E. Feldheim (J. Lond. Math. Soc. 13: 22 (1938)):

For ν = 0 the coefficient of H_{N1 + N2} is obviously unity. Special cases, such as

can be derived from Table 13.1 and agree with the general twofold product formula. This compact formula can be generalized to products of m Hermite polynomials, and this in turn yields a new closed form result for the integral I_m .

Let us begin with a new result for I₄ containing a product of four Hermite polynomials. Inserting the Feldheim identity for H_N1 H_N2 and H_N3 H_N4 and using orthogonality

for the remaining product of two Hermite polynomials yields

Here we use the notation M = (N₁ + N₂ + N₃ + N₄ )/2 and write the binomial coefficients explicitly, so

Now we return to the product expansion of m Hermite polynomials and the corresponding new result from it for I_m . We prove a generalized Feldheim identity,

where

by mathematical induction. Multiplying this by H_Nm+1 and using the Feldheim identity, we end up with the same formula for m + 1 Hermite polynomials, including the recursion relation

Its solution is

The limits of the summation indices are

The limits of the summation indices are

Example : APPLICATIONS OF THE PRODUCT FORMULAS

which agrees with our earlier result of Example 13.1.2. The last expression is based on the following observations. The power of 2 has the exponent N₁ + ν₂ = N . The factorials from the binomial coefficients are N₃ − ν₂ = (N₁ + N₃ − N₂)/2 = N − N₂,N₂ − ν₂ = (N₁ + N₂ − N₃)/2 = N − N₃ .

Next let us consider m = 4, where we do not order the Hermite indices Ni as yet. The reason is that the general Im expression was derived with a different grouping of the

Hermite polynomials than the separate calculation of I₄ with which we compare. That is why we’ll have to permute the indices to get the earlier result for I₄ . That is a general conclusion: Different groupings of the Hermite polynomials just give different permutations of the Hermite indices in the general result.

We have two summations over ν₂ and ν_m−1 = ν₃ , which is fixed by the constraint N₁ =

In the last expression we have substituted ν₃ and used

The Hermite polynomial product formula also applies to products of simple harmonic oscillator wave functions, with a different exponential weight function. To evaluate such integrals we use the generalized Feldheim identity for H_N2 ··· H_Nm in conjunction with the integral (see Gradshteyn and Ryzhik, p. 845, in the Additional Readings),

instead of the standard orthogonality integral for the remaining product of two Hermite polynomials. Here the hypergeometric function is the finite sum

This yields a result similar to The oscillator potential has also been employed extensively in calculations of nuclear structure (nuclear shell model) quark models of hadrons and the nuclear force.
There is a second independent solution of Eq. (13.13). This Hermite function of the second kind is an infinite series (Sections 9.5 and 9.6) and is of no physical interest, at least not yet.

In developing the properties of the Hermite polynomials, start at a number of different points, such as:
1. Hermite’s ODE, Eq. (13.13),
2. Rodrigues’ formula, Eq. (13.7),
3. Integral representation, Eq. (13.8),
4. Generating function, Eq. (13.1),
5. Gram–Schmidt construction of a complete set of orthogonal polynomials over (−∞, ∞) with a weighting factor of exp(−x ² ), Section 10.3.
Outline how you can go from any one of these starting points to all the other points.

Excercise

(a) Expand x ^2r in a series of even-order Hermite polynomials.

(b) Expand x ^2r +1 in a series of odd-order Hermite polynomials.

Hint. Use a Rodrigues representation and integrate by parts.

Evaluate

in terms of n and m and appropriate Kronecker delta functions.

(a) Using the Cauchy integral formula, develop an integral representation of H_n (x ) based on Eq. (13.1) with the contour enclosing the point z =−x .