Generating Functions — Hermite Polynomials
The Hermite polynomials (Fig. 13.1), Hn (x ), may be deﬁned by the generating function 2
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
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 Hn (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:
We also obtain from the generating function the important parity relation
by noting that Eq. (13.1) yields
The Rodrigues representation of Hn (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
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 Hm (x ) will survive:
Also, from the Maclaurin expansion, Eq. (13.4), we can derive our Hermite polynomial Hn (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.
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 ϕn (x) no longer a polynomial.
Substitution into Eq. (13.10) yields the differential equation for ϕn (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 φn (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 Hn (x )) without assuming the development that led to Eq. (13.13). The key aspect of the eigenvalue Eq. (13.13), −(2n + 1)ϕn (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 px 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
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 λn−1 = λn − 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
where c0 is an integration constant. The solution,
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
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 λn = n + 1/2, proving Eq. (13.13). The normalizations needed for Eq. (13.26) follow from Eqs. (13.25) and (13.23) and
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)
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
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 tj tk become
That is, from
so we obtain
The ni are all ﬁxed (making this case special and easy) because the Ni 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 HN1 + 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 Im .
Let us begin with a new result for I4 containing a product of four Hermite polynomials. Inserting the Feldheim identity for HN1 HN2 and HN3 HN4 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 Im . We prove a generalized Feldheim identity,
by mathematical induction. Multiplying this by HNm+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 νm−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 HN2 ··· HNm 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.
(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.
in terms of n and m and appropriate Kronecker delta functions.
(a) Using the Cauchy integral formula, develop an integral representation of Hn (x ) based on Eq. (13.1) with the contour enclosing the point z =−x .