Question

By power series, solve the Hermite differential equation $$y^{\prime \prime}-2 x y^{\prime}+2 p y=0$$ You should find an \(a_{0}\) series and an \(a_{1}\) series as for the Legendre equation in Section 2 Show that the \(a_{0}\) series terminates when \(p\) is an even integer, and the \(a_{1}\) series terminates when \(p\) is an odd integer. Thus for each integer \(n\), the differential equation (22.14) has one polynomial solution of degree \(n\). These polynomials with \(a_{0}\) or \(a_{1}\) chosen so that the highest order term is \((2 x)^{n}\) are the Hermite polynomials. Find \(H_{0}(x), H_{1}(x),\) and \(H_{2}(x) .\) Observe that you have solved an eigenvalue problem (see end of Section 2 ), namely to find values of \(p\) for which the given differential equation has polynomial solutions, and then to find the corresponding solutions (eigenfunctions).

Short answer

The Hermite polynomials are: \(H_0(x) = 1\), \(H_1(x) = 2x\), \(H_2(x) = 4x^2 - 2\).

Step-by-step solution

Express the solution as a power series

Assume the solution to the Hermite differential equation can be expressed as a power series: \[ y(x) = \sum_{n=0}^\infty a_n x^n \]

Find the derivatives

Compute the first and second derivatives of the power series: \[ y'(x) = \sum_{n=1}^\infty a_n n x^{n-1} \] \[ y''(x) = \sum_{n=2}^\infty a_n n (n-1) x^{n-2} \]

Substitute into the differential equation

Substitute the series, and its derivatives into the differential equation: \[ \sum_{n=2}^\infty a_n n (n-1) x^{n-2} - 2x \sum_{n=1}^\infty a_n n x^{n-1} + 2p \sum_{n=0}^\infty a_n x^n = 0 \]

Simplify and shift indices

Shift indices to combine like terms: \[ \sum_{n=0}^\infty [a_{n+2} (n+2)(n+1) - 2a_n (n+1) + 2pa_n] x^n = 0 \] Since this must hold for all x, the coefficients of each power of x must be zero.

Set up the recursion relation

Equate the coefficients of \(x^n\) to zero to get the recursion relation: \[ a_{n+2} = \frac{2(n-p)}{(n+2)(n+1)} a_n \]

Analyze the series

Notice that the series terminates when the numerator is zero, i.e. when \(n=p\). Thus, if \(p\) is even, the series with even indexed coefficients \(a_0\) terminates, and if \(p\) is odd, the series with odd indexed coefficients \(a_1\) terminates.

Find specific Hermite polynomials

Compute \(H_0(x)\), \(H_1(x)\), and \(H_2(x)\). For \(H_0(x)\), set \(p=0\), giving \(H_0(x) = 1\). For \(H_1(x)\), set \(p=1\), giving \(H_1(x) = 2x\). For \(H_2(x)\), set \(p=2\), giving \(H_2(x) = 4x^2 - 2\).

Key concepts

Power Series Solution

A **power series solution** is a method used to solve differential equations by expressing the solution as an infinite sum of terms. For the Hermite differential equation, we assume the solution can be written as a power series: \[ y(x) = \sum_{n=0}^\infty a_n x^n \] This means we represent the function in terms of coefficients \(a_n\) and powers of \(x\). The power series approach is useful because it transforms the differential equation into an algebraic equation involving the coefficients.
First, we compute the derivatives of the series needed for substitution into the Hermite differential equation.
Next, we substitute the series and its derivatives back into the original differential equation. This makes it possible to combine like terms.
The above steps simplify the differential equation into a form where it's possible to extract a **recursion relation**.

Hermite Polynomials

The **Hermite polynomials** are specific solutions to the Hermite differential equation. They are a set of orthogonal polynomials that occur in probability theory, quantum mechanics, and numerical analysis.
These polynomials are denoted by \(H_n(x)\) and derived by setting the parameter \(p\) to integer values.
The Hermite polynomials exhibit symmetry and have remarkable properties such as orthogonality and recursion relations, which help solve various kinds of problems involving the Hermite differential equation. For example, the polynomials can be used to find solutions to problems in quantum mechanics, like the quantum harmonic oscillator.
Specific Hermite polynomials derived in the example are:

• \(H_0(x) = 1\)
• \(H_1(x) = 2x\)
• \(H_2(x) = 4x^2 - 2\)

Eigenvalue Problem

The Hermite differential equation is also an **eigenvalue problem**. In this context, it means finding specific values of \(p\) (the eigenvalues) for which the differential equation has polynomial solutions (the eigenfunctions).
When solving the Hermite differential equation, we're essentially finding the eigenvalues that permit the series solution to terminate, yielding polynomial solutions.
For example, if \(p\) is an even integer, the power series with even-indexed coefficients \(a_0\) terminates, leading to a polynomial solution. Conversely, if \(p\) is odd, the series with odd-indexed coefficients \(a_1\) ends.
Therefore, solving the Hermite differential equation is about identifying values of \(p\) that result in these finite polynomials.

Recursion Relation

The recursion relation is vital in the process of solving the Hermite differential equation. This relationship between the coefficients of the power series allows us to determine all coefficients from the initial ones, \(a_0\) and \(a_1\). The recursion relation is:
\[ a_{n+2} = \frac{2(n-p)}{(n+2)(n+1)} a_n \]
It shows how each coefficient is connected to the previous ones. When \(n\) equals \(p\), the series will terminate, producing polynomial solutions if \(p\) is an integer.
By substituting this relation, terms cancel, and it helps to see when the series will naturally come to an end, ensuring our solution is a polynomial.

Polynomial Solutions

The Hermite differential equation has polynomial solutions for specific integer values of \(p\). We observe that these solutions come in two types: one associated with even coefficients (starting from \(a_0\)) and another with odd coefficients (starting from \(a_1\)).
The series terminates for such integer values, resulting in polynomial solutions simple but very useful functions in mathematical physics.
For instance, we've calculated the first few Hermite polynomials as follows based on different values of \(p\):

• \(H_0(x) = 1\), for \(p = 0\)
• \(H_1(x) = 2x\), for \(p = 1\)
• \(H_2(x) = 4x^2 - 2\), for \(p = 2\)

Each polynomial solution plays a significant role in various applications, including physics, where they describe quantum states in the harmonic oscillator model.