Evaluating The Integral Of Squared Hermite Polynomials

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The integral ${ \int_{-\infty}^{\infty} e^{-x^2} [H_n(x)]^2 \, dx }$ holds significant importance in various fields, including quantum mechanics and mathematical physics, particularly when dealing with the quantum harmonic oscillator. This integral involves the square of the Hermite polynomials, denoted by H_n(x), weighted by a Gaussian function e^{-x2}. Understanding its value requires delving into the properties of Hermite polynomials and their orthogonality relations. This comprehensive guide will walk you through the intricacies of evaluating this integral, providing a step-by-step explanation and highlighting the underlying mathematical concepts. We will explore the definition and properties of Hermite polynomials, derive the orthogonality relation, and finally, compute the value of the integral. This detailed exploration will not only help you understand the solution but also equip you with the knowledge to tackle similar problems involving special functions and integrals.

Hermite Polynomials: A Deep Dive

To effectively evaluate the integral, a strong understanding of Hermite polynomials is crucial. Hermite polynomials, denoted as H_n(x), are a set of orthogonal polynomials that play a pivotal role in various areas of mathematics, physics, and engineering. They are particularly important in quantum mechanics, where they appear in the solutions of the Schrödinger equation for the quantum harmonic oscillator. These polynomials can be defined in several ways, each offering a unique perspective on their properties and applications. One common definition is through the Rodrigues' formula, which provides an explicit expression for the n-th Hermite polynomial in terms of derivatives. Another definition is through a recurrence relation, which allows us to compute higher-order Hermite polynomials from lower-order ones. Additionally, Hermite polynomials can be defined using a generating function, which provides a compact way to represent the entire set of polynomials. Each of these definitions offers a unique perspective on the properties and applications of Hermite polynomials.

Definitions of Hermite Polynomials

1. Rodrigues' Formula: This formula provides an explicit expression for the n-th Hermite polynomial:

   ${ H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} }$

   This formula is particularly useful for deriving the explicit form of Hermite polynomials for specific values of n. It highlights the relationship between Hermite polynomials and the derivatives of the Gaussian function.

2. Recurrence Relation: This definition provides a recursive way to compute Hermite polynomials:

   ${ H_{n+1}(x) = 2xH_n(x) - 2nH_{n-1}(x) }$

   with initial conditions:

   ${ H_0(x) = 1, \quad H_1(x) = 2x }$

   This recurrence relation is highly efficient for computing Hermite polynomials of higher degrees. Starting from the initial polynomials H_0(x) and H_1(x), we can iteratively compute H_2(x), H_3(x), and so on.

3. Generating Function: The generating function provides a compact representation of the entire set of Hermite polynomials:

   ${ e^{2xt - t^2} = \sum_{n=0}^{\infty} \frac{H_n(x)}{n!} t^n }$

   This definition is particularly useful for deriving various properties and identities of Hermite polynomials. By manipulating the generating function, we can obtain expressions for integrals, derivatives, and other related quantities.

Properties of Hermite Polynomials

Hermite polynomials possess several important properties that make them invaluable in various mathematical and physical applications. These properties include orthogonality, parity, and specific values at x = 0. Understanding these properties is crucial for effectively working with Hermite polynomials and applying them to solve problems. The orthogonality property, in particular, plays a central role in evaluating the integral we are interested in.

1. Orthogonality: Hermite polynomials are orthogonal with respect to the weight function e^{-x2} over the interval (-\infty, \infty). This means that:

   ${ \int_{-\infty}^{\infty} H_m(x) H_n(x) e^{-x^2} dx = 0, \quad \text{if } m \neq n }$

   This property is fundamental to many applications of Hermite polynomials, including their use as basis functions in function expansions. The orthogonality relation simplifies many calculations and allows us to express functions as linear combinations of Hermite polynomials.

2. Parity: Hermite polynomials have a definite parity, meaning they are either even or odd functions. This property can be expressed as:

   ${ H_n(-x) = (-1)^n H_n(x) }$

   This means that if n is even, H_n(x) is an even function, and if n is odd, H_n(x) is an odd function. This property can be useful for simplifying integrals and other calculations involving Hermite polynomials.

3. Values at x = 0: The values of Hermite polynomials at x = 0 are given by:

   ${ H_n(0) = \begin{cases} 0, & \text{if } n \text{ is odd} \\ (-1)^{n/2} \frac{n!}{(n/2)!}, & \text{if } n \text{ is even} \end{cases} }$

   These values can be useful in various applications, such as evaluating series and approximating functions.

Evaluating the Integral: A Step-by-Step Approach

Now, let's dive into evaluating the integral ${ \int_{-\infty}^{\infty} e^{-x^2} [H_n(x)]^2 \, dx }$. This integral is a cornerstone in understanding the behavior of Hermite polynomials and their applications. The key to solving this integral lies in utilizing the orthogonality property of Hermite polynomials and a crucial normalization constant. We will break down the process into manageable steps, ensuring a clear understanding of each stage. First, we will revisit the orthogonality relation and highlight its significance. Then, we will focus on determining the normalization constant, which is essential for evaluating the integral when m = n. Finally, we will put everything together to arrive at the final result. This step-by-step approach will not only provide the solution but also offer insights into the mathematical techniques used in the process.

Utilizing the Orthogonality Relation

The orthogonality relation of Hermite polynomials is the cornerstone for evaluating the integral. As we discussed earlier, Hermite polynomials satisfy the following orthogonality condition:

${ \int_{-\infty}^{\infty} H_m(x) H_n(x) e^{-x^2} dx = 0, \quad \text{if } m \neq n }$

However, this relation only tells us what happens when m and n are different. To evaluate our integral, we need to consider the case when m = n. In this case, the integral is no longer zero, and we need to determine its value. This value is related to the normalization constant of the Hermite polynomials. The orthogonality relation provides a foundation, but the normalization constant is the key to unlocking the specific value of the integral we are interested in.

Determining the Normalization Constant

To find the value of the integral when m = n, we need to determine the normalization constant. This involves evaluating the integral:

${ \int_{-\infty}^{\infty} [H_n(x)]^2 e^{-x^2} dx }$

This integral represents the "length" or "norm" of the Hermite polynomial H_n(x) with respect to the weight function e^{-x2}. There are several ways to compute this integral, one of which involves using the generating function for Hermite polynomials. By carefully manipulating the generating function and utilizing the properties of Gaussian integrals, we can arrive at the normalization constant. This constant will provide the crucial link between the orthogonality relation and the specific value of the integral.

Let's use the generating function:

${ e^{2xt - t^2} = \sum_{n=0}^{\infty} \frac{H_n(x)}{n!} t^n }$

Multiply this by a similar expression with a different variable s:

${ e^{2xs - s^2} = \sum_{m=0}^{\infty} \frac{H_m(x)}{m!} s^m }$

Now, multiply these two series:

${ e^{2xt - t^2} e^{2xs - s^2} = \sum_{n=0}^{\infty} \sum_{m=0}^{\infty} \frac{H_n(x) H_m(x)}{n! m!} t^n s^m }$

Multiply both sides by e^{-x2} and integrate with respect to x from -\infty to \infty:

${ \int_{-\infty}^{\infty} e^{2xt - t^2} e^{2xs - s^2} e^{-x^2} dx = \int_{-\infty}^{\infty} \sum_{n=0}^{\infty} \sum_{m=0}^{\infty} \frac{H_n(x) H_m(x)}{n! m!} t^n s^m e^{-x^2} dx }$

Interchange the integral and the summation:

${ \int_{-\infty}^{\infty} e^{2xt - t^2} e^{2xs - s^2} e^{-x^2} dx = \sum_{n=0}^{\infty} \sum_{m=0}^{\infty} \frac{t^n s^m}{n! m!} \int_{-\infty}^{\infty} H_n(x) H_m(x) e^{-x^2} dx }$

The integral on the right-hand side is zero unless n = m, due to the orthogonality property. When n = m, we have:

${ \int_{-\infty}^{\infty} H_n(x) H_m(x) e^{-x^2} dx = \int_{-\infty}^{\infty} [H_n(x)]^2 e^{-x^2} dx }$

Let's denote this integral as I_n:

${ I_n = \int_{-\infty}^{\infty} [H_n(x)]^2 e^{-x^2} dx }$

So, the double summation reduces to a single summation:

${ \sum_{n=0}^{\infty} \sum_{m=0}^{\infty} \frac{t^n s^m}{n! m!} \int_{-\infty}^{\infty} H_n(x) H_m(x) e^{-x^2} dx = \sum_{n=0}^{\infty} \frac{(ts)^n}{(n!)^2} I_n }$

Now, let's evaluate the integral on the left-hand side:

${ \int_{-\infty}^{\infty} e^{2xt - t^2} e^{2xs - s^2} e^{-x^2} dx = \int_{-\infty}^{\infty} e^{-x^2 + 2x(t+s) - (t^2 + s^2)} dx }$

Complete the square in the exponent:

${ -x^2 + 2x(t+s) - (t^2 + s^2) = -(x - (t+s))^2 + (t+s)^2 - (t^2 + s^2) = -(x - (t+s))^2 + 2ts }$

So the integral becomes:

${ \int_{-\infty}^{\infty} e^{-(x - (t+s))^2 + 2ts} dx = e^{2ts} \int_{-\infty}^{\infty} e^{-(x - (t+s))^2} dx }$

The remaining integral is a Gaussian integral, which evaluates to \sqrt{\pi}:

${ e^{2ts} \int_{-\infty}^{\infty} e^{-(x - (t+s))^2} dx = e^{2ts} \sqrt{\pi} }$

Now, expand e^{2ts} as a power series:

${ e^{2ts} \sqrt{\pi} = \sqrt{\pi} \sum_{n=0}^{\infty} \frac{(2ts)^n}{n!} }$

Equating the coefficients of t^n s^n from both sides, we have:

${ \frac{I_n}{(n!)^2} = \sqrt{\pi} \frac{2^n}{n!} }$

Thus,

${ I_n = 2^n n! \sqrt{\pi} }$

Therefore, the normalization constant is:

${ \int_{-\infty}^{\infty} [H_n(x)]^2 e^{-x^2} dx = 2^n n! \sqrt{\pi} }$

The Final Result

Combining the orthogonality relation and the normalization constant, we arrive at the final result for the integral:

${ \int_{-\infty}^{\infty} e^{-x^2} H_m(x) H_n(x) dx = 2^n n! \sqrt{\pi} \delta_{mn} }$

where \delta_{mn} is the Kronecker delta, which is 1 if m = n and 0 if m \neq n. In our case, we are interested in the integral when m = n, so the result is:

${ \int_{-\infty}^{\infty} e^{-x^2} [H_n(x)]^2 dx = 2^n n! \sqrt{\pi} }$

This result confirms that the value of the integral ${ \int_{-\infty}^{\infty} e^{-x^2} [H_n(x)]^2 \, dx }$ is indeed 2^n n! \sqrt{\pi}. This value is crucial in various applications, particularly in quantum mechanics, where it appears in the normalization of the wave functions for the quantum harmonic oscillator. The detailed derivation we have presented provides a comprehensive understanding of how this result is obtained and highlights the importance of Hermite polynomials and their properties.

Conclusion

In conclusion, we have successfully evaluated the integral ${ \int_{-\infty}^{\infty} e^{-x^2} [H_n(x)]^2 \, dx }$ and found it to be equal to 2^n n! \sqrt{\pi}. This result is a testament to the power and elegance of Hermite polynomials and their orthogonality properties. We explored the definitions and properties of Hermite polynomials, emphasizing their orthogonality relation and how it simplifies the evaluation of integrals. The use of the generating function provided a powerful tool for computing the normalization constant, which is essential for obtaining the final result. This exploration not only provides the solution to the specific integral but also offers a framework for tackling similar problems involving special functions and integrals. The significance of this integral extends beyond pure mathematics, finding applications in various fields such as quantum mechanics, where it plays a crucial role in the study of the quantum harmonic oscillator. The detailed step-by-step approach presented in this guide ensures a clear understanding of the underlying concepts and techniques, making it a valuable resource for students, researchers, and anyone interested in the fascinating world of special functions and their applications.

This comprehensive guide has equipped you with a thorough understanding of Hermite polynomials and their applications in evaluating integrals. By mastering these concepts, you are well-prepared to tackle more advanced problems in mathematics, physics, and engineering.