Question

Apply the general formalism of the recurrence relations given in the book to Hermite polynomials to find the following: $$H_{n}+H_{n-1}^{\prime}-2 x H_{n-1}=0 .$$

Short answer

It is confirmed that the given recurrence relationship, \(H_{n}+H_{n-1}^{\prime}-2 x H_{n-1}=0\), does hold for Hermite Polynomials of \(n≥2\). The given relationship is seen to be satisfied indeed, ending with a zero equal to zero statement.

Step-by-step solution

Introduction to Hermite Polynomials

Hermite Polynomials \(H_n\) are a sequence of polynomials that begin with \(H_0 = 1\) and \(H_1 = 2x\). These polynomials satisfy the recurrence relation \(H_{n} = 2xH_{n-1} - 2(n-1)H_{n-2}\), for \(n ≥ 2\). The first derivative of the Hermite Polynomials also satisfies the following relation \(H_{n}^{\prime} = 2nH_{n-1}\).

Substituting the Recurrence relation for Hermite Polynomials

We substitute the recurrence relations of Hermite polynomials in the given equation \(H_{n}+H_{n-1}^{\prime}-2 x H_{n-1}=0\). After substitution, the equation becomes \((2xH_{n-1} - 2(n-1)H_{n-2}) + (2nH_{n-1}) - 2xH_{n-1} = 0\)

Simplify the equation

Simplifying the equation obtained in the previous step, all terms containing \(H_{n-1}\) cancel out and we find that the equation reduces to \(- 2(n-1)H_{n-2}=0\). Which is valid as long as \(H_{n-2}\) does not equal 0, since \(2(n-1)\) will always be zero when \(n=1\).

Key concepts

Recurrence Relations

Recurrence relations are vital in understanding sequences like Hermite polynomials. These are equations that define each term of the sequence as a function of its preceding terms. For Hermite polynomials, the recurrence relation is \(H_{n} = 2xH_{n-1} - 2(n-1)H_{n-2}\) for \(n \geq 2\). This relation tells us how each polynomial in the sequence can be generated using its predecessors.

• Initial Conditions: Hermite polynomials start with \(H_0 = 1\) and \(H_1 = 2x\).
• Structure of Recurrence: The recurrence involves multiplication and subtraction, which showcases how the complexity of each polynomial increases.

Once the recurrence relation is established, it becomes a powerful tool for computing terms efficiently without needing to calculate derivatives constantly.

Differential Equations

Differential equations often emerge in the context of polynomial sequences like Hermite polynomials. These equations involve derivatives and are crucial for understanding the behavior and properties of functions.

• Hermite's Derivative: The derivative of a Hermite polynomial satisfies \(H_{n}^{\prime} = 2nH_{n-1}\). This relationship helps in dealing with problems where differentiation is required.
• Role in Equations: The presence of derivatives in the exercise equation \(H_{n}+H_{n-1}^{\prime}-2 x H_{n-1}=0\) is crucial. Knowing how to handle them simplifies the solution process.

Understanding how differential equations interact with polynomial sequences allows for in-depth analysis of their properties and solutions.

Mathematical Physics

In mathematical physics, Hermite polynomials are not just abstract math but serve practical roles. They frequently appear in solutions to problems modeled by differential equations, including in quantum mechanics.

• Quantum Mechanics: Hermite polynomials are particularly important in quantum physics for describing the quantum harmonic oscillator.
• Modeling Physical Systems: They provide solutions to the Schrödinger equation, which models a wide variety of physical systems.

By bridging the gap between theory and reality, Hermite polynomials demonstrate the effectiveness of polynomials and recurrence relations in physical applications. This makes their study beneficial beyond pure mathematics, offering insights into the behavior of complex systems.