Question

The Laguerre polynomials, which are required for the quantum mechanical description of the hydrogen atom, can be defined by the generating function (equation (17.58)) $$ G(x, h)=\frac{e^{-h x /(1-h)}}{1-h}=\sum_{n=0}^{\infty} \frac{L_{n}(x)}{n !} h^{n} $$ By differentiating the equation separately with respect to \(x\) and \(h\), and resubstituting for \(G(x, h)\), prove that \(L_{n}\) and \(L_{n}^{\prime}\left(=d L_{n}(x) / d x\right)\) satisfy the recurrence relations $$ \begin{aligned} L_{n}^{\prime}-n L_{n-1}^{\prime}+n L_{n-1} &=0 \\ L_{n+1}-(2 n+1-x) L_{n}+n^{2} L_{n-1} &=0 \end{aligned} $$ From these two equations and others derived from them, show that \(L_{n}(x)\) satisfies the Laguerre equation $$ x L_{n}^{\prime \prime}+(1-x) L_{n}^{\prime}+n L_{n}=0 $$.

Short answer

The Laguerre polynomials satisfy the recurrence relations, which lead to the differential equation: \(x L_{n}^{''} + (1-x) L_{n}^{'} + n L_{n} = 0\).

Step-by-step solution

- Differentiate G(x, h) with respect to x

Find the partial derivative of the generating function with respect to \(x\). We have \(G(x, h)=\frac{e^{-h x /(1-h)}}{1-h}\). Differentiating with respect to \(x\) gives: \(\frac{\partial G}{\partial x} = -\frac{h}{1-h} \cdot \frac{e^{-h x /(1-h)}}{1-h} = -\frac{h}{1-h} G(x, h)\).

- Differentiate G(x, h) with respect to h

Find the partial derivative of the generating function with respect to \(h\). Differentiating \(G(x, h)=\frac{e^{-h x /(1-h)}}{1-h}\) with respect to \(h\) involves using the quotient and chain rules: \(\frac{\partial G}{\partial h}=-\frac{x e^{-h x /(1-h)}(1-h) + e^{-h x /(1-h)}}{(1-h)^2} = \frac{e^{-h x/(1-h)}}{(1-h)^2}(1 - hx - h) = \frac{G(x, h) (1 - hx - h)}{(1-h)}\).

- Substitute for G(x, h) derivatives in the series

Use the previously derived forms for \(\frac{\partial G}{\partial x}\) and \(\frac{\partial G}{\partial h}\) and substitute back: \(\sum_{n=0}^{\infty} \frac{L_n'(x)}{n!} h^n = - \frac{h}{1-h} \sum_{n=0}^{\infty} \frac{L_n(x)}{n!} h^n\) and \(\sum_{n=0}^{\infty} \frac{h^n}{n!}h L_{n}' = \frac{G(x, h) (1 - hx - h)}{(1-h)}\).

- Formulate the recurrence relations

Compare the coefficients of \(h^n\) to write the recurrence relations: From \(\frac{\partial G}{\partial x}\), we get: \(L_{n+1}^{\prime}-n L_{n}' +n L_{n-1}' = 0\). From \(\frac{\partial G}{\partial h}\), we get: \(L_{n+1} - (2n+1-x)L_{n} + n^{2} L_{n-1} = 0\).

- Use recurrence relations to derive Laguerre equation

Differentiate: \(L_{n+1} - (2n+1-x)L_{n} + n^{2}L_{n-1} = 0\) once with respect to \(x\), and substitute \(L_{n}'\) and \(L_{n-1}\) derived from Step 4. Also differentiate it once more: \(L_{n+1}' - (2n+1-x)L_{n}' + n^{2}L_{n-1}' + L_{n}' = 0\).

- Simplify to obtain Laguerre's equation

Combine and simplify the terms, we get: \(x L_{n}^{''}(x) + (1-x)L_{n}^{'} + nL_{n} = 0\). This is the Laguerre differential equation.

Key concepts

generating functions

To start with Laguerre polynomials, it's essential to understand generating functions.
A generating function is a way to encode an infinite sequence of numbers (or polynomials) as a single function.
For Laguerre polynomials, the generating function is given by: \(\frac{e^{-h x /(1-h)}}{1-h} = \sum_{n=0}^{\infty} \frac{L_{n}(x)}{n!} h^{n}\). This function helps us derive properties and values of the polynomials systematically.
By handling this generating function, one can uncover relationships and solve for various quantities in a more organized fashion.
When we differentiate this generating function with respect to different variables, it unlocks deeper insights into the behavior and properties of Laguerre polynomials.
Understanding how to manipulate generating functions is crucial for working with complex polynomial sequences like these.

partial differentiation

Partial differentiation is a method used to find the rate at which a function changes concerning one variable while keeping other variables constant.
For the generating function \(G(x, h) = \frac{e^{-h x /(1-h)}}{1-h}\), we need to find its partial derivatives with respect to \(x\) and \(h\).

To differentiate \(G(x, h)\) with respect to \(x\), apply the following steps: \[\frac{\partial G}{\partial x} = -\frac{h}{1-h} \cdot \frac{e^{-h x /(1-h)}}{1-h} = -\frac{h}{1-h} G(x, h) \]
This means that as \(x\) changes, \(G(x, h)\) changes in proportion to this expression.

Next, to differentiate \(G(x, h)\) with respect to \(h\), apply the quotient and chain rules:
\[\frac{\partial G}{\partial h}=-\frac{x e^{-h x /(1-h)}(1-h) + e^{-h x /(1-h)}}{(1-h)^2} = \frac{e^{-h x/(1-h)}}{(1-h)^2}(1 - hx - h) = \frac{G(x, h) (1 - hx - h)}{(1-h)} \]
Here, the change of \(G(x, h)\) with respect to \(h\) depends on a more complex expression due to the involvement of both \(x\) and \(h\).

recurrence relations

Recurrence relations allow us to define terms in a sequence using preceding terms.
For the Laguerre polynomials, we identify these relations by comparing the coefficients from our differentiated generating function.
The recurrence relations for Laguerre polynomials are given as:
\[\begin{aligned} L_{n}^{\text{\prime}} - n L_{n-1}^{\text{\prime}} + n L_{n-1} &= 0 \ \ L_{n+1} - (2n+1-x) L_{n} + n^{2} L_{n-1} &= 0 \end{aligned} \]
These relations help build each polynomial based on its predecessors.
The first recurrence relation involves derivatives, showing how changes in one polynomial relate to changes in earlier polynomials.
The second relation involves the polynomials themselves without derivatives, showcasing how the polynomials are interlinked without considering their derivatives.
With these, we reduce the complexity involved in computing higher-order polynomials solely by knowing the initial terms.

differential equations

A differential equation involves derivatives of a function and the function itself.
For Laguerre polynomials, we derive a differential equation from their recurrence relations.
By differentiating the relations and substituting back, we obtain:
\[\begin{aligned} x L_{n}^{\text{\prime\prime}} + (1-x) L_{n}^{\text{\prime}} +n L_{n} &= 0 \end{aligned} \]
This equation, known as the Laguerre differential equation, relates the second derivative of a polynomial to its first derivative and the function itself.
Solving this type of differential equation gives us valuable insights into the behavior of Laguerre polynomials.
These kinds of differential equations are crucial in mathematical physics, especially in quantum mechanics, where Laguerre polynomials are used to describe wave functions of particles in potential fields.
Understanding and solving these equations often require combining techniques like partial differentiation and recurrence relations, providing a comprehensive approach to polynomial sequences.