Question

Solve the following eigenvalue problem (see end of Section 2 and Problem 11): Given the differential equation $$y^{\prime \prime}+\left(\frac{\lambda}{x}-\frac{1}{4}-\frac{l(l+1)}{x^{2}}\right) y=0$$ where \(l\) is an integer \(\geq 0,\) find values of \(\lambda\) such that \(y \rightarrow 0\) as \(x \rightarrow \infty\), and find the corresponding eigenfunctions. Hint: let \(y=x^{l+1} e^{-x / 2} v(x),\) and show that \(v(x)\) satisfies the differential equation $$x v^{\prime \prime}+(2 l+2-x) v^{\prime}+(\lambda-l-1) v=0$$ Compare (22.26) to show that if \(\lambda\) is an integer \(>l,\) there is a polynomial solution \(v(x)=L_{\lambda-l-1}^{2 l+1}(x)\).

Short answer

\( \lambda \) must be an integer greater than \( l \), and the corresponding eigenfunctions are: \( y = x^{l+1} e^{-x/2} L_{\lambda-l-1}^{2 l+1}(x) \).

Step-by-step solution

Substitute the given hint into the differential equation

Given the transformation: \( y = x^{l+1} e^{-x/2} v(x) \), substitute it into the original differential equation: \( y^{\prime\prime} + \left( \frac{\lambda}{x} - \frac{1}{4} - \frac{l(l+1)}{x^2} \right) y = 0 \).

Compute the first and second derivatives of y

Calculate the first and second derivatives of \( y \): 1) \( y' = x^{l+1} e^{-x/2} v'(x) + e^{-x/2} (x^{l+1})' v(x) \) 2) \( y'' = x^{l+1} e^{-x/2} v''(x) + 2 e^{-x/2} (x^{l+1})' v'(x) + e^{-x/2} (x^{l+1})'' v(x) \).

Simplify the derivatives

By individually calculating and simplifying: 1) \( (x^{l+1})' = (l+1) x^l \) 2) \( (x^l e^{-x/2})' = x^l e^{-x/2}(-\frac{1}{2} x) \) 3) Substitute these simplified derivatives back into the equation.

Substitute simplified form in original equation

After the appropriate substitutions, the equation simplifies to: \( x v'' + (2l+2-x) v' + (\lambda - l - 1)v = 0 \). This is the differential equation for the function \( v(x) \).

Identify the standard equation

Compare \( x v'' + (2l+2-x) v' + (\lambda - l - 1)v = 0 \) to the generalized Laguerre differential equation: \( x L_n'' + (1 - x + 2l) L_n' + n L_n = 0 \).

Solve for eigenvalues

We identify \( n = \lambda - l - 1 \) and \( u = 2l + 1 \). If \( \lambda \) is an integer greater than \( l \), the solutions exist as polynomial functions: \( v(x) = L_{\lambda-l-1}^{2l+1}(x) \).

Determine eigenfunctions

From the expression \( y = x^{l+1} e^{-x/2} v(x) \), substitute \( v(x) \) to find the eigenfunctions: \( y(x) = x^{l+1} e^{-x/2} L_{\lambda-l-1}^{2 l+1}(x) \).

Key concepts

Differential Equations

Differential equations are mathematical equations that involve functions and their derivatives. They describe various physical phenomena such as motion, heat, and waves.
In the eigenvalue problem provided, we're dealing with a second-order linear differential equation:y'' + (λ/x - 1/4 - l(l+1)/x²) y = 0Such equations are often encountered in mathematical physics. Solving them can reveal the behavior of a system. Here, the given differential equation includes terms with both the function y(x) and its second derivative y''(x), representing complex interactions.

Eigenfunctions

Eigenfunctions are specific solutions to differential equations that are highly useful in various fields, like quantum mechanics.
When solving the given differential equation for specific values of λ (the eigenvalues), we get certain functions y(x) that satisfy it.To solve for eigenfunctions in this problem, we follow a transformation:y = x^{l+1} e^{-x/2} v(x)By substituting this into the differential equation and simplifying, we get a new equation for v(x). When λ is a specific integer greater than l, v(x) turns out to be a Laguerre polynomial. The eigenfunctions y(x) are then expressed in terms of these polynomials.

Laguerre Polynomials

Laguerre polynomials are a sequence of orthogonal polynomials that solve certain types of differential equations. In our problem, once we simplify the original equation, we end up with:x v'' + (2l+2-x) v' + (λ - l - 1)v = 0By comparing this with the standard form of the Laguerre differential equation, we find that v(x) can be substituted with the generalized Laguerre polynomial L_{λ-l-1}^{2l+1}(x). When λ is an integer greater than l, v(x) becomes a polynomial solution. Laguerre polynomials frequently appear in quantum mechanics, especially in the context of the hydrogen atom.

Eigenvalues

Eigenvalues are special values of λ for which the differential equation has non-trivial solutions (eigenfunctions).
In this problem, by setting v(x) as a Laguerre polynomial, we determine that eigenvalues λ must be integers greater than l.v(x) = L_{λ-l-1}^{2l+1}(x)This reveals that for each integer λ greater than l, there exists a corresponding eigenfunction. These combinations of eigenvalues and eigenfunctions are crucial in solving physical problems represented by differential equations, providing insight into the system’s behavior.