Question

Obtain an approximate formula for the Bessel function \(J_{m}(x)\) by the WKB method and give the limiting form of this expression for large \(x(x \geqslant m)\). Do not worry about getting the constant in front correct. You may assume \(m \geqslant \frac{1}{2}\).

Short answer

For large x, the Bessel function can be approximated as \( J_m(x) \rightarrow \frac{e^{ix}}{x^{1/2}} \).

Step-by-step solution

- Introduction to the WKB Method

The Wentzel-Kramers-Brillouin (WKB) method is used for approximating solutions to differential equations with varying coefficients. For Bessel functions, this method will help in finding an approximate solution for large values of x.

- Bessel's Differential Equation

Start with Bessel's differential equation: \[ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - m^2)y = 0 \] Consider the substitution: \( y = x^{-1/2} u(x) \), which transforms the leading order solutions.

- Applying the WKB Ansatz

Use the WKB solution form: \( u(x) = e^{i \theta(x)} \). Substitute into the differential equation to find the leading order term in the asymptotic series.

- Finding the Phase \( θ(x) \)

Solve for \( \theta(x) \) using the idea that \( \theta' = f(x) \). This leads to: \[ \theta(x) = \theta_0 \frac{\theta_0^2 - x^2}{\theta_0} \] Now integrate this expression to get \( \theta(x) \).

- Calculating the Approximate Form

From the previous steps, obtain the approximate WKB solution for large values of x: \[ J_m(x) \rightarrow \frac{1}{\theta_0} e^{i \theta(x)} \]

- Limiting Form for Large x

For large x (where \( x \rightarrow \theta_0 \)): \[ J_m(x) \rightarrow \frac{1}{\theta_0} \frac{e^{ix}}{x^{1/2}} \]

- Final Approximate Formula

Thus, the approximate formula for Bessel function \( J_{m}(x) \) by the WKB method for large x is: \[ J_m(x) \rightarrow \frac{e^{ix}}{x^{1/2}} \]

Key concepts

Bessel function

A Bessel function is a type of canonical solution to Bessel's differential equation, which appears in various physics problems, especially problems involving radial symmetry such as heat conduction and wave propagation.
Bessel functions are denoted as \(J_m(x)\), where \(m\) is the order of the function. For large arguments \(x\), these functions can be approximated using various methods, one of which is the WKB method.
Understanding Bessel functions is critical for solving physical problems with cylindrical symmetry, like the vibrational mode of a circular drumhead.

WKB method

The WKB (Wentzel-Kramers-Brillouin) method is an asymptotic analysis technique used to find approximate solutions to differential equations with varying coefficients.
This method is particularly handy when the argument of the function is large (like \( x \rightarrow \theta_0 \) in our case).
To apply the WKB method, we assume a solution form like \( u(x) = e^{i \theta(x)} \). This assumption transforms the differential equation into a form where the dominant parts can be solved more easily.
The WKB method has wide applications, particularly in quantum mechanics, where it helps to approximate wave functions.

Asymptotic analysis

Asymptotic analysis deals with the behavior of functions as the argument approaches a particular limit, often infinity.
In the context of the WKB method and Bessel functions, we deal with the case where \( x \) becomes very large. This allows us to simplify the problem by focusing on the dominating terms.
The key idea is to find simpler approximate forms of functions that are valid in the limit, making complex differential equations more tractable.
For Bessel functions, this means approximating \( J_m(x) \) for large \( x \) and reducing it to a simpler form like \( e^{ix} / x^{1/2} \).

Differential equations

Differential equations describe how a function changes based on its derivatives. For Bessel functions, we start with Bessel's differential equation:
\[ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - m^2)y = 0 \]
This equation has a combination of multiplicative terms and derivatives of different orders. The WKB method helps simplify such equations by focusing on main terms in conditions like large \(x\).
Differential equations are central in physics and engineering to model a wide range of phenomena including heat transfer, electromagnetic fields, and quantum mechanics.

Phase function

In the WKB method, the phase function \( \theta(x) \) represents the main term in the exponent of the guessed solution form \( u(x) = e^{i \theta(x)} \).
Finding \( \theta(x) \) involves solving an equation obtained from substituting the WKB ansatz into the original differential equation. For instance, we solve:
\[ \theta(x) = \theta_0 \frac{\theta_0^2 - x^2}{\theta_0} \]
The phase function encapsulates the oscillatory behavior of the solution and helps in deriving the approximate solution. For large \( x \), the simplification leads to forms such as \( e^{ix} / x^{1/2} \), which captures the main behavior of Bessel functions at large arguments.