Question

In the quantum mechanical study of the scattering of a particle by a potential, a Born-approximation solution can be obtained in terms of a function \(y(\mathbf{r})\) that satisfies an equation of the form $$ \left(-\nabla^{2}-K^{2}\right) y(\mathbf{r})=F(\mathbf{r}) $$ Assuming that \(y_{\mathbf{k}}(\mathbf{r})=(2 \pi)^{-3 / 2} \exp (i \mathbf{k} \cdot \mathbf{r})\) is a suitably normalised eigenfunction of \(-\nabla^{2}\) corresponding to eigenvalue \(-k^{2}\), find a suitable Green's function \(G_{K}\left(\mathbf{r}, \mathbf{r}^{\prime}\right)\) By taking the direction of the vector \(\mathbf{r}-\mathbf{r}^{\prime}\) as the polar axis for a \(\mathbf{k}\)-space integration, show that \(G_{K}\left(\mathbf{r}, \mathbf{r}^{\prime}\right)\) can be reduced to $$ \frac{1}{4 \pi\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} \int_{-\infty}^{\infty} \frac{w \sin w}{w^{2}-w_{0}^{2}} d w $$ where \(w_{0}=K\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\). (This integral can be evaluated using a contour integration (chapter 20 ) to give \(\left.\left(4 \pi\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)^{-1} \exp \left(i K\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right) .\right)\)

Short answer

The suitable Green's function is \[ G_K(\mathbf{r}, \mathbf{r}') = \frac{1}{4\pi |\mathbf{r} - \mathbf{r}'|} \exp(i K |\mathbf{r} - \mathbf{r}'|) \]

Step-by-step solution

Understand the Given Equation

Identify the form of the given differential equation: \[ (-abla^2 - K^2) y(\mathbf{r}) = F(\mathbf{r}) \] This equation involves finding a function y(\mathbf{r}) that satisfies this relation.

Recognize the Eigenfunction

Observe the given normalized eigenfunction: \[ y_{\mathbf{k}}(\mathbf{r}) = (2 \pi)^{-3/2} \exp(i \mathbf{k} \cdot \mathbf{r}) \] This eigenfunction corresponds to the eigenvalue \[ -k^2 \] of the operator \[ -abla^2 \]

Substitute the Eigenfunction

Substitute the normalized eigenfunction into the given differential equation: \[(-abla^2 - K^2) (2\pi)^{-3/2} \exp(i \mathbf{k} \cdot \mathbf{r}) \] Using the eigenvalue relation: \[ abla^2 \exp(i \mathbf{k} \cdot \mathbf{r}) = -k^2 \exp(i \mathbf{k} \cdot \mathbf{r}) \]

Define the Green's Function

Introduce the Green's function \[ G_K(\mathbf{r}, \mathbf{r}') \] that will satisfy the following equation: \[(-abla^2 - K^2) G_K(\mathbf{r}, \mathbf{r}') = \delta(\mathbf{r} - \mathbf{r}') \]

Fourier Transform

Express the Green's function using its Fourier transform: \[ G_K(\mathbf{r}, \mathbf{r}') = \frac{1}{(2\pi)^3} \int d^3k \; \frac{\exp(i \mathbf{k} \cdot (\mathbf{r} - \mathbf{r}'))}{k^2 - K^2} \]

Use Polar Coordinates for Integration

Take the direction of \( \mathbf{r} - \mathbf{r}' \) as the polar axis for the \( \mathbf{k} \)-space integration. Align the z-axis of spherical coordinates to \( \mathbf{r} - \mathbf{r}' \). This simplifies to: \[ G_K(\mathbf{r}, \mathbf{r}') = \frac{1}{(2\pi)^3} \int_0^{2\pi} d\phi \int_0^{\pi} \sin\theta \, d\theta \int_0^{\infty} \frac{k^2 \sin(kr)}{k^2 - K^2} dk \]

Simplify the Integral

By integrating over the angular parts (\( \phi \) and \( \theta \)), the integral simplifies to: \[ G_K(\mathbf{r}, \mathbf{r}') = \frac{1}{4\pi |\mathbf{r} - \mathbf{r}'|} \int_{-\infty}^{\infty} \frac{w \sin w}{w^2 - w_0^2} dw \] where \( w = k |\mathbf{r} - \mathbf{r}'| \) and \( w_0 = K |\mathbf{r} - \mathbf{r}'| \)

Evaluate the Integral

Using contour integration techniques (refer to Chapter 20), evaluate the integral: \[ \int_{-\infty}^{\infty} \frac{w \sin w}{w^2 - w_0^2} dw \] This evaluates to: \[ \exp(i K |\mathbf{r} - \mathbf{r}'|) \]

Final Solution for Green's Function

Combine the results to express the Green's function as: \[ G_K(\mathbf{r}, \mathbf{r}') = \frac{1}{4\pi |\mathbf{r} - \mathbf{r}'|} \exp(i K |\mathbf{r} - \mathbf{r}'|) \]

Key concepts

Quantum Mechanics

Quantum mechanics studies the behavior of particles at very small scales, such as electrons around an atom. It uses complex mathematical methods to describe the probability of a particle’s position and energy. The Born approximation is a method in quantum mechanics used to simplify the study of particle scattering. When a particle interacts with a potential field, this approximation helps estimate the scattering effect.
To understand particle scattering, we start with the Schrödinger equation, a foundational equation in quantum mechanics, used to describe how the quantum state of a system changes over time. Solutions to this equation give us information about the probable locations and energies of particles.
In this exercise, we examine the function y(𝑟) that fits into a differential equation, showing how a particle responds when encountering a potential. The eigenfunctions (solutions to the equation) represent wave-like states of particles, where the exponential term reveals the particle's wave nature.

Green's Function

Green's functions are powerful mathematical tools used to solve differential equations. They can be thought of as the response of a system to a point-like source of force or disturbance. In simpler terms, it helps in understanding how a system behaves in response to a small, localized input.
In this context, we seek a suitable Green's function, G_K(𝑟, 𝑟’), which helps us solve the differential equation involving the particle function y(𝑟). This function acts like a bridge, allowing us to link the source of the disturbance with the resulting system response.
By considering the direction of the vector between two points, we can use Green’s function to simplify and solve complex integrals in our equations. Finding and using this Green's function are crucial steps to unraveling the scattering behavior of the particle.

Fourier Transform

The Fourier transform is a mathematical technique that transforms a function into its constituent frequencies. This transformation is vital in physics and engineering, as it lets us analyze signals and functions in both the time (or space) domain and the frequency domain.
In our problem, we use the Fourier transform to express the Green’s function more conveniently. By transforming the differential equations and their solutions, we can handle them more easily in the frequency domain. Mathematically, it involves integrating functions with exponential terms, which can transform complex spatial dependencies into simpler algebraic forms.
The Fourier transform essentially decomposes our function into simpler, oscillatory components, making the integral and resulting equations more manageable.

Contour Integration

Contour integration is a method used in complex analysis, a branch of mathematics dealing with functions of complex numbers. This method involves integrating a function over a path, or contour, in the complex plane.
In our exercise, we use contour integration to evaluate an integral that arises when solving for the Green's function. The integral we encounter has a complex integrand, and contour integration allows us to find its value by considering paths around singularities (points where the integrand becomes undefined or infinite).
By deforming the contour and using residue theory, a part of contour integration, we can calculate integrals more efficiently and often find exact solutions. This mathematical technique is both elegant and practical for solving integrals in quantum mechanics and other fields.