Question

Normalize the total energy eigenfunction for the rectangular two-dimensional box, \\[ \psi_{n_{n} n_{y}}(x, y)=N \sin \left(\frac{n_{x} \pi x}{a}\right) \sin \left(\frac{n_{y} \pi y}{b}\right) \\] in the interval \(0 \leq x \leq a, 0 \leq y \leq b\)

Short answer

The normalized wavefunction for the given rectangular two-dimensional box is: \[ \psi_{n_x n_y}(x, y) = \frac{2}{\sqrt{ab}} \sin\left( \frac{n_x \pi x}{a} \right) \sin\left( \frac{n_y \pi y}{b} \right) \]

Step-by-step solution

Square the wavefunction and multiply by normalization constant squared

First, we need to square the given wavefunction and multiply it with the normalization constant squared, N^2: \[ \left| \psi_{n_x n_y}(x, y) \right|^2 = N^2 \sin^2\left(\frac{n_x \pi x}{a}\right) \sin^2\left(\frac{n_y \pi y}{b}\right) \]

Integrate the squared wavefunction

Now, we integrate the squared wavefunction with respect to x from \(0\) to \(a\), and with respect to y from \(0\) to \(b\): \[ \int_{0}^{a} \int_{0}^{b} N^2 \sin^2\left(\frac{n_x \pi x}{a}\right) \sin^2\left(\frac{n_y \pi y}{b}\right) dx dy \] Let's solve the integral: \[ N^2 \int_{0}^{a} \sin^2\left(\frac{n_x \pi x}{a}\right) dx \int_{0}^{b} \sin^2\left(\frac{n_y \pi y}{b}\right) dy \] Using the following integral: \[ \int_{0}^{L} \sin^2\left(\frac{m \pi x}{L}\right)dx = \frac{L}{2} \] We obtain: \[ N^2 \frac{a}{2} \frac{b}{2} = N^2 \frac{ab}{4} \]

Solve for the normalization constant N

Now, we will set the previous result equal to 1 and solve for N: \[ N^2 \frac{ab}{4} = 1 \] \[ N^2 = \frac{4}{ab} \] Taking the square root of both sides, we find the normalization constant, N: \[ N = \frac{2}{\sqrt{ab}} \] #Solution# The normalized wavefunction for the given rectangular two-dimensional box is: \[ \psi_{n_x n_y}(x, y) = \frac{2}{\sqrt{ab}} \sin\left( \frac{n_x \pi x}{a} \right) \sin\left( \frac{n_y \pi y}{b} \right) \]

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the scale of atoms and subatomic particles. It is a mathematical framework for the dual particle-wave nature of these particles, where the behavior of quantum systems is fundamentally different from classical systems due to the quantum nature of the matter. One of the core components of quantum mechanics is the quantization of energy, momentum and angular momentum, and the discretization of certain physical properties.

Quantum mechanics often seems counterintuitive, challenging our classical notions of reality. However, it's indispensable for understanding how the universe works on the smallest scales and has led to technologies such as semiconductors, lasers, and MRI machines.

Wavefunction

The wavefunction is a central concept in quantum mechanics representing the quantum state of a particle or system of particles. It is a mathematical function that encodes the probabilities of the outcomes of measurements made on the system, such as the probability of finding a particle in a certain position or with a certain momentum.

The properties of wavefunctions are governed by Schrödinger's equation. Importantly, wavefunctions must be normalized, meaning that their squared modulus, representing probability density, must integrate to one across all space. This is to ensure that the probability of finding the particle somewhere in space is certain.

Rectangular Two-Dimensional Box

In quantum mechanics, a rectangular two-dimensional box is a theoretical model used to simplify the complex reality of particle confinement. This model is a part of the 'particle in a box' problems, which are idealized systems where particles are free to move within a defined area or volume but cannot escape it - a concept that parallels the behavior of electrons in quantum wells.

The boundaries of the box are considered to be infinitely high potential walls, which the particles cannot penetrate. Solutions to this model involve wavefunctions that satisfy the model's boundary conditions - zero at the walls and continuous inside the box. They form standing waves, displaying the characteristic quantization of energy levels within the box.

Eigenfunctions

Eigenfunctions are a key concept when dealing with operators in quantum mechanics. An eigenfunction of an operator is a non-zero function that, when the operator is applied to it, results in a scalar multiple of the original function. This scalar is known as the eigenvalue and is often associated with a measurable physical property, such as energy or momentum.

In the context of the rectangular two-dimensional box problem, the eigenfunctions describe the permissible wavefunctions that satisfy both the Schrödinger equation and the boundary conditions of the system. These wavefunctions can only take specific forms that lead to quantized energy levels, and each wavefunction corresponds to a particular energy state of the particle trapped within the box.