# 3: The Schrödinger Equation and the Particle-in-a-Box Model

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- 92364

The particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems. In classical systems, for example a ball trapped inside a large box, the particle can move at any speed within the box and it is no more likely to be found at one position than another. However, when the well becomes very narrow (on the scale of a few nanometers), quantum effects become important. The particle may only occupy certain positive energy levels. The particle in a box model provides one of the very few problems in quantum mechanics which can be solved analytically, without approximations. This means that the observable properties of the particle (such as its energy and position) are related to the mass of the particle and the width of the well by simple mathematical expressions. Due to its simplicity, the model allows insight into quantum effects without the need for complicated mathematics. It is one of the first quantum mechanics problems taught in undergraduate physics courses, and it is commonly used as an approximation for more complicated quantum systems.

- 3.1: The Schrödinger Equation
- Erwin Schrödinger posited an equation that predicts both the allowed energies of a system as well as address the wave-particle duality of matter. Schrödinger equation for de Broglie's matter waves cannot be derived from some other principle since it constitutes a fundamental law of nature. Its correctness can be judged only by its subsequent agreement with observed phenomena (a posteriori proof).

- 3.2: Linear Operators in Quantum Mechanics
- An operator is a generalization of the concept of a function. Whereas a function is a rule for turning one number into another, an operator is a rule for turning one function into another.

- 3.3: The Schrödinger Equation is an Eigenvalue Problem
- To every dynamical variable \(a\) in quantum mechanics, there corresponds an eigenvalue equation, usually written \[\hat{A}\psi=a\psi\label{3.3.2}\] The \(a\) eigenvalues represents the possible measured values of the \(A\) operator.

- 3.4: The Quantum Mechanical Free Particle
- The simplest system in quantum mechanics has the potential energy V=0 everywhere. This is called a free particle since it has no forces acting on it. We consider the one-dimensional case, with motion only in the x-direction. We discuss that the wavefunction can be a linear combination of eigenfunctions and wavepackets can be constructed of eigenstates to generate a localized particle picture that a single eigenstate does not posess.

- 3.5: Wavefunctions Have a Probabilistic Interpretation
- The most commonly accepted interpretation of the wavefunction that the square of the module is proportional to the probability density (probability per unit volume) that the electron is in the volume dτ located at r. Since the wavefunction represents the wave properties of matter, the probability amplitude P(x,t) will also exhibit wave-like behavior.

- 3.6: The Energy of a Particle in a Box is Quantized
- The particle in the box model system is the simplest non-trivial application of the Schrödinger equation, but one which illustrates many of the fundamental concepts of quantum mechanics.

- 3.7: Wavefunctions Must Be Normalized
- The probability of a measurement of \(x\) yielding a result between \(-\infty\) and \(+\infty\) is 1. So wavefunctions should be normalized if possible.

- 3.8: The Average Momentum of a Particle in a Box is Zero
- From the mathematical expressions for the wavefunctions and energies for the particle-in-a-box, we can answer a number of interesting questions. Key to addressing these questions is the formulation and use of expectation values. This is demonstrated in the module and used in the context of evaluating average properties (energy, position, and momentum of the particle in a box).

- 3.9: The Uncertainty Principle Redux - Estimating Uncertainties from Wavefunctions
- The operators x and p are not compatible and there is no measurement that can precisely determine both x and p simultaneously. The uncertainty principle is a consequence of the wave property of matter. A wave has some finite extent in space and generally is not localized at a point. Consequently there usually is significant uncertainty in the position of a quantum particle in space.

- 3.10: Particle in a Finite Box and Tunneling (Optional)
- The quantum effects of tunneling are introduced within the context of a particle in a 1D box with finite height walls.

- 3.11: A Particle in a Two-Dimensional Box
- A particle in a 2-dimensional box is a fundamental quantum mechanical approximation describing the translational motion of a single particle confined inside an infinitely deep well from which it cannot escape.

- 3.12: A Particle in a Three-Dimensional Box
- The 1D particle in the box problem can be expanded to consider a particle within a 3D box for three lengths \(a\), \(b\), and \(c\). When there is NO FORCE (i.e., no potential) acting on the particles inside the box. Motion and hence quantization properties of each dimension is independent of the other dimensions. This Module introduces the concept of degeneracy where multiple wavefunctions (different quantum numbers) have the same energy.

- 3.E: The Schrödinger Equation and a Particle in a Box (Exercises)
- These are homework exercises to accompany the chapter.

Thumbnail: The quantum wavefunction of a particle in a 2D infinite potential well of dimensions \(L_x\) and \(L_y\). The wavenumbers are \(n_x=2\) and \(n_y=2\). (Public Domain; Inductiveload).