3: The Schrödinger Equation and a Particle in a Box
- Page ID
- 11780
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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
- This page covers the introduction and derivation of the Schrödinger equation in quantum mechanics, comparing it to the Bohr model and highlighting wave-particle duality. It details the three-dimensional Schrödinger equation, discussing wavefunction amplitude, potential energy dependence, and the Laplacian operator.
- 3.2: Linear Operators in Quantum Mechanics
- This page covers the role of operators in quantum mechanics, particularly the Hamiltonian, in the time-independent Schrödinger Equation. It explains how operators transform functions, the significance of commutation, and the properties of linear and hermitian operators. Integration by parts is discussed as a technique to show the equivalence of certain integral expressions.
- 3.3: The Schrödinger Equation is an Eigenvalue Problem
- This page discusses core concepts in quantum mechanics, focusing on operators, eigenvalue problems, and the Schrödinger equation. It explains the connection between quantum observables and operators, notably the Hamiltonian operator, which defines total energy. The wavefunction represents a quantum system's state, with eigenstates linked to measurable quantities. Only some wavefunctions are eigenstates, but they can be expressed as superpositions.
- 3.4: Wavefunctions Have a Probabilistic Interpretation
- This page explains the probabilistic interpretation of wavefunctions in quantum mechanics, focusing on Born's 1926 interpretation, which states that the square of a wavefunction's modulus represents probability density for locating particles. It highlights essential characteristics of valid wavefunctions, including being single-valued, finite, and continuous.
- 3.5: The Energy of a Particle in a Box is Quantized
- This page explores the particle-in-a-box model, illustrating fundamental quantum concepts like quantized energy levels and wavefunction properties. It discusses the normalized form of eigenfunctions, highlighting symmetry in probability density and increasing nodes with energy levels. The significance of orthogonality among eigenstates is emphasized, as well as characteristics of wavefunctions, such as real properties and inner products confirming orthogonality.
- 3.6: Wavefunctions Must Be Normalized
- This page explains the calculation of probabilities in quantum mechanics using wavefunctions, highlighting the importance of their absolute square as a probability density. It includes examples for one to three-dimensional scenarios and illustrates normalization to ensure total probabilities sum to one.
- 3.7: The Average Momentum of a Particle in a Box is Zero
- This page discusses expectation values in quantum and classical mechanics, focusing on how to calculate average properties like kinetic and potential energies in a particle-in-a-box model. It highlights the significance of zero-point energy and the behavior of wavefunctions. The text also addresses the position expectation value and average momentum, noting the symmetry that results in zero average momentum across states.
- 3.8: The Uncertainty Principle - Estimating Uncertainties from Wavefunctions
- This page explains Heisenberg's Uncertainty Principle, highlighting the relationship between position and momentum uncertainties, which cannot be measured simultaneously. It discusses calculations using a Gaussian wavefunction, confirming that the product of uncertainties adheres to the principle. An exercise is presented involving a particle in a one-dimensional box to further explore these concepts.
- 3.9: A Particle in a Three-Dimensional Box
- This page explores the quantum mechanics of a particle in a 3D box, applying the Time-Independent Schrödinger Equation and discussing wavefunctions expressed through quantum numbers. It examines energy levels, including non-degenerate ground states and degenerate excited states, highlighting the role of symmetry and accidental degeneracies.
- 3.E: The Schrödinger Equation and a Particle in a Box (Exercises)
- These are homework exercises to accompany Chapter 3 of McQuarrie and Simon's "Physical Chemistry: A Molecular Approach" Textmap.
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).