Summary of One-Dimensional Systems

Quantum Mechanics A

Schrödinger Equation The most fundamental equation of quantum mechanics; given a Hamiltonian ${\mathcal {H}}$, it describes how a state $|\Psi \rangle$ evolves in time.

Physical Basis of Quantum Mechanics

Basic Concepts and Theory of Motion UV Catastrophe (Black-Body Radiation) Photoelectric Effect Stability of Matter Double Slit Experiment Stern-Gerlach Experiment The Principle of Complementarity The Correspondence Principle The Philosophy of Quantum Theory

Schrödinger Equation

Brief Derivation of Schrödinger Equation Relation Between the Wave Function and Probability Density Stationary States Heisenberg Uncertainty Principle Some Consequences of the Uncertainty Principle

Operators, Eigenfunctions, and Symmetry

Linear Vector Spaces and Operators Commutation Relations and Simultaneous Eigenvalues The Schrödinger Equation in Dirac Notation Transformations of Operators and Symmetry Time Evolution of Expectation Values and Ehrenfest's Theorem

Motion in One Dimension

One-Dimensional Bound States Oscillation Theorem The Dirac Delta Function Potential Scattering States, Transmission and Reflection Motion in a Periodic Potential Summary of One-Dimensional Systems

Discrete Eigenvalues and Bound States

Harmonic Oscillator Spectrum and Eigenstates Analytical Method for Solving the Simple Harmonic Oscillator Coherent States Charged Particles in an Electromagnetic Field WKB Approximation

Time Evolution and the Pictures of Quantum Mechanics

The Heisenberg Picture: Equations of Motion for Operators The Interaction Picture The Virial Theorem

Angular Momentum

Commutation Relations Angular Momentum as a Generator of Rotations in 3D Spherical Coordinates Eigenvalue Quantization Orbital Angular Momentum Eigenfunctions

Central Forces

General Formalism Free Particle in Spherical Coordinates Spherical Well Isotropic Harmonic Oscillator Hydrogen Atom WKB in Spherical Coordinates

The Path Integral Formulation of Quantum Mechanics

Feynman Path Integrals The Free-Particle Propagator Propagator for the Harmonic Oscillator

Continuous Eigenvalues and Collision Theory

Differential Cross Section and the Green's Function Formulation of Scattering Central Potential Scattering and Phase Shifts Coulomb Potential Scattering

1. Zero Potential

Physical Example: Proton in beam from cyclotron

Significant Feature: Results used for other systems

2. Step Potential (energy below top)

Physical Example: Conduction electron near surface of metal

Significant Feature: Penetration of excluded region

3. Step Potential (energy above top)

Physical Example: Neutron trying to escape nucleus

Significant Feature: Partial reflection at potential discontinuity

4. Barrier Potential (energy below top)

Physical Example: $\alpha \!$ particle trying to escape Coulomb barrier

Significant Feature: Tunneling

5. Barrier Potential (energy above top)

Physical Example: Electron scattering from negatively ionized atom

Significant Feature: No reflection at certain energies

6. Infinite Square Well Potential

Physical Example: Molecule strictly confined to box

Significant Feature: Approximation to finite square well

7. Finite Square Well Potential

Physical Example: Neutron bound in nucleus

Significant Feature: Energy quantization

8. Periodic Potential

Physical Example: Electron in lattice

Significant Feature: Energy band and energy gap

Problem

An electron is moving freely inside of a one-dimensional box with walls at $x=0\!$ and $x=a.\!$ If the electron is initially in the ground state of the box and we suddenly increase the size of the box by moving the right-hand wall instantaneously from $x=a\!$ to $x=4a,\!$ then calculate the probability of finding the electron in

(a) the ground state of the new box, and

(b) the first excited state of the new box.

Solution