Schrödinger Equation

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

In this chapter, we introduce the Schrödinger equation, the most fundamental equation in quantum mechanics. Given a Hamiltonian ${\mathcal {H}}$, this equation describes how the wave function evolves in time. As a simple example of this equation, let us consider a single particle in a potential $V({\textbf {r}},t)$. The Hamiltonian for this system is

${\mathcal {H}}={\frac {p^{2}}{2m}}+V({\textbf {r}},t).$

To obtain the corresponding Schrödinger equation, we make the replacements, ${\textbf {p}}\rightarrow {\frac {\hbar }{i}}\nabla$ and $H\rightarrow i\hbar {\frac {\partial }{\partial t}}$. This turns both sides of the above equation into operators, which, when applied to the wave function $\Psi ({\textbf {r}},t)$, yields the Schrödinger equation,

$-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi +V({\textbf {r}},t)\Psi =i\hbar {\frac {\partial \Psi }{\partial t}}.$

If the potential does not depend on time, then the solution can be written in the form,

$\Psi ({\textbf {r}},t)=e^{-iEt/\hbar }\psi ({\textbf {r}}),$

where $\psi ({\textbf {r}})$ satisfies the time-independent Schrödinger equation,

$-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +V({\textbf {r}})\psi =E\psi .$

As stated before, the wave function is a measure of the probability that a particle is in a given state; in fact, the probability density of finding a particle at a position ${\textbf {r}}$ at a given time $t$ is $|\Psi ({\textbf {r}},t)|^{2}$, or $|\psi ({\textbf {r}})|^{2}$ for the time-independent case. Because of this, the wave function must be normalized such that

$\int d^{3}{\textbf {r}}\,|\Psi ({\textbf {r}},t)|^{2}=1,$

or

$\int d^{3}{\textbf {r}}\,|\psi ({\textbf {r}})|^{2}=1$

in the time-independent case. We will also find that the Schrödinger equation respects conservation of probability, as expected.

In Dirac bra-ket notation, the time-dependent Schrödinger equation is

$i\hbar {\frac {\partial }{\partial t}}|\Psi (t)\rangle ={\mathcal {H}}|\Psi (t)\rangle ,$

while the time-independent equation is

${\mathcal {H}}|\psi \rangle =E|\psi \rangle .$

The single-particle case discussed before may be obtained from these more general equations by projecting them into position space.

Chapter Contents

• Original Idea of Schrödinger Equation
• Brief Derivation of Schrödinger Equation
• Stationary States
• Conservation of Probability
• States, Dirac Bra-Ket Notation
• Heisenberg Uncertainty Relations
• Some Consequences of the Uncertainty Principle