Brief Derivation of 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

Imagine a particle constrained to move along the $x$-axis, subject to a potential energy $V(x,t)\!$. Classically, we could model this system by writing down its Hamiltonian $H,\!$ given by

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

We then employ Hamilton's equations of motion,

${\dot {p}}=-{\frac {\partial H}{\partial x}},\,{\dot {x}}={\frac {\partial H}{\partial p}},$

where a dot denotes a time derivative, to determine the motion of the particle. Applying these equations to the above Hamiltonian, we can recover Newton's second law,

$m{\ddot {a}}=-{\frac {\partial V}{\partial x}}=F.$

Now by applying the appropriate initial conditions for our particle, we obtain a solution for the trajectory of the particle. As we will see, the above relation is only an approximation to actual physical reality. As we attempt to describe increasingly smaller objects, we enter the quantum mechanical regime, where we can no longer neglect the particles' wave properties. Allowing $\displaystyle {p\rightarrow {\frac {\hbar }{i}}{\frac {\partial }{\partial x}}}$ and $\displaystyle {H\rightarrow i\hbar {\frac {\partial }{\partial t}}},$ we can use the Hamiltonian for a classical particle above to find an equation that describes this wave nature. We find that the wave function $\Psi (x,t)\!$ satisfies the Schrödinger equation for a scalar potential $V(x,t)\!$ in one dimension:

$i\hbar {\frac {\partial }{\partial t}}\Psi (x,t)=\left[-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+V(x,t)\right]\Psi (x,t)$

A similar equation may be derived in multiple dimensions:

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

Given a solution which satisfies the above Schrödinger equation, quantum mechanics provides a mathematical description of the laws obeyed by the probability amplitudes associated with quantum motion.

We can also generalize the Schrödinger equation to a system which contains $N\!$ particles. We assume that the wave function is $\Psi ({\textbf {r}}_{1},{\textbf {r}}_{2},\ldots ,{\textbf {r}}_{N},t)$ and that the Hamiltonian of the system can be expressed as

$H=\sum _{k=1}^{N}{\frac {{\textbf {p}}_{k}^{2}}{2m_{k}}}+V({\textbf {r}}_{1},{\textbf {r}}_{2},\ldots ,{\textbf {r}}_{N}).$

Making a similar replacement as before, we may obtain the Schrödinger equation for a many-particle system:

$i\hbar {\frac {\partial }{\partial t}}\Psi ({\textbf {r}}_{1},{\textbf {r}}_{2},\ldots ,{\textbf {r}}_{N},t)=\left[\sum _{k=1}^{N}{\frac {\hbar ^{2}}{2m_{k}}}\nabla _{k}^{2}+V({\textbf {r}}_{1},{\textbf {r}}_{2},\ldots ,{\textbf {r}}_{N})\right]\Psi ({\textbf {r}}_{1},{\textbf {r}}_{2},\ldots ,{\textbf {r}}_{N},t)$

Here, $\nabla _{k}^{2}$ is the Lapalacian operator that acts on the coordinates of particle $k.\!$