AcademyQuantum Wave Functions
Academy
One-Dimensional Schrodinger Equation
Level 1 - Physics topic page in Quantum Wave Functions.
Principle
The Schrodinger equation evolves the wave function by applying the total-energy operator to it.
Notation
\(\hbar\)
reduced Planck constant
\(\mathrm{J\,s}\)
\(m\)
particle mass
\(\mathrm{kg}\)
\(V(x)\)
potential energy function
\(\mathrm{J}\)
\(\hat H\)
Hamiltonian operator
\(\mathrm{J}\)
\(E\)
energy eigenvalue
\(\mathrm{J}\)
\(\psi(x,t)\)
time-dependent wave function
m^{-1/2}
Method
Derivation 1: Energy operator
In one dimension, total energy is kinetic plus potential energy.
Classical energy form
\[E=\frac{p^2}{2m}+V(x)\]
Momentum operator
\[\hat p=-i\hbar\frac{\partial}{\partial x}\]
Hamiltonian
\[\hat H=-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+V(x)\]
Derivation 2: Time evolution
The Hamiltonian generates time evolution of the state.
Time-dependent equation
\[i\hbar\frac{\partial\psi}{\partial t}=\hat H\psi\]
Full one-dimensional form
\[i\hbar\frac{\partial\psi}{\partial t}= -\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}+V(x)\psi\]
Derivation 3: Stationary states
When \(V\) has no time dependence, energy eigenstates separate into space and time factors.
Separated form
\[\psi(x,t)=\phi(x)e^{-iEt/\hbar}\]
Time-independent equation
\[-\frac{\hbar^2}{2m}\frac{d^2\phi}{dx^2}+V(x)\phi=E\phi\]
Rules
These equations use a nonrelativistic particle in one spatial dimension.
Time-dependent Schrodinger equation
\[i\hbar\frac{\partial\psi}{\partial t}=\left[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+V(x)\right]\psi\]
Time-independent Schrodinger equation
\[-\frac{\hbar^2}{2m}\frac{d^2\phi}{dx^2}+V(x)\phi=E\phi\]
Stationary-state time factor
\[\psi(x,t)=\phi(x)e^{-iEt/\hbar}\]
Examples
Question
If
\[V=0\]
what spatial equation does a stationary state satisfy?Answer
It satisfies
\[d^2\phi/dx^2=-(2mE/\hbar^2)\phi\]
so free-particle stationary states are sinusoidal or complex exponential.Checks
- The kinetic-energy operator contains a second spatial derivative.
- The equation is linear, so superpositions of solutions are also solutions.
- Energy eigenvalues come from boundary conditions and normalizability.
- A stationary state may have time-dependent phase without time-dependent density.