AcademyAtomic Quantum Structure
Academy
Three-Dimensional Schrodinger Equation
Level 1 - Physics topic page in Atomic Quantum Structure.
Principle
The three-dimensional Schrodinger equation applies the same energy-operator idea as the one-dimensional equation, but the kinetic-energy operator uses the full spatial curvature of the wave function.
For a time-independent potential, stationary states solve an eigenvalue equation for allowed energies and spatial wave functions.
Notation
\(\psi(\vec r,t)\)
time-dependent wave function
m^{-3/2}
\(\phi(\vec r)\)
stationary-state spatial wave function
m^{-3/2}
\(U(\vec r)\)
potential energy
\(\mathrm{J}\)
\(\nabla^2\)
three-dimensional Laplacian
\(\mathrm{m^{-2}}\)
\(\hat H\)
Hamiltonian operator
\(\mathrm{J}\)
\(E\)
energy eigenvalue
\(\mathrm{J}\)
Method
Derivation 1: Build the Hamiltonian
In three dimensions, the kinetic-energy operator uses the Laplacian.
Classical energy
\[E=\frac{p_x^2+p_y^2+p_z^2}{2m}+U(\vec r)\]
Laplacian
\[\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\]
Hamiltonian
\[\hat H=-\frac{\hbar^2}{2m}\nabla^2+U(\vec r)\]
Derivation 2: Separate stationary states
If the potential is not time dependent, energy eigenstates separate into a spatial wave function and a phase factor.
Separated state
\[\psi(\vec r,t)=\phi(\vec r)e^{-iEt/\hbar}\]
Time-independent equation
\[-\frac{\hbar^2}{2m}\nabla^2\phi+U\phi=E\phi\]
Derivation 3: Normalize probability
Probability density is integrated over volume, not just along a line.
Probability density
\[\rho(\vec r)=|\psi(\vec r,t)|^2\]
Normalization
\[\int |\psi(\vec r,t)|^2\,dV=1\]
Rules
Three-dimensional TDSE
\[i\hbar\frac{\partial\psi}{\partial t}=\left[-\frac{\hbar^2}{2m}\nabla^2+U(\vec r)\right]\psi\]
Three-dimensional TISE
\[-\frac{\hbar^2}{2m}\nabla^2\phi+U(\vec r)\phi=E\phi\]
Laplacian
\[\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\]
Normalization
\[\int|\psi|^2\,dV=1\]
Examples
Question
A free particle has
\[\phi=e^{i\vec k\cdot\vec r}\]
What does \[\nabla^2\phi\]
equal?Answer
\[\nabla^2\phi=-k^2\phi\]
so \[E=\frac{\hbar^2k^2}{2m}\]
Checks
- The Laplacian replaces the one-dimensional second derivative.
- Normalization uses a volume integral.
- Energy eigenvalues come from boundary conditions and normalizability.
- A stationary state's probability density can be time independent even though its phase changes.