AcademyQuantum Wave Functions
Academy
Particle in a Box
Level 1 - Physics topic page in Quantum Wave Functions.
Principle
Hard-wall boundary conditions force standing waves, so a confined particle has discrete energies.
Notation
\(L\)
box width
\(\mathrm{m}\)
\(n\)
positive integer quantum number
1
\(k_n\)
allowed wave number
\(\mathrm{m^{-1}}\)
\(\lambda_n\)
allowed de Broglie wavelength
\(\mathrm{m}\)
\(\psi_n\)
nth normalized spatial eigenfunction
m^{-1/2}
\(E_n\)
nth energy eigenvalue
\(\mathrm{J}\)
Method
Derivation 1: Boundary conditions
For an infinite square well, the wave function must vanish at both walls.
General sinusoidal form
\[\psi(x)=A\sin kx+B\cos kx\]
Left wall
\[\psi(0)=0\Rightarrow B=0\]
Right wall
\[\psi(L)=0\Rightarrow \sin(kL)=0\]
Derivation 2: Quantization
The right-wall condition selects only integer half-wavelengths.
Allowed wave number
\[k_n=\frac{n\pi}{L}\]
Allowed wavelength
\[\lambda_n=\frac{2L}{n}\]
Derivation 3: Energy levels
Kinetic energy is set by the allowed momentum magnitude.
Momentum magnitude
\[p_n=\hbar k_n\]
Energy
\[E_n=\frac{p_n^2}{2m}=\frac{n^2\pi^2\hbar^2}{2mL^2}\]
Rules
For a particle in an infinite well from \(0\) to \(L\):
Eigenfunctions
\[\psi_n(x)=\sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)\]
Energy levels
\[E_n=\frac{n^2\pi^2\hbar^2}{2mL^2}\]
Center expectation
\[\langle x\rangle=\frac{L}{2}\]
Examples
Question
What is the ratio
\[E_3/E_1\]
for an infinite square well?Answer
Because
\[E_n\propto n^2\]
\[E_3/E_1=9\]
Checks
- The quantum number starts at \(n=1\), not \(n=0\).
- The ground-state energy is nonzero because confinement forces curvature.
- Larger boxes have more closely spaced energy levels.
- The probability density is symmetric about the center for every stationary state.