Particle-in-a-Box Hamiltonian

Hamiltonian

For a particle on $0<x<L$ with infinite walls,

$$\hat H = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2},$$

with boundary conditions

$$\psi(0)=\psi(L)=0.$$

The energy spectrum is

$$E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}, \qquad n=1,2,3,\ldots$$

Assumptions

• One-dimensional nonrelativistic particle.
• Infinite hard-wall confinement at $x=0$ and $x=L$.
• The wavefunction vanishes at the walls.
• No potential inside the interval.

Common Mistakes

• Reusing the free-particle continuum spectrum after imposing walls.
• Starting the quantum number at $n=0$ for the infinite well.
• Forgetting that changing the interval shifts phases and eigenfunctions but not the energy scale.
• Treating finite barriers as infinite without checking penetration corrections.

Canonical Links

• Infinite Square Well
• Boundary Conditions
• Particle-in-a-Box Spectrum
• Free Particle Hamiltonian

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.