How to Solve the Particle in a Box Problem

In quantum mechanics, the particle in a box is a conceptually simple problem in position space that illustrates the quantum nature of particles by only allowing discrete values of energy. In this problem, we start from the Schrödinger equation, find the energy eigenvalues, and proceed to impose normalization conditions to derive the eigenfunctions associated with those energy levels.


  1. Begin with the time-independent Schrödinger equation. The Schrödinger equation is one of the fundamental equations in quantum mechanics that describes how quantum states evolve in time. The time-independent equation is an eigenvalue equation, and thus, only certain eigenvalues of energy exist as solutions.
  2. Substitute the Hamiltonian of a free particle into the Schrödinger equation.
    • In the one-dimensional particle in a box scenario, the Hamiltonian is given by the following expression. This is familiar from classical mechanics as the sum of the kinetic and potential energies, but in quantum mechanics, we assume that position and momentum are operators.
    • In position space, the momentum operator is given by
    • Meanwhile, we let
    • Rearranging terms and defining a constant
  3. Solve the above equation. This equation is familiar from classical mechanics as the equation describing simple harmonic motion.
    • The theory of differential equations tells us that the general solution to the above equation is of the following form, where
    • Of course, the solution is valid only up to an overall phase, which does change with time, but phase changes do not affect any of our observables, including energy. Therefore, for our purposes, we will write the wavefunction as only varying with position
  4. Impose boundary conditions. Remember that
  5. Take the determinant of the matrix and evaluate. In order for the above homogeneous equation to have nontrivial solutions, the determinant must vanish. This is a standard result from linear algebra. If you are not familiar with this background, you may treat this as a theorem.
    • The sine function is 0 only when its argument is an integer multiple of
    • Recall that
    • These are the energy eigenvalues of the particle in a box. Because
    • The energy of the particle can only take on positive values, even at rest. The ground-state energy
    • We will now proceed to derive the energy eigenfunctions.
  6. Write out the wavefunction with the unknown constant. We know from the constraint of the wavefunction at
  7. Normalize the wavefunction. Normalizing will determine the constant
  8. Arrive at the wavefunction. This is the description of a particle inside a box, surrounded by infinite potential energy walls. While


  • When normalizing, substituting an appropriate integer for
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