Bound-State Solutions of the Schrödinger Equation by Numerical Integration

This Demonstration shows the mathematical solution of the time-independent Schrödinger equation for four potentials, the harmonic oscillator , the V-shaped potential , the anharmonic oscillator , and a square-well potential. The wavefunction is arbitrarily fixed at . You can obtain linearly independent solutions by numerical integration for different values of the derivative and the energy level. The vertical dashed lines indicate the locations of the classical turning points.

For any value of ,it is always possible to tune in a value of such that goes to zero either in the limit as or in the limit as . The goal is to find the discrete values of (the eigenvalues of the Hamiltonian) such that an acceptable solution (the eigenfunction) exists that goes to zero for both limits and .

For instance, the eigenvalues of the harmonic oscillator are , , , , …., while the first five eigenvalues corresponding to the V-shaped potential are , , , , and .

Units are such that , , and the proportionality coefficients in the potential functions are as indicated.

This Demonstration is inspired by section 5.7 of [1].

Snapshot 1: Harmonic oscillator with . The particular solution with behaves well in the limit as but not in the limit as . Therefore, is not an eigenvalue.

Snapshot 2: Harmonic oscillator with . The particular solution with behaves well in the limit as but not in the limit as . Therefore, is not an eigenvalue.

Snapshot 3: Harmonic oscillator with . The particular solution with behaves well in both limits as and . Therefore, is an eigenvalue (first excited state).

Snapshot 4: V-shaped potential with . The particular solution with behaves well in both limits as and . Therefore, is an eigenvalue (ground state).

Snapshot 5: V-shaped potential with . The particular solution with behaves well in both limits as and . Therefore, is an eigenvalue (fourth excited state).

Reference

[1] R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, New York: Wiley, 1985.