# Bound-State Solutions of the Schrödinger Equation by Numerical Integration

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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.

Contributed by: Andrés Santos
(December 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

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.

## Permanent Citation