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.

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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.

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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.



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