Numerical Solutions to the Time-Independent 1-D Schr ̈odinger Equation
A1A
Abstract
Using the Numerov algorithm, the Schr ̈odinger equation was solved for the square well, harmonic and
linear potentials. The wavefunctions were integrated using the Numerov algorithm, which required an
initial trial energy, so a function was written to determine the eigenstates of the potential, allowing the
numerical solutions to be obtained without any knowledge of the analytic solutions. The uncertainty
relation was verified in the case of the square well and harmonic potentials, and it was observed that forlarge eigenstates, the harmonic potential behaves like the square well potential. The matrix Numerov algorithm was then used to solve the Schr ̈odinger equation in a much more elegant manner.
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