Notes on Cambridge Maths

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Harmonic Oscillator in QM

Schrodinger Equation with
TDSE:

TISE:

Introduce and .
(NOTE: )
So we write:

Now look for solutions of form which gives:

Now look for powerseries solution which gives

The series converges iff the recurrence terminates, hence eigenvalues are or . Eigenfunctions are Hermite polynomials:

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