Ladder Operators for the Harmonic Oscillator

The Hamiltonian for the linear harmonic oscillator can be written , in units with . The eigenstates are given by , , , where is a Hermite polynomial. An alternative reformulation of the problem can be based on the representation in terms of ladder operators and . The step-down or annihilation operator acts on the eigenfunctions according to , with . The step-up or creation operator satisfies
In this Demonstration, the eigenfunction is plotted in black. Also shown is either in red or in blue.

THINGS TO TRY

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
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DETAILS

Snapshot 1: annihilates the ground state:
Snapshots 2 and 3: raises the states and by one level
Reference: S. M. Blinder, Introduction to Quantum Mechanics, Amsterdam: Elsevier, 2004 pp. 66–68.
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