Ladder Operators for the Harmonic Oscillator

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

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

[more]

In this Demonstration, the eigenfunction is plotted in black. Also shown is either in red or in blue.

[less]

Contributed by: S. M. Blinder (March 2011)
Open content licensed under CC BY-NC-SA


Snapshots


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.



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send