9887

Energy Levels of a Quantum Harmonic Oscillator in Second Quantization Formalism

This Demonstration shows the application of the second quantization formalism for understanding the quantized energy levels of a 1D harmonic oscillator. The raising (creation) and lowering (destruction or annihilation) operators respectively add and subtract quanta to the ground state or any other state . In this way one can move up and down the energy scale of allowed eigenvalues , with the eigenfunctions given by the Hermite polynomials, since the following recursion relations hold from quantum mechanics: , , with and for the definition of a vacuum. All these relations can be deduced from the ground state by the relation .
They also obey the eigenvalue equation , where is the number operator that gives the number of quanta added to the ground state (GS). The Hamiltonian for the harmonic oscillator is given by and the raising and lowering operators are related to the position and momentum operators by ) and ), with and . The raising and lowering operators are also called ladder operators, because they move up and down the equally spaced energy levels as if on a ladder.
In this Demonstration you can do this by setting the slider to a particular starting energy level (by default, gives the ground state energy) and clicking the corresponding buttons, "raise: " and "lower: ". To go back to the beginning, click the "reset " button. When you reach the vacuum state, , annihilating the state.

THINGS TO TRY

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

Snapshot 1: ground state (GS) of the harmonic oscillator: starting and current energy set at the same level, zero quanta added to GS
Snapshot 2: starting energy and current energy set at ; two quanta added to the GS
Snapshot 3: starting energy set at and raising operator button clicked; reached state
A. Messiah, "The Harmonic Oscillator," Quantum Mechanics, New York: Dover Publications, 1999 pp. 432-461.
J. M. Feagin, Quantum Methods with Mathematica, New York: Springer–Verlag, 2002.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+