# Energy Levels of a Quantum Harmonic Oscillator in Second Quantization Formalism

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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 .

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Contributed by: Jessica Alfonsi (University of Padova, Italy) (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## 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.