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
.
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.
Permanent Citation