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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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.

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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.



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