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Energy Levels of a Quantum Harmonic Oscillator in Second Quantization Formalism
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 =(n+1/2)ℏω of allowed eigenvalues , with the eigenfunctions given by the Hermite polynomials, since the following recursion relations hold from quantum mechanics: n>=n+1>, , with and for the definition of a vacuum. All these relations can be deduced from the ground state by the relation .
†
a
a
|0>
|n>
E
n
n=0,1,2,…,∞
|n>
†
a
1/2
(n+1)
an>=n-1>
1/2
n
n≠0
a|0>=0
n>=0>
-1/2
n!
n
†
a
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.
N|n>=n|n>
N=a
†
a
H=(N+1/2)ℏω
†
a
a
Q
P
a=
2
2(q+ip†
a
2
2(q-ipq=Q
1/2
ℏ
mω
p=P
1/2
(mℏω)
†
a
a
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: ψ" and "lower: ". To go back to the beginning, click the "reset " button. When you reach the vacuum state, , annihilating the state.
n=0
†
a
aψ
ψ
a|0>=0