WOLFRAM|DEMONSTRATIONS PROJECT

Energy Levels of a Quantum Harmonic Oscillator in Second Quantization Formalism

​
set start level
0
1
2
3
raise:
†
a
ψ
lower: a ψ
reset ψ
This Demonstration shows the application of the second quantization formalism for understanding the quantized energy levels of a 1D harmonic oscillator. The raising (creation)
†
a
and lowering (destruction or annihilation)
a
operators respectively add and subtract quanta to the ground state
|0>
or any other state
|n>
. In this way one can move up and down the energy scale
E
n
=(n+1/2)ℏω
of allowed eigenvalues
n=0,1,2,…,∞
, with the eigenfunctions
|n>
given by the Hermite polynomials, since the following recursion relations hold from quantum mechanics:
†
a
n>=
1/2
(n+1)
n+1>
,
an>=
1/2
n
n-1>
, with
n≠0
and
a|0>=0
for the definition of a vacuum. All these relations can be deduced from the ground state by the relation
n>=
-1/2
n!
n
†
a
0>
.
They also obey the eigenvalue equation
N|n>=n|n>
, where
N=
†
a
a
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
H=(N+1/2)ℏω
and the raising
†
a
and lowering
a
operators are related to the position
Q
and momentum operators
P
by
a=
2
2(q+ip
) and
†
a
=
2
2(q-ip
), with
q=
1/2
ℏ
mω
Q
and
p=
1/2
(mℏω)
P
. The raising
†
a
and lowering
a
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,
n=0
gives the ground state energy) and clicking the corresponding buttons, "raise:
†
a
ψ
" and "lower:
aψ
". To go back to the beginning, click the "reset
ψ
" button. When you reach the vacuum state,
a|0>=0
, annihilating the state.