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The excited state n ought to be the ground state acted on by n creation
| i
operators,or writing a recursive definition
a† n =ξ(n)n+1 , for n 0, (4.52)
| i | i ≥
where ξ(n) is an asyet undetermined normalization. This equationagreeswith
the previous formula for 1 when n=0 if we demand ξ(0)=1.
| i
On the other hand, acting with the annihilation operator on the state n
| i
should yield the state n 1
| − i
an =η(n)n 1 . (4.53)
| i | − i
A careful analysis yields the coefficients ξ(n) = √n+1 and η(n) = √n, so
that we have
a n =√n+1 n+1 (4.54)
†
| i | i
an =√n n 1 . (4.55)
| i | − i
Equation (4.54) record the action of a as a raising operator, its action on
†
a state is to increase the quantum number n by 1. Likewise, Equation (4.55)
recordthe actionofa asalowering operator,its actionona stateis to decrease
the quantum number n by 1.
The name creation operator originates in the quantum theory of the elec-
tromagnetic field. The frequency modes of an electromagnetic field can be
described by harmonic oscillators. In a quantum description of the electromag-
netic field, a mode with frequency f is corresponds to a harmonic oscillator
with ω =2πf. The intensity of the field corresponds to the number of photons
in the mode, and the number of photons is precisely the quantum number n
of the harmonic oscillator. As will be shown below, the energy in the mode is
¯hω(n+ 1). In this context, the action of the creation operator a is to create a
2 †
newphotoninthe frequencymode. Correspondingly,the actionofthe lowering
or annihilationoperatora is to decreasethe quantumnumber n or annihilate a
photon in the mode.
Next, it follows that a a is a number operator, counting the excitation level
†
of the state. Using the equations (4.54) and (4.55) we get
a†an =a†√nn 1 =√na† n 1 =√n√nn =nn
| i | − i | − i | i | i
In these calculations we are using the fact that numbers commutes with
operators,i.e. the order can be interchanged freely.
85
Sometimes the number operator is denoted by N, writing N = a a. Not
†
surprisingly, the states n are eigenstates of the number operator, or
| i
N n =nn . (4.56)
| i | i
The states n are furthermore eigenstates of the Hamiltonian since it can
| i
be written in terms of the number operator as H =¯hω(N + 1),
2
1 1
H n =¯hω(N + )n =h¯ω(n+ )n . (4.57)
| i 2 | i 2 | i
Fromthisequationwecanreadoftheenergyspectrumofthelinearharmonic
oscillator,
1
E =¯hω(n+ ). (4.58)
n
2
We havethe followingsetofformulas,summarizingthis algebraictreatment
of the harmonic oscillator
[a,a ]=1
†
a0 =0
| i
a n|n == √√ nn n+1 1|n+1 i.
a†
i
| i | − i
a an =nn
 †
| i | i
Admittedly there is quite a lot of handwaving going into this ’derivation’ of
the harmonic oscillator spectrum, but this has more to do with the pedestrian
approach of this section, than to the method as such. This algebraic approach
to solving the harmonic oscillator can be made more rigorous. But it is clearly
moreabstractthanthewave-functionapproach. Thereadermightwonderwhat
is the actual content of an abstract equation such as a n =√n+1 n .
†
| i | i
One, computer science oriented, way of thinking of equations like these is
to regard them purely syntactically. Then, wherever we see the combination of
symbols a n we are allowedto replace them by √n+1 n . Of course,quite a
†
| i | i
few more rules are needed in order to ’calculate’ syntactically with this model,
butinprinciple,thewholetheorycanbephrasedentirelyabstractlyintermsof
formalsyntacticrules. Semanticscanbeaddedtothemodelbyinterpretingthe
operatorsandthestates. Onsuchinterpretionisintermsofconfigurationspace,
derivativesandwavefunctions. Anotheroneisintermsof(infinitedimensional)
matricesandvectors. Thiscanalsoberegardedasmakingadistinctionbetween
interfaceandimplementation. Seeninthisway,theequationa n =√n+1 n
†