Chapter 8
Raising and lowering
operators
This is an introduction to raising and lowering operators, also known
creation and annihilation operators. The idea and the usefulness of
them will be demonstrated with the harmonic oscillator as an example, but the method is far more general.
We begin with the Hamiltonian of the harmonic oscillator,
p2
1
+ mω 2 x2 ,
(8.1)
2m 2
where all the symbols have their usual meaning. Inspired by the classical identity
r
r
1
1
mω
mω
x
,
(8.2)
H=ω x
− ip √
+ ip √
2
2
2mω
2mω
H=
we will calculate the quantum mechanical version of the right hand side of this
equation. It differs from the classical for two reason. First it is an operator
that operates on a state, instead of being a function. Secondly, the position and
momentum operators do not commute. We have
r
r
1
1
mω
mω
~ω x
− ip √
+ ip √
x
=
(8.3)
2~
2~
2m~ω
2m~ω
p2
1
iω
1
=
+ mω 2 x2 + [x, p] = H − ~ω,
2m 2
2
2
because [x, p] = i~. We now introduce the notation
r
1
mω
a† = x
− ip √
(8.4)
2~
2m~ω
r
1
mω
+ ip √
(8.5)
a=x
2~
2m~ω
23
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CHAPTER 8. RAISING AND LOWERING OPERATORS
and we can therefore write the Hamiltonian as
1
.
H = ~ω a† a +
2
(8.6)
Let’s consider some of the properties of a and a† . Both of the operators x and
p are hermitian and the conjugate of a is therefore found by complex conjugation
of the complex unit i. That gives (a)† = a† , which looks almost trivial but this is
because we have chosen the names cleverly from the start. Note that neither of
the operators are hermitian. (Excercise: Show that the Hamiltonian in Eq.(8.6)
is still hermitian.) Next we have a look at the commutation properties of the
two operators. First, because x commutes with itself and similar for p, we have
[a, a† ] =
1
1
[x, −ip] +
[ip, x] = 1.
2~
2~
(8.7)
With this commutator and Eq.(8.6) we can calculate the commutator of H with
a and a† ;
[H, a] = [~ωa† a, a] = −~ωa.
(8.8)
We can calculate the analogous one with a† in a similar way. Alternatively we
can conjugate Eq.(8.8)
[H, a]† = −~ωa† .
(8.9)
This is calculated as
[H, a]† = (Ha − aH)† = (Ha)† − (aH)† = a† H † − a† H † = [a† , H] = −~ωa† .
(8.10)
or
[H, a† ] = ~ωa† .
(8.11)
So far this has all been an excercise in rewriting. Let us now show how
the new operators work on an energy-eigenstate, φE , with energy E. If you
prefer, you can think of φE as a wavefunction but this is not strictly speaking
necessary. In fact, the formalism is more general than that and it is good to
keep in mind that the operator really operates on a quantum state and not a
specific representation of this state. With one of the commutators above we
have
[H, a]φE = −~ωaφE ⇒ HaφE − EaφE = −~aφE .
(8.12)
After collecting terms we have
H(aφE ) = (E − ~ω)(aφE ).
(8.13)
This means that aφE is an eigenfunction of H with energy E − ~ω. The term
lowering operator used for a is therefore appropriate. We can repeat the calculation with a† instead of a and we would get that a† φE is an energy eigenstate
with an energy E + ~ω. Thus, a† is a raising operator.
We will now find the energy of the lowest state. First we realize that the
energy is always positive, with the definition of the Hamiltonian we have used,
25
beacuse it consists of two non-negative terms, proportional to x2 and p2 . This
means that there is a lowest energy state which we will denote φ0 . Application
of a on this state gives a state with lower energy (see Eq.(8.13)), contrary to
our assumptions. The only solution is that the state must vanish. In terms of
wavefunctions, this must mean that it is identical zero everywhere,
aφ0 = 0.
If we use the Hamiltonian on this state we have
1
1
†
Hφ0 = a a + ~ω φ0 = ~ωφ0 .
2
2
(8.14)
(8.15)
The lowest energy for a stationary state is therefore ~ω/2. If the raising operator
is applied to this state it will increase the energy with ~ω. This was precisely
the amount with which it was lowered by application of the lowering operator.
We can repeat the argument for excited states also. We therefore conclude that
the raising and lowering operators operate on the same ladder of eigenstates.
This is also seen from the Hamiltonian, the lowering operator lowers the energy
and converts the state to the corresponding lower energy eigenstate, and the
raising operator returns the state and energy to the initial values.
As the final result we can find the factor with which one changes the normalization of the state produced by application of the raising and lowering operators. This may sound as a quantity of minor interest but it is actually of great
importance when one formulates theories based on these types of operators.
We know from the Hamiltonian and the previous studies of the actions of a
and a† that
a† aφn = nφn .
(8.16)
Formulated in the bra-ket notation the same equation reads
a† a|ni = n|ni.
(8.17)
If we take the inner product of both sides of this equation with the bra hn| we
get
hn|a† a|ni = hn|n|ni = nhn|ni = n.
(8.18)
The left hand side of the equation can be evaluated when we observe that
(a|ni)† = hn|a† .
(8.19)
This makes it possible to write the left hand side as the norm of the vector a|ni,
(hn|a)a|ni = n.
(8.20)
a|ni = cn |n − 1i,
(8.21)
If we write the action of a as
2
we therefore
√ have that |cn | = n. We will chose the phase factor to be unity and
have cn = n. With this result we can determine the corresponding coefficient
26
CHAPTER 8. RAISING AND LOWERING OPERATORS
for the raising operator in a number of different ways. One may for example
compare with the action of a† a as extracted from the action of the Hamiltonian:
√
n|n − 1i ⇒
(8.22)
a|ni =
√ †
†
a a|ni =
na |n − 1i.
The left hand side of this equation is n|ni and we therefore conclude that
√
a† |ni = n + 1|n + 1i,
(8.23)
which should be compared with
a|ni =
√
n|n − 1i.
(8.24)