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We will not derive this solution to the harmonic oscillator.4 Instead we
will introduce a more abstract, and more powerful formalism, which is also
the standard formalism used in almost all applications of harmonic oscillators.
This is the method of annihilation and creationoperators and in the process of
introducing them we will alsointroduce the very useful notationon bra and ket
vectors invented by Dirac.
4.2.2 Operators for momentum and position
One step in the quantizationofa classicalsystemis to replaceclassicaldynam-
ical variables with operators. Momentum p and position x are replaced by the
momentum operator and the position operator respectively, often denoted by pˆ
and xˆ.5 Explicit representations of these operators are
∂
pˆ= i¯h (4.23)
− ∂x
xˆ=x, (4.24)
whichisthesamerepresentationoftheoperatorsasinequations(4.3)and(4.4).
This particularrepresentationis valid in the configurationspace representation
ofthe statesusingx-spacewave-functions. Note thatmomentumisrepresented
by a differential operator, and position by a multiplication. A consequence
of this is that the order of application of the operators matters. A simple
calculation illustrates this. Consider applying first xˆ and then pˆ to a state ψ,
and then applying these operators in the reverse order, i.e. first pˆand then xˆ:
∂ ∂ ∂
pˆxˆψ = i¯h xψ = i¯hψ i¯hx ψ = i¯h 1+x ψ,
− ∂x − − ∂x − ∂x
(cid:0) (cid:1) (cid:0) (cid:1)
∂
xˆpˆψ = i¯hx ψ.
− ∂x
Then subtract these expressions to get
(xˆpˆ pˆxˆ)ψ =i¯hψ.
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The state ψ is completely arbitrary here, and can be removed, yielding the
operator equation
xˆpˆ pˆxˆ=i¯h. (4.25)
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4Anystandardtextbook onquantum mechanics containsthecalculations[27,?].
5Pronouncedp-hatandx-hat.
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This combination of operators is so important in quantum mechanics and
so frequently occuring that a special notation is introduced. A commutator
bracket, or simply a commutator, between two operators Aˆ and Bˆ is defined by
[Aˆ,Bˆ]=AˆBˆ BˆAˆ. (4.26)
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Using this notation, equation (4.25) can be written
[xˆ,pˆ]=i¯h. (4.27)
This is a fundamental equation relating the operators xˆ and pˆand it holds
whatever representation is used. Therefore it can be used as a quantization
condition.
For completeness, we also record the trivial commutators
[xˆ,xˆ]=0 (4.28)
[pˆ,pˆ]=0. (4.29)
In general, an operator always commutes with itself.
4.2.3 Commutators
If the commutator between two operators is non-zero, i.e if [Aˆ,Bˆ] = 0, the
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operators are said to be non-commuting. In that case, as we have seen, the
order in which they are applied to a quantum state matters. Sometimes, when
there is no particular ordering to prefer, or when the ordering is ambiguous, a
symmetrical ordering is chosen as a kind of default ordering
1
(AˆBˆ) = (AˆBˆ+BˆAˆ). (4.30)
sym
2
4.2.4 A note on classical dynamics
Classical dynamics for a particle in one dimension of space is governed by
Newton’s equation
ma=F, (4.31)
where a denotes the acceleration. Acceleration is the time derivative of
velocity,
dv
a= ,
dt
which in its turn is related to the particle momentum through the equation
p
v = . (4.32)
m
The force F is determined by the potential V
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dV
F = .
−dx
Combining these four equations while assuming that the mass is constant
(the normal case), yields
dp dV
= . (4.33)
dt −dx
Since the kinetic energy K does not depend on x, it is possible to replace
the potentialenergyV inthis formulawiththe totalenergyE.Itis practicalto
changenotationslightlyanduseH forthetotalenergyalsointheclassicalcase,
reserving E for the quantum mechanical energy eigenvalues. Thus we write for
the total energy H =K+V. This yields the dynamical equation
dp dH
= . (4.34)
dt −dx
At this stage one might worry that the original Newton equation (4.31) is
a second order differential equation (remember, the acceleration a is a second
order derivative with respect to time), and (4.34) is a first order differential
equation. Something is missing.