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There are a few popular formulations of quantum theory. One of them uses
configurationspacewavefunctionsandtheirconjugatemomentumspaceFourier
transforms. This is one of the traditional formulations, originating with Erwin
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Schr¨odinger,anditistraditionallycalledwavemechanics. Anotherformulation,
contemporary with Scro¨dinger’s, is Heisenberg’s matrix mechanics. These are
the originalformulations of quantum mechanics and they were almost immedi-
ately shown to be equivalent. In the context of a theoretical study of quantum
computation another formulation, somewhat more abstract, and which can be
consideredas a generalizationofthe other formulations,due to P.A.M Dirac,is
more appropriate. A very readable account of this formulation is Dirac’s own
classic book [30].
A fourth formulation, the path integral formulation , developed by R.P.N
Feynman in the 1950’s[31], will not be mentioned here. It is of extreme useful-
nessinmoderntheoreticalphysics,butits methods does notseemto be needed
in quantum computation.
4.1 Quantum mechanics in one space dimension
Asourintroductiontoquantummechanicswewillstudyaparticlemovinginone
dimension of space under the influence of a potential. The state of the particle
is described by the wave-function ψ(x,t), where x is the space coordinate and
t is the time. The states of a system can be described in different ways. This
particularrepresentationis calledthe configuration space representation, where
configuration refers to using space to parameterize the state. The dynamics of
the state is governedby the Schr¨odinger equation [32]
∂
i¯h ψ =Hψ. (4.1)
∂t
In this equation, ¯h is a physical constant which sets the scale of quantum
phenomena.1H is the Hamiltonian operator. The equation equates the time
rateofchangeofthewavefunctionwiththeactionoftheHamiltonian,thusthe
dynamics of the state is encoded in the form of H. In quantum computation,
the ’program’ of the quantum computer can be regarded as encoded in the
Hamiltonian. But more on this later on.
The Hamiltonianisrelatedtothe classicalenergyofthe system. Inclassical
physics, a particle has a mechanical energy consisting of kinetic energy K and
potential energy V, and the total energy is E = K +V. The kinetic energy is
given by
p2
K = , (4.2)
2m
where p is the particle momentum, classically related to the velocity v
throughp=mv wheremistheparticlemass. Thus,thekineticenergycanalso
be written as
mv2
K = ,
2
1It’snumericalvalueis1.054 10−34 Js.
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a formula perhaps more readily recognized by non-physicists. However, the
first form is the fundamental one.
The potential energy depends on the forces acting on the particle. Forces
are not further analyzed in this context, and a formula is simply given for V.
In general, it is a function of space and time, but we will only consider time-
independent potentials.
Quantization is performed via the heuristic rules
replace x by x
·
∂
replace p by i¯h ,
− ∂x
or more concisely
x x (4.3)
−→ ·
∂
p i¯h (4.4)
−→− ∂x
In these rules, the left hand sides should be thought of as classical physics
entities, whereas the right hand sides stands for the corresponding quantum
mechanical operators. An operator can be either multiplication by a function
f oradifferentialoperatorD(asinthesecondrule)actingonthestate.2 Ifthis
·
sounds confusing, this is not the proper time for worry. It is best just to carry
on in order to get a little bit more used to the quantum mechanical machinery.
If one applies these rules to the classical energy, one gets the Hamiltonian,
or in formulas
1 ∂ 2 ¯h2 ∂2
E H = i¯h +V(x)= +V(x). (4.5)
−→ 2m (cid:16)− ∂x (cid:17) −2m∂x2
The Schr¨odinger equation now becomes
∂ ¯h2 ∂2
i¯h ψ = ψ+V(x)ψ. (4.6)
∂t −2m∂x2
This is a partialdifferentialequationgoverningthe time developmentof the
system. This simple example captures most of the main features of quantum
mechanics in this formulation.
A more realistic system would be in three spatial dimensions. The force
acting on the particle is given by the potential, examples of which could be
the coulomb field from an atomic nucleus on an electron, forces from other
electrons and perhaps time-dependent electromagnetic fields. But we will stick
to this simple one-dimensional system and solve the equation in two cases; the
square well potential and the harmonic oscillator.
The first steps in the solution are general and does not depend on the form
of the potential, except that it is assumed to be independent of time. The
2Otherrepresentations ofquantum mechanical operatorswillappearsubsequently.
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method is the standard separation of variables method used in solving partial