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Thus we get back the secular equation (5.55) explicitly. Solving this equa-
tion,wegetthe n,notnecessarilydistinct, eigenvaluesofthe originalmatrixA.
Note that the eigenvalues does not depend on the diagonalizing matrix D.
Diagonalization of hermitean operators
SupposeAisahermiteanoperator. Thenitisdiagonalizableandcanbewritten
as in equation (5.56),
A= λ i i.
i
| ih |
Xi
Taking the hermitean conjugate, we get
A = (λ i i) = λ i i,
† i † ∗i|
| ih | ih |
Xi Xi
since, obviously (i i) = i i for each i. But A=A so that we must have
† †
| ih | | ih |
λ∗i|i ihi |= λ |i ihi |.
i
Xi Xi
This is only possible if all eigenvalues are real numbers, or λ†i =λ i.
Thus, hermitean operators have real eigenvalues. And conversely, if an op-
erator have all eigenvalues real, then it is hermitean.
Simultaneous diagonalization theorem
SupposetwooperatorsAandB arediagonalinthesamebasis. Thenitiseasily
shown that they commute. This follows since the product of two diagonal ma-
trices is itself diagonal,and the elements on the diagonal is simply the product
of the diagonal elements of A and B.
A 0 0 B 0 0
11 11
··· ···
0 A 0 0 B 0
 22  22 
. . ··· . . . ··· . =
. . . . . .
. . . . . .
  
 0 0 A  0 0 B 
 ··· nn ··· nn
113
A B 0 0
11 11
···
0 A B 0
 22 22 
. . ··· .
. . .
. . .
 
 0 0 A B 
 ··· nn nn
The converse is also true, if two operators commute, then they are simulta-
neously diagonalizable in the same basis. For a proof, see [39].
5.5 Quantum dynamics
The time evolution, or dynamics, of a closed quantum system can be described
intworelatedways. Asystemisclosedifthereisnointeractionwiththerestof
the world. In practice, this might not be a realistic assumption. In principle it
is not possible to isolate one piece of the world from the rest, there are always
interactions between system and environment. The assumption is that either
this interaction can be arbitrarily weak or controlled. The usefulness of the
closedness assumption is that all of the systems dynamics is encoded in the
Hamiltonian.
Schro¨dinger equation
Traditionally, the dynamics is described by the Schro¨dinger equation. This is
a (first order) differential equation in the time variable t, equating the time
derivative of the state to the action of the Hamiltonian operator on the state
dψ
i¯h | i =H ψ , (5.59)
dt | i
where ¯h is a fundamental physical constant setting the scale of quantum phe-
nomena.2 In theoreticalcontexts its value is often taken to be 1. This amounts
to a choice of measuring units, where the ’natural’ scale of phenomena is taken
to be submicroscopic.
The form of the Hamiltonian depends on the representation chosen for the
states. In the configuration space representation presented in chapter 4, the
statesarewavefunctions,andtheHamiltonianisapartialdifferentialoperator.
In quantumcomputation contexts, H will be a matrix acting on superpositions
of computational basis states.
FindingtheproperHamiltonianforaphysicalsystemisingeneraladifficult
problem. Itisnotgenerallyconsideredasaquestionwithinquantummechanics
itself, since as we havepointed out, quantum mechanics is just a frameworkfor
formulating physical theories. However this last point might very well change
as our understanding of fundamental physics develops.3
Beingahermiteanoperator,H canbediagonalized. SincetheHamiltonianis
physicallyrelatedtotheenergyofthesystem,thecorrespondingeigenvaluesand
2Itsvalueis6.626 10−34 Js
3Itmightbetheca·
sethatawouldbe”theoryofeverything”comespackagedwithquantum
mechanicsasaninseparablepart.
114
eigenstatesarereferredtoasenergyeigenvaluesandenergyeigenstates. Naming
theeigenvalueswithE andeigenstates n ,wehavethespectraldecomposition
n
| i
H = E n n. (5.60)
n
| ih |
Xn