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Commutators
Certain combinations of operators often occur in quantum mechanics. One is
the commutator between two operators A and B. It is defined as
[A,B]=AB BA. (5.49)
−
Since, in general, operators don’t commute, the commutator is in general
non-zero. Sets of hermitean operators that do commute among themselves are
especially important in that they can represent sets of physical quantities that
can be measured simultaneously.
5.3 Transformations and symmetries
Unitaryoperatorseffectsymmetrytransformationsofquantumstatesandquan-
tum operators. Symmetries are transformations of states that do not affect
observable quantities, i.e they do not change the results of measurements.
A first look at measurement
A measurement always results in a number. It is a fundamental property of
quantummechanicsthatthestatesthemselvesarenotobservableormeasurable.
Essentiallythe only wayto getnumbersoutofquantummechanicsis by taking
innerproductsofstates. Sincethestatesaredescribedbyvectorsitisreasonable
to suspect that physical quantities are described by linear operators. Thus
the result of a measurement is in some way related to inner products of the
form φAφ . Such inner products are often called diagonal matrix elements
h | | i
in analogy to (5.34). Furthermore, if A is an hermitean operator, φAφ is
h | | i
109
a real number which is interpreted as the expectation value for the quantity
represented by A. The theory of measurement will developed in section 5.6
after some more terminology is introduced.
Symmetry transformations
Inordertostudysymmetrytransformations,suppose ψ and φ aretwoquan-
| i | i
tum states. Acting on these states with the unitary operator U we get the
transformed states U ψ and U φ . It is customary to write transformations as
| i | i
ψ ψ =U ψ . (5.50)
′
| i→| i | i
Likewise, the transformation of a bra vector is
φ φ′ = φU†. (5.51)
h |→h | h |
Thatthisisthecorrectformofatransformationofabravectorfollowsfrom
the equation (U φ ) = φU applied to the transformation of a ket vector.
† †
| i h |
With U a unitary operator, the inner product between the states φ and
| i
ψ is unaffected by this transformation. We get
| i
φψ φU U ψ = φU 1U ψ = φψ .
† −
h | i→h | | i h | | i h | i
The correspondingformfora transformationofa linearoperatorcanbe de-
rived by demanding the matrix element φAψ to be invariant under a trans-
h | | i
formation. We know how to transform states of the form ψ . Consider trans-
| i
formingstates ofthe formAψ , i.e statesactedonby alinearoperatorA. The
| i
transformed state is UAψ , which can be expanded as
| i
UAψ =UA(U 1U)ψ =(UAU 1)U ψ .
− −
| i | i | i
In this way we separate the transformation of the state Aψ into a transfor-
| i
mation of the state ψ and the operator A. Thus it is natural to define a
| i
transformation of a linear operator A as
A A =UAU 1 =UAU . (5.52)
′ − †
→
That this is a reasonable definition is born out by calculating the transfor-
mation of the matrix element φAψ
h | | i
φAψ ( φU†)UAU†(U ψ )
h | | i→ h | | i
= φ(U U)A(U U)ψ )= φAψ ),
† †
h | | i h | | i
which shows the invariance of the matrix element under the transformation.
Soalthoughstatesandoperatorsareaffectedbysymmetrytransformations,
measurable quantities are not; this is the essence of symmetry in quantum me-
chanics.
110
5.4 Eigenvectors and eigenvalues
An eigenvector of a linear operator A on a vector space is a non-zero vector x
| i
such that
Ax =λx (5.53)
| i | i
where the eigenvalue λ is a complex number. Introducing the identity operator
on the right hand side of the equation, it can be rewritten as a proper matrix
equation
(A λI)x =0. (5.54)
− | i
Fromthe theoryoflinearequationsitfollowsthatthis equationhasno non-
zero solutions x unless the determinant of the matrix A λI is zero. If the
| i −
determinant is zero, then the vector x is identically zero since the equation
| i