Operator formalism

19
Quantum operators
Although the previous chapter was principally concerned with the use of linear
operators and their eigenfunctions in connection with the solution of given
differential equations, it is of interest to study the properties of the operators
themselves and determine which of them follow purely from the nature of the
operators, without reference to specific forms of eigenfunctions.
19.1 Operator formalism
The results we will obtain in this chapter have most of their applications in the
field of quantum mechanics and our descriptions of the methods will reflect this.
In particular, when we discuss a function ψ that depends upon variables such as
space coordinates and time, and possibly also on some non-classical variables, ψ
will usually be a quantum-mechanical wavefunction that is being used to describe
the state of a physical system. For example, the value of |ψ|2 for a particular
set of values of the variables is interpreted in quantum mechanics as being the
probability that the system’s variables have that set of values.
To this end, we will be no more specific about the functions involved than
attaching just enough labels to them that a particular function, or a particular
set of functions, is identified. A convenient notation for this kind of approach
is that already hinted at, but not specifically stated, in subsection 17.1, where
the definition of an inner product is given. This notation, often called the Dirac
notation, denotes a state whose wavefunction is ψ by | ψ; since ψ belongs to a
vector space of functions, | ψ is known as a ket vector. Ket vectors, or simply kets,
must not be thought of as completely analogous to physical vectors. Quantum
mechanics associates the same physical state with keiθ | ψ as it does with | ψ
for all real k and θ and so there is no loss of generality in taking k as 1 and θ
as 0. On the other hand, the combination c1 | ψ1 + c2 | ψ2 , where | ψ1 and | ψ2 648
19.1 OPERATOR FORMALISM
represent different states, is a ket that represents a continuum of different states
as the complex numbers c1 and c2 are varied.
If we need to specify a state more closely – say we know that it corresponds
to a plane wave with a wave number whose magnitude is k – then we indicate
this with a label; the corresponding ket vector would be written as | k. If we also
knew the direction of the wave then | k would be the appropriate form. Clearly,
in general, the more labels we include, the more precisely the corresponding state
is specified.
The Dirac notation for the Hermitian conjugate (dual vector) of the ket vector
| ψ is written as ψ| and is known as a bra vector; the wavefunction describing this
∗
state
∗ is ψ , the complex conjugate of ψ. The inner product of two wavefunctions
ψ φ dv is then denoted by ψ| φ or, more generally if a non-unit weight function
ρ is involved, by
(19.1)
ψ| ρ| φ, evaluated as
ψ ∗ (r)φ(r)ρ(r) dr.
Given the (contrived) names for the two sorts of vectors, an inner product like
ψ| φ becomes a particular type of ‘bra(c)ket’. Despite its somewhat whimsical
construction, this type of quantity has a fundamental role to play in the interpretation of quantum theory, because expectation values, probabilities and transition
rates are all expressed in terms of them. For physical states the inner product of
the corresponding ket with itself, with or without an explicit weight function, is
non-zero, and it is usual to take
ψ|ψ = 1.
Although multiplying a ket vector by a constant does not change the state
described by the vector, acting upon it with a more general linear operator A
results (in general) in a ket describing a different state. For example, if ψ is a state
that is described in one-dimensional x-space by the wavefunction ψ(x) = exp(−x2 )
and A is the differential operator ∂/∂x, then
| ψ1 = A| ψ ≡ | Aψ
is the ket associated with the state whose wavefunction is ψ1 (x) = −2x exp(−x2 ),
clearly a different state. This allows us to attach a meaning to an expression such
as φ|A| ψ through the equation
φ|A| ψ = φ| ψ1 ,
(19.2)
i.e. it is the inner product of | ψ1 and | φ. We have already used this notation in
equation (19.1), but there the effect of the operator A was merely multiplication
by a weight function.
If it should happen that the effect of an operator acting upon a particular ket
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QUANTUM OPERATORS
is to produce a scalar multiple of that ket, i.e.
A| ψ = λ| ψ,
(19.3)
then, just as for matrices and differential equations, | ψ is called an eigenket or,
more usually, an eigenstate of A, with corresponding eigenvalue λ; to mark this
special property the state will normally be denoted by | λ, rather than by the
more general | ψ. Taking the Hermitian conjugate of this ket vector eigenequation
gives a bra vector equation,
ψ|A† = λ∗ ψ|.
(19.4)
It should be noted that the complex conjugate of the eigenvalue appears in this
equation. Should the action of A on |ψ produce an unphysical state (usually
one whose wavefunction is identically zero, and is therefore unacceptable as a
quantum-mechanical wavefunction because of the required probability interpretation) we denote the result either by 0 or by the ket vector | ∅ according to
context. Formally, | ∅ can be considered as an eigenket of any operator, but one
for which the eigenvalue is always zero.
If an operator A is Hermitian (A† = A) then its eigenvalues are real and
the eigenstates can be chosen to be orthogonal; this can be shown in the same
way as in chapter 17 (but using a different notation). As indicated there, the
reality of their eigenvalues is one reason why Hermitian operators form the
basis of measurement in quantum mechanics; in that formulation of physics, the
eigenvalues of an operator are the only possible values that can be obtained when
a measurement of the physical quantity corresponding to the operator is made.
Actual individual measurements must always result in real values, even if they
are combined in a complex form (x + iy or reiθ ) for final presentation or analysis,
and using only Hermitian operators ensures this. The proof of the reality of the
eigenvalues using the Dirac notation is given below in a worked example.
In the same notation the Hermitian property of an operator A is represented
by the double equality
A φ|ψ = φ|A| ψ = φ|A ψ.
It should be remembered that the definition of an Hermitian operator involves
specifying boundary conditions that the wavefunctions considered must satisfy.
Typically, they are that the wavefunctions vanish for large values of the spatial
variables upon which they depend; this deals with most physical systems since
they are nearly all formally infinite in extent. Some model systems require the
wavefunction to be periodic or to vanish at finite values of a spatial variable.
Depending on the nature of the physical system, the eigenvalues of a particular
linear operator may be discrete, part of a continuum, or a mixture of both. For
example, the energy levels of the bound proton–electron system (the hydrogen
atom) are discrete, but if the atom is ionised and the electron is free, the energy
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19.1 OPERATOR FORMALISM
spectrum of the system is continuous. This system has discrete negative and
continuous positive eigenvalues for the operator corresponding to the total energy
(the Hamiltonian).
Using the Dirac notation, show that the eigenvalues of an Hermitian operator are real.
Let | a be an eigenstate of Hermitian operator A corresponding to eigenvalue a, then
⇒
⇒
A| a = a| a,
a|A| a = a|a| a = aa| a,
and
a|A† = a∗ a|,
a|A† | a = a∗ a| a,
a|A| a = a∗ a| a,
since A is Hermitian.
Hence,
(a − a∗ )a| a = 0,
⇒ a = a∗ , since a| a = 0.
Thus a is real. It is not our intention to describe the complete axiomatic basis of quantum
mechanics, but rather to show what can be learned about linear operators, and
in particular about their eigenvalues, without recourse to explicit wavefunctions
on which the operators act.
Before we proceed to do that, we close this subsection with a number of results,
expressed in Dirac notation, that the reader should verify by inspection or by
following the lines of argument sketched in the statements. Where a sum over
a complete set of eigenvalues is shown, it should be replaced by an integral for
those parts of the eigenvalue spectrum that are continuous. With the notation
that |an is an eigenstate of Hermitian operator A with non-degenerate eigenvalue
an (or, if an is k-fold degenerate, then a set of k mutually orthogonal eigenstates
has been constructed and the states relabelled), we have the following results.
A| an = an | an ,
am |an = δm n
(orthonormality of eigenstates),
A(cn | an + cm | am ) = cn an | an + cm am | am (linearity).
(19.5)
(19.6)
The definitions of the sum and product of two operators are
(A + B)| ψ ≡ A| ψ + B| ψ,
(19.7)
AB| ψ ≡ A(B| ψ)
(19.8)
⇒
Ap | an = apn | an .
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(= BA| ψ in general),
(19.9)
QUANTUM OPERATORS
If A| an = a| an for all N1 ≤ n ≤ N2 , then
N2
| ψ =
dn | an satisfies A| ψ = a| ψ for any set of di .
n=N1
For a general state | ψ,
| ψ =
∞
cn | an , where cn = an |ψ.
(19.10)
n=0
This can also be expressed as the operator identity,
1=
∞
| an an |,
(19.11)
n=0
in the sense that
| ψ = 1 | ψ =
∞
| an an |ψ =
n=0
It also follows that
1 = ψ|ψ =
∞
c∗m am |
m=0
∞
cn | an .
n=0
∞
cn |an =
∞
c∗m cn δm n =
m,n
n=0
∞
n=0
|cn |2 .
(19.12)
Similarly, the expectation value of the physical variable corresponding to A is
ψ|A| ψ =
∞
c∗m am | A| an cn =
m,n
=
∞
∞
c∗m am | an | an cn
m,n
c∗m cn an δm n =
m,n
∞
|cn |2 an .
(19.13)
n=0
19.1.1 Commutation and commutators
As has been noted above, the product AB of two linear operators may or may
not be equal to the product BA. That is
AB| ψ is not necessarily equal to BA| ψ.
If A and B are both purely multiplicative operators, multiplication by f(r)
and g(r) say, then clearly the order of the operations is immaterial, the result
| f(r)g(r)ψ being obtained in both cases. However, consider a case in which A
is the differential operator ∂/∂x and B is the operator ‘multiply by x’. Then the
wavefunction describing AB| ψ is
∂ψ
∂
(xψ(x)) = ψ(x) + x ,
∂x
∂x
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19.1 OPERATOR FORMALISM
whilst that for BA| ψ is simply
x
∂ψ
,
∂x
which is not the same.
If the result
AB| ψ = BA| ψ
is true for all ket vectors | ψ, then A and B are said to commute; otherwise they
are non-commuting operators.
A convenient way to express the commutation properties of two linear operators
is to define their commutator, [ A, B ], by
[ A, B ] | ψ ≡ AB| ψ − BA| ψ.
(19.14)
Clearly two operators that commute have a zero commutator. But, for the example
given above we have that
∂ψ
∂
∂ψ
, x ψ(x) = ψ(x) + x
− x
= ψ(x) = 1 × ψ
∂x
∂x
∂x
or, more simply, that
∂
, x = 1;
∂x
(19.15)
in words, the commutator of the differential operator ∂/∂x and the multiplicative
operator x is the multiplicative operator 1. It should be noted that the order of
the linear operators is important and that
[ A, B ] = − [ B, A ] .
(19.16)
Clearly any linear operator commutes with itself and some other obvious zero
commutators (when operating on wavefunctions with ‘reasonable’ properties) are:
[ A, I ] , where I is the identity operator;
[ An , Am ] , for any positive integers n and m;
[ A, p(A) ] , where p(x) is any polynomial in x;
[ A, c ] , where A is any linear operator and c is any constant;
[ f(x), g(x) ] , where the functions are mutiplicative;
[ A(x), B(y) ] , where the operators act on different variables, with
∂ ∂
,
as a specific example.
∂x ∂y
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QUANTUM OPERATORS
Simple identities amongst commutators include the following:
[ A, B + C ] = [ A, B ] + [ A, C ] ,
(19.17)
[ A + B, C ] = [ A, C ] + [ B, C ] ,
(19.18)
[ A, BC ] = ABC − BCA + BAC − BAC
= (AB − BA)C + B(AC − CA)
= [ A, B ] C + B [ A, C ] ,
(19.19)
[ AB, C ] = A [ B, C ] + [ A, C ] B.
(19.20)
If A and B are two linear operators that both commute with their commutator, prove that
[ A, B n ] = nB n−1 [ A, B ] and that [ An , B ] = nAn−1 [ A, B ].
Define Cn by Cn = [ A, B n ]. We aim to find a reduction formula for Cn :
Cn = A, B B n−1
= [ A, B ] B n−1 + B A, B n−1 , using (19.19),
= B n−1 [ A, B ] + B A, B n−1 , since [ [ A, B ] , B ] = 0,
= B n−1 [ A, B ] + BCn−1 , the required reduction formula,
= B n−1 [ A, B ] + B{B n−2 [ A, B ] + BCn−2 }, applying the formula,
= 2B n−1 [ A, B ] + B 2 Cn−2
= ···
= nB n−1 [ A, B ] + B n C0 .
However, C0 = [ A, I ] = 0 and so Cn = nB n−1 [ A, B ].
Using equation (19.16) and interchanging A and B in the result just obtained, we find
[ An , B ] = − [ B, An ] = −nAn−1 [ B, A ] = nAn−1 [ A, B ] ,
as stated in the question. As the power of a linear operator can be defined, so can its exponential;
this situation parallels that for matrices, which are of course a particular set of
operators that act upon state functions represented by vectors. The definition
follows that for the exponential of a scalar or matrix, namely
exp A =
∞
An
n=0
n!
.
(19.21)
Related functions of A, such as sin A and cos A, can be defined in a similar way.
Since any linear operator commutes with itself, when two functions of it
are combined in some way, the result takes a form similar to that for the
corresponding functions of scalar quantities. Consider, for example, the function
f(A) defined by f(A) = 2 sin A cos A. Expressing sin A and cos A in terms of their
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19.1 OPERATOR FORMALISM
defining series, we have
f(A) = 2
∞
∞
(−1)m A2m+1 (−1)n A2n
.
(2m + 1)!
(2n)!
m=0
n=0
Writing m + n as r and replacing n by s, we have
f(A) = 2
=2
∞
r=0
∞
2r+1
A
r
s=0
(−1)s
(−1)r−s
(2r − 2s + 1)! (2s)!
(−1)r cr A2r+1 ,
r=0
where
cr =
r
s=0
1
1
=
(2r − 2s + 1)! (2s)!
(2r + 1)!
r
2r+1
C2s .
s=0
By adding the binomial expansions of 22r+1 = (1 + 1)2r+1 and 0 = (1 − 1)2r+1 , it
can easily be shown that
22r+1 = 2
r
2r+1
C2s
⇒
cr =
s=0
22r
.
(2r + 1)!
It then follows that
2 sin A cos A = 2
∞
(−1)r A2r+1 22r
r=0
(2r + 1)!
=
∞
(−1)r (2A)2r+1
r=0
(2r + 1)!
= sin 2A,
a not unexpected result.
However, if two (or more) linear operators that do not commute are involved,
combining functions of them is more complicated and the results less intuitively
obvious. We take as a particular case the product of two exponential functions
and, even then, take the simplified case in which each linear operator commutes
with their commutator (so that we may use the results from the previous worked
example).
If A and B are two linear operators that both commute with their commutator, show that
exp(A) exp(B) = exp(A + B + 12 [ A, B ] ).
We first find the commutator of A and exp λB, where λ is a scalar quantity introduced for
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