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Chapter 3
Linear Algebra In Dirac Notation
3.1 Hilbert Space and Inner Product
In Ch. 2 it was noted that quantum wave functions form a linear space in the sense that multiplying a function by a complex number or adding two wave functions together produces another wave function. It was also pointed out that a particular quantum state can be represented either by a wave function ψ(x) which depends upon the position variable x, or by an alternative function ψˆ(p) of the momentum variable p. It is convenient to employ the Dirac symbol |ψ〉, known as a “ket”, to denote a quantum state without referring to the particular function used to represent it. The kets, which we shall also refer to as vectors to distinguish them from scalars, which are complex numbers, are the elements of the quantum Hilbert space H. (The real numbers form a subset of the complex numbers, so that when a scalar is referred to as a “complex number”, this includes the possibility that it might be a real number.) If α is any scalar (complex number), the ket corresponding to the wave function αψ(x) is denoted by α|ψ〉, or sometimes by |ψ〉α, and the ket corresponding to φ(x) + ψ(x) is denoted by |φ〉 + |ψ〉 or |ψ〉 + |φ〉, and so forth. This correspondence could equally well be expressed using momentum wave functions, because the Fourier transform, (2.15) or (2.16), is a linear relationship between ψ(x) and ψˆ(p), so that αφ(x) + βψ(x) and α φˆ(p) + β ψˆ(p) correspond to the same quantum state α|ψ〉+β|φ〉. The addition of kets and multiplication by scalars obey some fairly obvious rules:
α
β|ψ〉
= (αβ)|ψ〉, (α + β)|ψ〉 = α|ψ〉 + β|ψ〉, α
|φ〉 + |ψ〉
= α|φ〉 + α|ψ〉, 1 |ψ〉 = |ψ〉.
Multiplying any ket by the number 0 yields the unique zero vector or zero ket, which will, because there is no risk of confusion, also be denoted by 0. The linear space H is equipped with an inner product
I
|ω〉, |ψ〉
= 〈ω|ψ〉 (3.2)
which assigns to any pair of kets |ω〉 and |ψ〉 a complex number. While the Dirac notation 〈ω|ψ〉, already employed in Ch. 2, is more compact than the one based on I
, it is, for purposes of exposition, useful to have a way of writing the inner product which clearly indicates how it depends on two different ket vectors.
23
24 CHAPTER 3. LINEAR ALGEBRA IN DIRAC NOTATION
An inner product must satisfy the following conditions:
- Interchanging the two arguments results in the complex conjugate of the original expression:
I
|ψ〉, |ω〉
[
I
|ω〉, |ψ〉
)]∗
- The inner product is linear as a function of its second argument:
I
|ω〉, α|φ〉 + β|ψ〉
= αI
|ω〉, |φ〉
|ω〉, |ψ〉
- The inner product is an antilinear function of its first argument:
I
α|φ〉 + β|ψ〉, |ω〉
= α∗I
|φ〉, |ω〉
|ψ〉, |ω〉
- The inner product of a ket with itself,
I
|ψ〉, |ψ〉
= 〈ψ|ψ〉 = ‖ψ‖^2 (3.6)
is a positive (greater than zero) real number unless |ψ〉 is the zero vector, in which case 〈ψ|ψ〉 = 0. The term “antilinear” in the third condition refers to the fact that the complex conjugates of α and β appear on the right side of (3.5), rather than α and β themselves, as would be the case for a linear function. Actually, (3.5) is an immediate consequence of (3.3) and (3.4)—simply take the complex conjugate of both sides of (3.4), and then apply (3.3)—but it is of sufficient importance that it is worth stating separately. The reader can check that the inner products defined in (2.3) and (2.24) satisfy these conditions. (There are some subtleties associated with ψ(x) when x is a continuous real number, but we must leave discussion of these matters to books on functional analysis.) The positive square root ‖ψ‖ of ‖ψ‖^2 in (3.6) is called the norm of |ψ〉. As already noted in Ch. 2, α|ψ〉 and |ψ〉 have exactly the same physical significance if α is a non-zero complex number. Consequently, as far as the quantum physicist is concerned, the actual norm, as long as it is positive, is a matter of indifference. By multiplying a non-zero ket by a suitable constant, one can always make its norm equal to 1. This process is called normalizing the ket, and a ket with norm equal to 1 is said to be normalized. Normalizing does not produce a unique result, because eiφ|ψ〉, where φ is an arbitrary real number or phase, has precisely the same norm as |ψ〉. Two kets |φ〉 and |ψ〉 are said to be orthogonal if 〈φ|ψ〉 = 0, which by (3.3) implies that 〈ψ|φ〉 = 0.
3.2 Linear Functionals and the Dual Space
Let |ω〉 be some fixed element of H. Then the function
J
|ψ〉
= I
|ω〉, |ψ〉
assigns to every |ψ〉 in H a complex number in a linear manner,
J
α|φ〉 + β|ψ〉
= αJ
|φ〉
|ψ〉
as a consequence of (3.4). Such a function is called a linear functional. There are many different linear functionals of this sort, one for every |ω〉 in H. In order to distinguish them we could place
26 CHAPTER 3. LINEAR ALGEBRA IN DIRAC NOTATION
3.3 Operators, Dyads
A linear operator, or simply an operator A is a linear function which maps H into itself. That is, to each |ψ〉 in H, A assigns another element A
|ψ〉
in H in such a way that
A
α|φ〉 + β|ψ〉
= αA
|φ〉
|ψ〉
whenever |φ〉 and |ψ〉 are any two elements of H, and α and β are complex numbers. One custom- arily omits the parentheses and writes A|φ〉 instead of A
|φ〉
where this will not cause confusion, as on the right (but not the left) side of (3.15). In general we shall use capital letters, A, B, and so forth, to denote operators. The letter I is reserved for the identity operator which maps every element of H to itself: I|ψ〉 = |ψ〉. (3.16)
The zero operator which maps every element of H to the zero vector will be denoted by 0. The inner product of some element |φ〉 of H with the ket A|ψ〉 can be written as ( |φ〉
A|ψ〉 = 〈φ|A|ψ〉, (3.17)
where the notation on the right side, the “sandwich” with the operator between a bra and a ket, is standard Dirac notation. It is often referred to as a “matrix element”, even when no matrix is actually under consideration.( (Matrices are discussed in Sec. 3.6.) One can write 〈φ|A|ψ〉 as 〈φ|A
|ψ〉
, and think of it as the linear functional or bra vector
〈φ|A (3.18)
acting on or evaluated at |ψ〉. In this sense it is natural to think of a linear operator A on H as inducing a linear map of the dual space H†^ onto itself, which carries 〈φ| to 〈φ|A. This map can also, without risk of confusion, be denoted by A, and while one could write it as A
〈φ|
, in Dirac notation 〈φ|A is more natural. Sometimes one speaks of “the operator A acting to the left”. Dirac notation is particularly convenient in the case of a simple type of operator known as a dyad, written as a ket followed by a bra, |ω〉〈τ |. Applied to some ket |ψ〉 in H, it yields
|ω〉〈τ |
|ψ〉
= |ω〉〈τ |ψ〉 = 〈τ |ψ〉|ω〉. (3.19)
Just as in (3.9), the first equality is “obvious” if one thinks of the product of 〈τ | with |ψ〉 as 〈τ |ψ〉, and since the latter is a scalar it can be placed either after or in front of the ket |ω〉. Setting A in (3.17) equal to the dyad |ω〉〈τ | yields
〈φ|
|ω〉〈τ |
|ψ〉 = 〈φ|ω〉〈τ |ψ〉, (3.20)
where the right side is the product of the two scalars 〈φ|ω〉 and 〈τ |ψ〉. Once again the virtues of Dirac notation are evident in that this result is an almost automatic consequence of writing the symbols in the correct order. The collection of all operators is itself a linear space, since a scalar times an operator is an operator, and the sum of two operators is also an operator. The operator αA + βB applied to an element |ψ〉 of H yields the result: ( αA + βB
|ψ〉 = α
A|ψ〉
B|ψ〉
3.3. OPERATORS, DYADS 27
where the parentheses on the right side can be omitted, since
αA
|ψ〉 is equal to α
A|ψ〉
, and both can be written as αA|ψ〉. The product AB of two operators A and B is the operator obtained by first applying B to some ket, and then A to the ket which results from applying B:
AB
|ψ〉
= A
B
|ψ〉
Normally the parentheses are omitted, and one simply writes AB|ψ〉. However, it is very important to note that operator multiplication, unlike multiplication of scalars, is not commutative: in general, AB 6 = BA, since there is no particular reason to expect that A
B
|ψ〉
will be the same element of H as B
A
|ψ〉
In the exceptional case in which AB = BA, that is, AB|ψ〉 = BA|ψ〉 for all |ψ〉, one says that these two operators commute with each other, or (simply) commute. The identity operator I commutes with every other operator, IA = AI = A, and the same is true of the zero operator, A0 = 0A = 0. The operators in a collection {A 1 , A 2 , A 3 ,.. .} are said to commute with each other provided Aj Ak = AkAj (3.23)
for every j and k. Operator products follow the usual distributive laws, and scalars can be placed anywhere in a product, though one usually moves them to the left side:
A(γC + δD) = γAC + δAD, (αA + βB)C = αAC + βBC.
In working out such products it is important that the order of the operators, from left to right, be preserved: one cannot (in general) replace AC with CA. The operator product of two dyads |ω〉〈τ | and |ψ〉〈φ| is fairly obvious if one uses Dirac notation:
|ω〉〈τ | · |ψ〉〈φ| = |ω〉〈τ |ψ〉〈φ| = 〈τ |ψ〉|ω〉〈φ|, (3.25)
where the final answer is a scalar 〈τ |ψ〉 multiplying the dyad |ω〉〈φ|. Multiplication in the reverse order will yield an operator proportional to |ψ〉〈τ |, so in general two dyads do not commute with each other. Given an operator A, if one can find an operator B such that
AB = I = BA, (3.26)
then B is called the inverse of the operator A, written as A−^1 , and A is the inverse of the operator B. On a finite-dimensional Hilbert space one only needs to check one of the equalities in (3.26), as it implies the other, whereas on an infinite-dimensional space both must be checked. Many operators do not posses inverses, but if an inverse exists, it is unique. The antilinear dagger operation introduced earlier, (3.11) and (3.12), can also be applied to operators. For a dyad one has: ( |ω〉〈τ |
= |τ 〉〈ω|. (3.27)
3.4. PROJECTORS AND SUBSPACES 29
3.4 Projectors and Subspaces
A particular type of Hermitian operator called a projector plays a central role in quantum theory. A projector is any operator P which satisfies the two conditions
P 2 = P, P †^ = P. (3.34)
The first of these, P 2 = P , defines a projection operator which need not be Hermitian. Hermitian projection operators are also called orthogonal projection operators, but we shall call them projec- tors. Associated with a projector P is a linear subspace P of H consisting of all kets which are left unchanged by P , that is, those |ψ〉 for which P |ψ〉 = |ψ〉. We shall say that P projects onto P, or is the projector onto P. The projector P acts like the identity operator on the subspace P. The identity operator I is a projector, and it projects onto the entire Hilbert space H. The zero operator 0 is a projector which projects onto the subspace consisting of nothing but the zero vector. Any non-zero ket |φ〉 generates a one-dimensional subspace P, often called a ray or (by quantum physicists) a pure state, consisting of all scalar multiples of |φ〉, that is to say, the collection of kets of the form {α|φ〉}, where α is any complex number. The projector onto P is the dyad
P = [φ] = |φ〉〈φ|/〈φ|φ〉, (3.35)
where the right side is simply |φ〉〈φ| if |φ〉 is normalized, which we shall assume to be the case in the following discussion. The symbol [φ] for the projector projecting onto the ray generated by |φ〉 is not part of standard Dirac notation, but it is very convenient, and will be used throughout this book. Sometimes, when it will not cause confusion, the square brackets will be omitted: φ will be used in place of [φ]. It is straightforward to show that the dyad (3.35) satisfies the conditions in (3.34) and that P (α|φ〉) = |φ〉〈φ|(α|φ〉) = α|φ〉〈φ|φ〉 = α|φ〉, (3.36)
so that P leaves the elements of P unchanged. When it acts on any vector |χ〉 orthogonal to |φ〉, 〈φ|χ〉 = 0, P produces the zero vector:
P |χ〉 = |φ〉〈φ|χ〉 = 0|φ〉 = 0. (3.37)
The properties of P in (3.36) and (3.37) can be given a geometrical interpretation, or at least one can construct a geometrical analogy using real numbers instead of complex numbers. Consider the two dimensional plane shown in Fig. 3.1, with vectors labeled using Dirac kets. The line passing through |φ〉 is the subspace P. Let |ω〉 be some vector in the plane, and suppose that its projection onto P, along a direction perpendicular to P, Fig. 3.1(a), falls at the point α|φ〉. Then
|ω〉 = α|φ〉 + |χ〉, (3.38)
where |χ〉 is a vector perpendicular (orthogonal) to |φ〉, indicated by the dashed line. Applying P to both sides of (3.38), using (3.36) and (3.37), one finds that
P |ω〉 = α|φ〉. (3.39)
That is, P on acting on any point |ω〉 in the plane projects it onto P along a line perpendicular to P, as indicated by the arrow in Fig. 3.1(a). Of course, such a projection applied to a point
30 CHAPTER 3. LINEAR ALGEBRA IN DIRAC NOTATION
P
|φ〉
|ω〉
|χ〉
α|φ〉
(a)
P
|φ〉
|ω〉
Q|ω〉
|η〉
Q|η〉
(b)
Figure 3.1: Illustrating (a) an orthogonal (perpendicular) projection onto P; (b) a nonorthogonal projection represented by Q.
already on P leaves it unchanged, corresponding to the fact that P acts as the identity operation for elements of this subspace. For this reason, P
P
|ω〉
is always the same as P
|ω〉
, which is equivalent to the statement that P 2 = P. It is also possible to imagine projecting points onto P along a fixed direction which is not perpendicular to P, as in Fig. 3.1(b). This defines a linear operator Q which is again a projection operator, since elements of P are mapped onto themselves, and thus Q^2 = Q. However, this operator is not Hermitian (in the terminology of real vector spaces, it is not symmetrical), so it is not an orthogonal (“perpendicular”) projection operator. Given a projector P , we define its complement, written as ∼P or P˜ , also called the negation of P (see Sec. 4.4), to be the projector defined by
P˜ = I − P. (3.40)
It is easy to show that P˜ satisfies the conditions for a projector in (3.34), and that
P P˜ = 0 = P P.˜ (3.41)
From (3.40) it is obvious that P is, in turn, the complement (or negation) of P˜. Let P and P⊥ be the subspaces of H onto which P and P˜ project. Any element |ω〉 of P⊥^ is orthogonal to any element |φ〉 of P:
〈ω|φ〉 =
|ω〉
|φ〉 =
P |ω〉
P |φ〉
= 〈ω| P P˜ |φ〉 = 0, (3.42)
because P P˜ = 0. Here we have used the fact that P and P˜ act as identity operators on their respective subspaces, and the third equality makes use of (3.33). As a consequence, any element |ψ〉 of H can be written as the sum of two orthogonal kets, one belonging to P and one to P ⊥:
|ψ〉 = I|ψ〉 = P |ψ〉 + P˜ |ψ〉. (3.43)
Using (3.43), one can show that P⊥^ is the orthogonal complement of P, the collection of all elements of H which are orthogonal to every ket in P. Similarly, P is the orthogonal complement (P ⊥)⊥^ of P⊥.
32 CHAPTER 3. LINEAR ALGEBRA IN DIRAC NOTATION
Let R be the subspace of H consisting of all linear combinations of kets belonging to an orthonormal collection {|φj 〉}, that is, all elements of the form
|ψ〉 =
j
σj |φj 〉, (3.50)
where the σj are complex numbers. Then the projector R onto R is the sum of the corresponding dyad projectors: R =
j
|φj 〉〈φj | =
j
[φj ]. (3.51)
This follows from the fact that, in light of (3.49), R acts as the identity operator on a sum of the form (3.50), whereas R|ω〉 = 0 for every |ω〉 which is orthogonal to every |φj 〉 in the collection, and thus to every |ψ〉 of the form (3.50) If every element of H can be written in the form (3.50), the orthonormal collection is said to form an orthonormal basis of H, and the corresponding decomposition of the identity is
I =
j
|φj 〉〈φj | =
j
[φj ]. (3.52)
A basis of H is a collection of linearly independent kets which span H in the sense that any element of H can be written as a linear combination of kets in the collection. Such a collection need not consist of normalized states, nor do they have to be mutually orthogonal. However, in this book we shall for the most part use orthonormal bases, and for this reason the adjective “orthonormal” will sometimes be omitted when doing so will not cause confusion.
3.6 Column Vectors, Row Vectors, and Matrices
Consider a Hilbert space H of dimension n, and a particular orthonormal basis. To make the notation a bit less cumbersome, let us label the basis kets as |j〉 rather than |φj 〉. Then (3.48) and (3.52) take the form
〈j|k〉 = δjk, (3.53) I =
j
|j〉〈j|, (3.54)
and any element |ψ〉 of H can be written as
|ψ〉 =
j
σj |j〉. (3.55)
By taking the inner product of both sides of (3.55) with |k〉, one sees that
σk = 〈k|ψ〉, (3.56)
and therefore (3.55) can be written as
|ψ〉 =
j
〈j|ψ〉|j〉 =
j
|j〉〈j|ψ〉. (3.57)
3.6. COLUMN VECTORS, ROW VECTORS, AND MATRICES 33
The form on the right side with the scalar coefficient 〈j|ψ〉 following rather than preceding the ket |j〉 provides a convenient way of deriving or remembering this result since (3.57) is the obvious equality |ψ〉 = I|ψ〉 with I replaced with the dyad expansion in (3.54). Using the basis {|j〉}, the ket |ψ〉 can be conveniently represented as a column vector of the coefficients in (3.57): (^)
〈 1 |ψ〉 〈 2 |ψ〉 · · · 〈n|ψ〉
.^ (3.58)
Because of (3.57), this column vector uniquely determines the ket |ψ〉, so as long as the basis is held fixed there is a one-to-one correspondence between kets and column vectors. (Of course, if the basis is changed, the same ket will be represented by a different column vector.) If one applies the dagger operation to both sides of (3.57), the result is
〈ψ| =
j
〈ψ|j〉〈j|, (3.59)
which could also be written down immediately using (3.54) and the fact that 〈ψ| = 〈ψ|I. The numerical coefficients on the right side of (3.59) form a row vector ( 〈ψ| 1 〉, 〈ψ| 2 〉,... 〈ψ|n〉
which uniquely determines 〈ψ|, and vice versa. This row vector is obtained by “transposing” the column vector in (3.58)—i.e., laying it on its side—and taking the complex conjugate of each element, which is the vector analog of 〈ψ| =
|ψ〉
. An inner product can be written as a row vector times a column vector, in the sense of matrix multiplication:
〈φ|ψ〉 =
j
〈φ|j〉〈j|ψ〉. (3.61)
This can be thought of as 〈φ|ψ〉 = 〈φ|I|ψ〉 interpreted with the help of (3.54). Given an operator A on H, its jk matrix element is
Ajk = 〈j|A|k〉, (3.62)
where the usual subscript notation is on the left, and the corresponding Dirac notation, see (3.17), is on the right. The matrix elements can be arranged to form a square matrix
〈 1 |A| 1 〉 〈 1 |A| 2 〉 · · · 〈 1 |A|n〉 〈 2 |A| 1 〉 〈 2 |A| 2 〉 · · · 〈 2 |A|n〉 · · · · · · · · · · · · · · · · · · · · · · · · 〈n|A| 1 〉 〈n|A| 2 〉 · · · 〈n|A|n〉
with the first or left index j of 〈j|A|k〉 labeling the rows, and the second or right index k labeling the columns. It is sometimes helpful to think of such a matrix as made up of a collection of n
3.7. DIAGONALIZATION OF HERMITIAN OPERATORS 35
An eigenvalue is said to be degenerate if it occurs more than once in (3.69), and its multiplicity is the number of times it occurs in the sum. An eigenvalue which only occurs once (multiplicity of one) is called nondegenerate. The identity operator has only one eigenvalue, equal to 1, whose multiplicity is the dimension n of the Hilbert space. A projector has only two distinct eigenvalues: 1 with multiplicity equal to the dimension m of the subspace onto which it projects, and 0 with multiplicity n − m. The basis which diagonalizes A is unique only if all its eigenvalues are non-degenerate. Otherwise this basis is not unique, and it is sometimes more convenient to rewrite (3.69) in an alternative form in which each eigenvalue appears just once. The first step is to suppose that, as is always possible, the kets |αj 〉 have been indexed in such a fashion that the eigenvalues are a non-decreasing sequence: a 1 ≤ a 2 ≤ a 3 ≤ · · ·. (3.71)
The next step is best explained by means of an example. Suppose that n = 5, and that a 1 = a 2 < a 3 < a 4 = a 5. That is, the multiplicity of a 1 is two, that of a 3 is one, and that of a 4 is two. Then (3.69) can be written in the form
A = a 1 P 1 + a 3 P 2 + a 4 P 3 , (3.72)
where the three projectors
P 1 = |α 1 〉〈α 1 | + |α 2 〉〈α 2 |, P 2 = |α 3 〉〈α 3 |, P 3 = |α 4 〉〈α 4 | + |α 5 〉〈α 5 |
form a decomposition of the identity. By relabeling the eigenvalues as
a′ 1 = a 1 , a′ 2 = a 3 , a′ 3 = a 4 , (3.74)
it is possible to rewrite (3.72) in the form
A =
j
a′ j Pj , (3.75)
where no two eigenvalues are the same:
a′ j 6 = a′ k for j 6 = k. (3.76)
Generalizing from this example, we see that it is always possible to write a Hermitian operator in the form (3.75) with eigenvalues satisfying (3.76). If all the eigenvalues of A are nondegenerate, each Pj projects onto a ray or pure state, and (3.75) is just another way to write (3.69). One advantage of using the expression (3.75), in which the eigenvalues are unequal, in preference to (3.69), where some of them can be the same, is that the decomposition of the identity {Pj } which enters (3.75) is uniquely determined by the operator A. On the other hand, if an eigenvalue of A is degenerate, the corresponding eigenvectors are not unique. In the example in (3.72) one could replace |α 1 〉 and |α 2 〉 by any two normalized and mutually orthogonal kets |α′ 1 〉 and |α′ 2 〉 belonging to the two-dimensional subspace onto which P 1 projects, and similar considerations apply to |α 4 〉 and |α 5 〉. We shall call the (unique) decomposition of the identity {Pj }which allows a Hermitian
36 CHAPTER 3. LINEAR ALGEBRA IN DIRAC NOTATION
operator A to be written in the form (3.75) with eigenvalues satisfying (3.76) the decomposition corresponding to or generated by the operator A. If {A, B, C,.. .} is a collection of Hermitian operators which commute with each other, (3.23), they can be simultaneously diagonalized in the sense that there is a single orthonormal basis |φj 〉 such that A =
j
aj [φj ], B =
j
bj [φj ], C =
j
cj [φj ], (3.77)
and so forth. If instead one writes the operators in terms of the decompositions which they generate, as in (3.75), A =
j
a′ j Pj , B =
k
b′ kQk, C =
l
c′ lRl, (3.78)
and so forth, the projectors in each decomposition commute with the projectors in the other decompositions: Pj Qk = QkPj , etc.
3.8 Trace
The trace of an operator A is the sum of its diagonal matrix elements:
Tr(A) =
j
〈j|A|j〉. (3.79)
While the individual diagonal matrix elements depend upon the orthonormal basis, their sum, and thus the trace, is independent of basis and depends only on the operator A. The trace is a linear operation in that if A and B are operators, and α and β are scalars,
Tr(αA + βB) = αTr(A) + βTr(B). (3.80)
The trace of a dyad is the corresponding inner product,
Tr
|φ〉〈τ |
j
〈j|φ〉〈τ |j〉 = 〈τ |φ〉, (3.81)
as is clear from (3.61). The trace of the product of two operators A and B is independent of the order of the product,
Tr(AB) = Tr(BA), (3.82)
and the trace of the product of three or more operators is not changed if one makes a cyclic permutation of the factors:
Tr(ABC) = Tr(BCA) = Tr(CAB), Tr(ABCD) = Tr(BCDA) = Tr(CDAB) = Tr(DABC),
and so forth; the cycling is done by moving the operator from the first position in the product to the last, or vice versa. By contrast, Tr(ACB) is, in general, not the same as Tr(ABC), for
38 CHAPTER 3. LINEAR ALGEBRA IN DIRAC NOTATION
whose trace is equal to 1. In quantum physics a positive operator with trace equal to 1 is called a density matrix. The terminology is unfortunate, because a density matrix is an operator, not a matrix, and the matrix for this operator depends on the choice of orthonormal basis. However, by now the term is firmly embedded in the literature, and attempts to replace it with something more rational, such as “statistical operator”, have not been successful. If C is any operator, then C†C is a positive operator, since for any |ψ〉, 〈ψ|C†C|ψ〉 = 〈φ|φ〉 ≥ 0 , (3.90)
where |φ〉 = C|ψ〉. Consequently, Tr(C†C) is non-negative. If Tr(C†C) = 0, then C†C = 0, and 〈ψ|C†C|ψ〉 vanishes for every |ψ〉, which means that C|ψ〉 is zero for every |ψ〉, and therefore C = 0. Thus for any operator C it is the case that
Tr(C†C) ≥ 0 , (3.91)
with equality if and only if C = 0. The product AB of two positive operators A and B is, in general, not Hermitian, and therefore not a positive operator. However, if A and B commute, AB is positive, as can be seen from the fact that there is an orthonormal basis in which the matrices of both A and B, and therefore also AB are diagonal. This result generalizes to the product of any collection of commuting positive operators. Whether or not A and B commute, the fact that they are both positive means that Tr(AB) is a real, non-negative number,
Tr(AB) = Tr(BA) ≥ 0 , (3.92)
equal to 0 if and only if AB = BA = 0. This result does not generalize to a product of three or more operators: if A, B, and C are positive operators that do not commute with each other, there is in general nothing one can say about Tr(ABC). To derive (3.92) it is convenient to first define the square root A^1 /^2 of a positive operator A by means of the formula A^1 /^2 =
j
aj [αj ], (3.93)
where √aj is the positive square root of the eigenvalue aj in (3.69). Then when A and B are both positive, one can write
Tr(AB) = Tr(A^1 /^2 A^1 /^2 B^1 /^2 B^1 /^2 ) = Tr(A^1 /^2 B^1 /^2 B^1 /^2 A^1 /^2 ) = Tr(C†C) ≥ 0 ,
where C = B^1 /^2 A^1 /^2. If Tr(AB) = 0, then, see (3.91), C = 0 = C†, and both BA = B^1 /^2 CA^1 /^2 and AB = A^1 /^2 C†B^1 /^2 vanish.
3.10 Functions of Operators
Suppose that f (x) is an ordinary numerical function, such as x^2 or ex. It is sometimes convenient to define a corresponding function f (A) of an operator A, so that the value of the function is also an operator. When f (x) is a polynomial
f (x) = a 0 + a 1 x + a 2 x^2 + · · · apxp, (3.95)
3.10. FUNCTIONS OF OPERATORS 39
one can write f (A) = a 0 I + a 1 A + a 2 A^2 + · · · apAp, (3.96)
since the square, cube, etc. of any operator is is itself an operator. When f (x) is represented by a power series, as in log(1 + x) = x − x^2 /2 + · · · , the same procedure will work provided the series in powers of the operator A converges, but this is often not a trivial issue. An alternative approach is possible in the case of operators which can be diagonalized in some orthonormal basis. Thus if A is written in the form (3.69), one can define f (A) to be the operator
f (A) =
j
f (aj ) [αj ], (3.97)
where f (aj ) on the right side is the value of the numerical function. This agrees with the previous definition in the case of polynomials, but allows the use of much more general functions. As an example, the square root A^1 /^2 of a positive operator A as defined in (3.93) is obtained by setting f (x) =
x for x ≥ 0 in (3.97). Note that in order to use (3.97), the numerical function f (x) must be defined for any x which is an eigenvalue of A. For a Hermitian operator these eigenvalues are real, but in other cases, such as the unitary operators discussed in Sec. 7.2, the eigenvalues may be complex, so for such operators f (x) will need to be defined for suitable complex values of x.
