Griffiths R.B. Consistent Quantum Theory

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74 Probabilities and physical variables

about a system when its actual state is not known. For example, if the probability

of some state is close to 1, one can be fairly conﬁdent, but not absolutely cer-

tain, that a system chosen at random will be in this state and not in some other

state.

Rather than imagining the ensemble to be a large collection of systems, it is

sometimes useful to think of it as made up of the outcomes of a large number of

experiments carried out at successive times, with care being taken to ensure that

these are independent in the sense that the outcome of any one experiment is not

allowed to inﬂuence the outcome of later experiments. Thus instead of a large

collection of dice, one can think of a single die which is rolled a large number of

times. The fraction of experiments leading to the result s is then the probability p

The outcome of any particular experiment in the sequence is not known in advance,

but a knowledge of the probabilities provides partial information.

Probability theory as a mathematical discipline does not tell one how to choose

a probability distribution. Probabilities are sometimes obtained directly from ex-

perimental data. In other cases, such as the Boltzmann distribution for systems in

thermal equilibrium, the probabilities express well-established physical laws. In

some cases they simply represent a guess. Later we shall see how to use the dy-

namical laws of quantum theory to calculate various quantum probabilities. The

true meaning of probabilities is a subject about which there continue to be disputes,

especially among philosophers. These arguments need not concern us, for prob-

abilities in quantum theory, when properly employed with a well-deﬁned sample

space, obey the same rules as in classical physics. Thus the situation in quantum

physics is no worse (or better) than in the everyday classical world.

Conditional probabilities play a fundamental role in probabilistic reasoning and

in the application of probability theory to quantum mechanics. Let A and B be two

events, and suppose that Pr(B)>0. The conditional probability of A given B is

deﬁned to be

Pr(A | B) = Pr( A ∧ B)/ Pr(B), (5.23)

where A∧B is the event “A AND B” represented by the product ABof the classical

indicators, or of the quantum projectors. Hence one can also write Pr(AB) in place

of Pr( A ∧ B). The intuitive idea of a conditional probability can be expressed in

the following way. Given an ensemble, consider only those members in which B

occurs (is true). These comprise a subensemble of the original ensemble, and in

this subensemble the fraction of systems with property A is given by Pr(A | B)

rather than by Pr(A), as in the original ensemble. For example, in the case of a

die, let B be the property that s is even, and A the property s ≤ 3. Assuming

equal probabilities for all outcomes, Pr(A) = 1/2. However, Pr(A | B) = 1/3,

5.4 Probabilities and ensembles 75

corresponding to the fact that of the three possibilities s = 2, 4, 6 which constitute

the compound event B, only one is less than or equal to 3.

If B is held ﬁxed, Pr(A | B) as a function of its ﬁrst argument A behaves like

an “ordinary” probability distribution. For example, if we use s to indicate points

in the sample space, the numbers Pr(s | B) are nonnegative, and



Pr(s | B) =

1. One can think of Pr( A | B) with B ﬁxed as obtained by setting to zero the

probabilities of all elements of the sample space for which B is false (does not

occur), and multiplying the probabilities of those elements for which B is true

by a common factor, 1/ Pr(B), to renormalize them, so that the probabilities of

mutually-exclusive sets of events sum to one. That this is a reasonable procedure

is evident if one imagines an ensemble and thinks about the subensemble of cases

in which B occurs. It makes no sense to deﬁne a probability conditioned on B if

Pr(B) = 0, as there is no way to renormalize zero probability by multiplying it by

a constant in order to get something ﬁnite.

In the case of quantum systems, once an appropriate sample space has been de-

ﬁned the rules for manipulating probabilities are precisely the same as for any other

(“classical”) probabilities. The probabilities must be nonnegative, they must sum

to 1, and conditional probabilities are deﬁned in precisely the manner discussed

above. Sometimes it seems as if quantum probabilities obey different rules from

what one is accustomed to in classical physics. The reason is that quantum the-

ory allows a multiplicity of sample spaces, that is, decompositions of the identity,

which are often incompatible with one another. In classical physics a single sam-

ple space is usually sufﬁcient, and in cases in which one may want to use more

than one, for example alternative coarse grainings of the phase space, the different

possibilities are always compatible with each other. However, in quantum theory

different sample spaces are generally incompatible with one another, so one has

to learn how to choose the correct sample space needed for discussing a particular

physical problem, and how to avoid carelessly combining results from incompati-

ble sample spaces. Thus the difﬁculties one encounters in quantum mechanics have

to do with choosing a sample space. Once the sample space has been speciﬁed, the

quantum rules are the same as the classical rules.

There have been, and no doubt will continue to be, a number of proposals for

introducing special “quantum probabilities” with properties which violate the usual

rules of probability theory: probabilities which are negative numbers, or complex

numbers, or which are not tied to a Boolean algebra of projectors, etc. Thus far,

none of these proposals has proven helpful in untangling the conceptual difﬁculties

of quantum theory. Perhaps someday the situation will change, but until then there

seems to be no reason to abandon standard probability theory, a mode of reasoning

which is quite well understood, both formally and intuitively, and replace it with

some scheme which is deﬁcient in one or both of these respects.

76 Probabilities and physical variables

5.5 Random variables and physical variables

In ordinary probability theory a random variable is a real-valued function V de-

ﬁned everywhere on the sample space. For example, if s is the number of spots

when a die is rolled, V(s) = s is an example of a random variable, as is V(s) =

/6. For coin tossing, V(H) =+1/2, V(T) =−1/2 is an example of a random

variable.

If one regards the x, p phase plane for a particle in one dimension as a sample

space, then any real-valued function V(x, p) is a random variable. Examples of

physical interest include the position, the momentum, the kinetic energy, the po-

tential energy, and the total energy. For a particle in three dimensions the various

components of angular momentum relative to some origin are also examples of

random variables.

In classical mechanics the term physical variable is probably more descriptive

than “random variable” when referring to a function deﬁned on the phase space,

and we shall use it for both classical and quantum systems. However, thinking of

physical variables as random variables, that is, as functions deﬁned on a sample

space, is particularly helpful in understanding what they mean in quantum theory.

The quantum counterpart of the function V representing a physical variable in

classical mechanics is a Hermitian or self-adjoint operator V = V

†

on the Hilbert

space. Thus position, energy, angular momentum, and the like all correspond to

speciﬁc quantum operators. Generalizing from this, we shall think of any self-

adjoint operator on the Hilbert space as representing some (not necessarily very

interesting) physical variable. A quantum physical variable is often called an ob-

servable. While this term is not ideal, given its association with somewhat con-

fused and contradictory ideas about quantum measurements, it is widely used in

the literature, and in this book we shall employ it to refer to any quantum physical

variable, that is, to any self-adjoint operator on the quantum Hilbert space, without

reference to whether it could, in practice or in principle, be measured.

To see how self-adjoint operators can be thought of as random variables in the

sense of probability theory, one can make use of a fact discussed in Sec. 3.7: if

V = V

†

, then there is a unique decomposition of the identity {P

}, determined by

the operator V, such that, see (3.75),

V =





, (5.24)

where the v



are eigenvalues of V, and v



= v



for j = k. Since any decomposition

of the identity can be regarded as a quantum sample space, one can think of the

collection {P

} as the “natural” sample space for the physical variable or operator

V. On this sample space the operator V behaves very much like a real-valued

function: to P

it assigns the value v



,toP

the value v



, and so forth. That

5.5 Random variables and physical variables 77

(5.24) can be interpreted in this way is suggested by the fact that for a discrete

sample space S, an ordinary random variable V can always be written as a sum of

numbers times the elementary indicators deﬁned in (5.6),

V(s) =



(s), (5.25)

where v

= V(r). Since quantum projectors are analogous to classical indicators,

and the indicators on the right side of (5.25) are associated with the different ele-

ments of the sample space, there is an obvious and close analogy between (5.24)

and (5.25).

The only possible values for a quantum observable V are the eigenvalues v



(5.24) or, equivalently, thev

in (5.32), just as the only possible valuesof a classical

random variable are the v

in (5.25). In order for a quantum system to possess the

value v for the observable V, the property “V = v” must be true, and this means

that the system is in an eigenstate of V. That is to say, the quantum system is

described by a nonzero ket |ψ such that

V|ψ=v|ψ, (5.26)

or, more generally, by a nonzero projector Q such that

VQ= v Q. (5.27)

In order for (5.27) to hold for a projector Q onto a space of dimension 2 or more,

the eigenvalue v must be degenerate, and if v = v



, then

Q = Q, (5.28)

where P

is the projector in (5.24) corresponding to v



Let us consider some examples, beginning with a one-dimensional harmonic

oscillator. Its (total) energy corresponds to the Hamiltonian operator H, which can

be written in the form

H =



(n + 1/2)

hω [φ

], (5.29)

where the corresponding decomposition of the energy was introduced in (5.17).

The Hamiltonian can thus be thought of as a function which assigns to the projector

[φ

], or to the subspace of multiples of |φ

, the energy (n + 1/2)

hω. In the case

of a spin-half particle the operator for the z component of spin angular momentum

divided by

h is

] −

−

]. (5.30)

It assigns to [z

] the value +1/2, and to [z

−

] the value −1/2. Next think of a toy

78 Probabilities and physical variables

model in which the sites are labeled by an integer m, and suppose that the distance

between adjacent sites is the length b. Then the position operator will be given by

B =



mb[m]. (5.31)

The position operator x for a “real” quantum particle in one dimension is a com-

plicated object, and writing it in a form equivalent to (5.24) requires replacing the

sum with an integral, using mathematics which is outside the scope of this book.

In all the examples considered thus far, the P

are projectors onto one-dimen-

sional subspaces, sotheycan be written as dyads, and (5.24) is equivalent to writing

V =



|ν

ν



[ν

], (5.32)

where the eigenvalues in (5.32) are identical to those in (5.24), except that the sub-

script labels may be different. As discussed in Sec. 3.7, (5.24) and (5.32) will be

different if one or more of the eigenvalues of V are degenerate, that is, if a partic-

ular eigenvalue occurs more than once on the right side of (5.32). For instance, the

energy eigenvalues of atoms are often degenerate due to spherical symmetry, and

in this case the projector P

for the jth energy level projects onto a space whose

dimension is equal to the multiplicity (or degeneracy) of the level. When such

degeneracies occur, it is possible to construct nontrivial reﬁnements of the decom-

position {P

} in the sense discussed in Sec. 5.3, by writing one or more of the P

as a sum of two or more nonzero projectors. If {Q

} is such a reﬁnement, it is

obviously possible to write

V =





, (5.33)

where the extra prime allows the eigenvalues in (5.33) to carry different subscripts

from those in (5.24). One can again think of V as a random variable, that is, a

function, on the ﬁner sample space {Q

}. Note that when it is possible to reﬁne

a quantum sample space in this manner, it is always possible to reﬁne it in many

different ways which are mutually incompatible. Whereas any one of these sample

spaces is perfectly acceptable so far as the physical variable V is concerned, one

will make mistakes if one tries to combine two or more incompatible sample spaces

in order to describe a single physical system; see the comments in Sec. 5.3.

On the other hand, V cannot be deﬁned as a physical (“random”) variable on a

decomposition which is coarser than {P

}, since one cannot assign two different

eigenvalues to the same projector or subspace. (To be sure, one might deﬁne a

“coarse” version of V, but that would be a different physical variable.) Nor can V

be deﬁned as a physical or random variable on a decomposition which is incom-

patible with {P

}, in the sense discussed in Sec. 5.3. It may, of course, be possible

5.6 Averages 79

to approximate V with an operator which is a function on an alternative decompo-

sition, but such approximations are outside the scope of the present discussion.

5.6 Averages

The average V of a random variable V(s) on a sample space S is deﬁned by the

formula

V=



s∈S

V(s). (5.34)

That is, the probabilities are used to weight the values of V at the different sample

points before adding them together. One way to justify the weights in (5.34) is

to imagine an ensemble consisting of a very large number N of systems. If V is

evaluated for each system, and the results are then added together and divided by

N, the outcome will be (5.34), because the fraction of systems in the ensemble in

state s is equal to p

Random variables form a real linear space in the sense that if U(s) and V(s) are

two random variables, so is the linear combination

uU(s) + vV(s), (5.35)

where u and v are real numbers. The average operation deﬁned in (5.34) is a

linear functional on this space, since

uU(s) + vV(s)=uU+vV. (5.36)

Another property of is that when it is applied to a positive random variable

W(s) ≥ 0, the result cannot be negative:

W≥0. (5.37)

In addition, the average of the identity is 1,

I=1, (5.38)

because the probabilities {p

} sum to 1.

The linear functional is obviously determined once the probabilities {p

} are

given. Conversely, a functional deﬁned on the linear space of random variables

determines a unique probability distribution, since one can use averages of the

elementary indicators in (5.6),

=P

, (5.39)

in order to deﬁne positive probabilities which sum to 1 in view of (5.9) and (5.38).

80 Probabilities and physical variables

In a similar way, the probability of a compound event A is equal to the average of

its indicator:

Pr(A) =A. (5.40)

Averages for quantum mechanical physical (random) variables follow precisely

the same rules; the only differences are in notation. One starts with a sample space

} of projectors which sum to I, and a set of nonnegative probabilities {p

}

which sum to 1. A random variable on this space is a Hermitian operator which

can be written in the form

V =



, (5.41)

where the different eigenvalues appearing in the sum need not be distinct. That is,

the sample space could be either the “natural” space associated with the operator

V as discussed in Sec. 5.5, or some reﬁnement. The average

V=



(5.42)

is formally equivalent to (5.34).

A probability distribution on a given sample space can only be used to calculate

averages of random variables deﬁned on this sample space; it cannot be used, at

least directly, to calculate averages of random variables which are deﬁned on some

other sample space. While this is rather obvious in ordinary probability theory, its

quantum counterpart is sometimes overlooked. In particular, the probability distri-

bution associated with {P

} cannot be used to calculate the average of a self-adjoint

operator S whose natural sample space is a decomposition {Q

} incompatible with

}. Instead one must use a probability distribution for the decomposition {Q

An alternative way of writing (5.41) is the following. The positive operator

ρ =



/ Tr(P

) (5.43)

has a trace equal to 1, so it is a density matrix, as deﬁned in Sec. 3.9. It is easy to

show that

V=Tr(ρV) (5.44)

by applying the orthogonality conditions (5.10) to the product ρV. Note that ρ and

V commute with each other. The formula (5.44) is sometimes used in situations in

which ρ and V do not commute with each other. In such a case ρ is functioning as

a pre-probability, as will be explained in Ch. 15.

Composite systems and tensor products

6.1 Introduction

A composite system is one involving more than one particle, or a particle with in-

ternal degrees of freedom in addition to its center of mass. In classical mechanics

the phase space of a composite system is a Cartesian product of the phase spaces

of its constituents. The Cartesian product of two sets A and B is the set of (or-

dered) pairs {(a, b)}, where a is any element of A and b is any element of B.For

three sets A, B, and C the Cartesian product consists of triples {(a, b, c)}, and so

forth. Consider two classical particles in one dimension, with phase spaces x

, p

and x

, p

. The phase space for the composite system consists of pairs of points

from the two phase spaces, that is, it is a collection of quadruples of the form

, p

, x

, p

, which can equally well be written in the order x

, x

, p

. This is

formally the same as the phase space of a single particle in two dimensions, a col-

lection of quadruples x, y, p

, p

. Similarly, the six-dimensional phase space of a

particle in three dimensions is formally the same as that of three one-dimensional

particles.

In quantum theory the analog of a Cartesian product of classical phase spaces

is a tensor product of Hilbert spaces. A particle in three dimensions has a Hilbert

space which is the tensor product of three spaces, each corresponding to motion

in one dimension. The Hilbert space for two particles, as long as they are not

identical, is the tensor product of the two Hilbert spaces for the separate particles.

The Hilbert space for a particle with spin is the tensor product of the Hilbert space

of wave functions for the center of mass, appropriate for a particle without spin,

with the spin space, which is two-dimensional for a spin-half particle.

Not only are tensor products used in quantum theory for describing a composite

system at a single time, they are also very useful for describing the time develop-

ment of a quantum system, as we shall see in Ch. 8. Hence any serious student

82 Composite systems and tensor products

of quantum mechanics needs to become familiar with the basic facts about tensor

products, and the corresponding notation, which is summarized in Sec. 6.2.

Special rules apply to the tensor product spaces used for identical quantum par-

ticles. For identical bosons one uses the symmetrical subspace of the Hilbert space

formed by taking a tensor product of the spaces for the individual particles, while

for identical fermions one uses the antisymmetrical subspace. The basic procedure

for constructing these subspaces is discussed in various introductory and more ad-

vanced textbooks (see references in the Bibliography), but the idea behind it is

probably easiest to understand in the context of quantum ﬁeld theory, which lies

outside the scope of this book. While we shall not discuss the subject further, it

is worth pointing out that there are a number of circumstances in which the fact

that the particles are identical can be ignored — that is, one makes no signiﬁcant

error by treating them as distinguishable — because they are found in different lo-

cations or in different environments. For example, identical nuclei in a solid can be

regarded as distinguishable as long as it is a reasonable physical approximation to

assume that they are approximately localized, e.g., found in a particular unit cell,

or in a particular part of a unit cell. In such cases one can construct the tensor

product spaces in a straightforward manner using the principles described below.

6.2 Deﬁnition of tensor products

Given two Hilbert spaces A and B, their tensor product A ⊗ B can be deﬁned

in the following way, where we assume, for simplicity, that the spaces are ﬁnite-

dimensional. Let {|a

 : j = 1, 2,...m} be an orthonormal basis for the m-

dimensional space A, and {|b

 : p = 1, 2,...n} an orthonormal basis for the

n-dimensional space B, so that

a

=δ

, b

=δ

. (6.1)

Then the collection of mn elements

⊗|b

 (6.2)

forms an orthonormal basis of the tensor product A ⊗ B, which is the set of all

linear combinations of the form

|ψ=





⊗|b



, (6.3)

where the γ

are complex numbers.

Given kets

|a=



, |b=



 (6.4)

6.2 Deﬁnition of tensor products 83

in A and B, respectively, their tensor product is deﬁned as

|a⊗|b=





⊗|b



, (6.5)

which is of the form (6.3) with

= α

. (6.6)

The parentheses in (6.3) and (6.5) are not really essential, since



α|a



⊗|b is

equal to α



|a⊗|b



, and we shall henceforth omit them when this gives rise to

no ambiguities. The deﬁnition (6.5) implies that the tensor product operation ⊗ is

distributive:

|a⊗





+β





= β



|a⊗|b



+β



|a⊗|b



,





+α





⊗|b=α



⊗|b+α



⊗|b.

(6.7)

An element of A ⊗B which can be written in the form |a⊗|b is called a prod-

uct state, and states which are not product states are said to be entangled. When

several coefﬁcients in (6.3) are nonzero, it may not be readily apparent whether the

corresponding state is a product state or entangled, that is, whether or not γ

can

be written in the form (6.6). For example,

1.0|a

⊗|b

+0.5|a

⊗|b

−1.0|a

⊗|b

−0.5|a

⊗|b

 (6.8)

is a product state



−|a



⊗



+0.5|b



, whereas changing the sign of the

last coefﬁcient yields an entangled state:

1.0|a

⊗|b

+0.5|a

⊗|b

−1.0|a

⊗|b

+0.5|a

⊗|b

. (6.9)

The linear functional or bra vector corresponding to the product state |a⊗|b

is written as



|a⊗|b



†

=a|⊗b|, (6.10)

where the A ⊗ B order of the factors on either side of ⊗ does not change when

the dagger operation is applied. The result for a general linear combination (6.3)

follows from (6.10) and the antilinearity of the dagger operation:

ψ|=



|ψ



†



∗

a

|⊗b

|. (6.11)

Consistent with these formulas, the inner product of two product states is given by



|a⊗|b



†





⊗|b





=a|a



·b|b



, (6.12)

and of a general state |ψ, (6.3), with another state

|ψ



=





⊗|b

, (6.13)