Download Time in quantum mechanics

Time in quantum mechanics
Jan Hilgevoord
University of Utrecht
David Atkinson
University of Groningen
CLARENDON PRESS
2011
.
OXFORD
0.1
The problem of time in quantum mechanics
Many physicists believe that time constitutes a serious problem in quantum
mechanics. The difficulty was epitomized in Wolfgang Pauli’s Handbook article
of 1933 (Pauli 1933: 140):
We conclude therefore that the introduction of an operator t must be renounced
as a matter of principle, and that time t must necessarily be considered as an
ordinary number (‘c-number’) in wave mechanics. (Translation: JH and DA).
Pauli’s article signalled the conclusion of a period of rapid development of the
new quantum theory that had been inaugurated by Heisenberg’s revolutionary
paper of 1925 (Heisenberg 1925). An essential feature of quantum theory is that
the dynamical variables of classical mechanics are represented by self-adjoint
operators on a Hilbert space. The basic example is that of the position and
momentum variables q and p of a particle, which are replaced by self-adjoint
operators satisfying the commutation relation
p q − qp = −ih̄ ,
(0.1)
h
, and h is Planck’s constant. If one assumes q and p to have
where h̄ = 2π
continuous eigenvalues running from −∞ to +∞ on the real axis, the well-known
unique solution of (0.1) is q = q, p = −ih̄ d/dq, on a suitable space of functions
(up to unitary equivalence, and modulo irreducibility).
In a second famous article (Heisenberg 1927) Heisenberg sought to clarify
the physical meaning of relation (0.1) by considering experiments in which the
position and momentum of a particle could be measured. He concluded that
these quantities could not both be measured with arbitrary precision in the
same experiment. This was expressed by the uncertainty relation
δp δq ∼ h̄
(0.2)
where δp and δq are the precisions, or uncertainties, with which the values of p
and q are known. Actually, by introducing appropriate definitions of the quantities δp and δq, relation (0.2) can be shown to be a direct consequence of relation
(0.1). For example, in terms of the standard deviation as a measure of uncertainty, the inequality
∆p∆q ≥ 21 h̄
(0.3)
can be derived. In the same article two more commutation relations are presented:
Et − tE = −ih̄
or
Jw − wJ = −ih̄ .
(0.4)
The quantities J and w are the conjugate variables that typically appear in the
description of periodic systems. Classically J and w, like p and q, form a pair
of conjugate variables, and so the second of the relations (0.4) is analogous to
(0.1). The meaning of the first equation (0.4) is however much less clear. In some
of the passages following equations (0.4), Heisenberg identifies E with J and t
with w, calling t a ‘phase’, that is, t is considered to be an internal variable
2
of the system. This interpretation reflects the ‘or’ between the equations (0.4).
But in other passages, and notably in the examples leading up to the uncertainty
relation δE δt ∼ h̄, t is treated as a classical time parameter, clearly contradicting
(0.4).
Why did Heisenberg present the ‘same’ formula (0.4) in two different guises?
In classical mechanics the time parameter is sometimes turned into an internal
dynamical variable conjugate to (minus) the Hamiltonian of the system. Heisenberg may have had this in mind in connection with the first equation (0.4),
although the minus sign in that equation would not then be correct. The notation also suggests a connection to Eq.(0.1). In relativity theory the momentum
p and energy E of a particle are the components of a four-vector, and it is quite
common to consider the position q of the particle and the time parameter t to
form a four-vector also. The first equation (0.4) might then be seen as a natural
complement of Eq.(0.1), as dictated by relativity theory. We shall come back to
this matter in the next section.
At about the same time it had become clear that Eqs.(0.4) can be mathematically problematic. In many cases (but not all) the range of the eigenvalues
of J is the positive real axis, while the eigenvalues of w are angles in the interval
[0, 2π]. It can be shown that self-adjoint operators whose eigenvalues satisfy these
conditions cannot also satisfy the second relation (0.4). Also, it is not possible for
the first equation (0.4) to be correct if E is bounded from below, and if the eigenvalue spectrum of t is the whole real axis (the latter condition being a necessary
requirement if t is to be interpreted as the time parameter). Since in many cases
the energy operator is known to be a well-behaved self-adjoint operator that is
indeed bounded from below, it seems to follow that an acceptable time operator
does not exist in quantum mechanics. Whence Pauli’s verdict mentioned at the
beginning; and as a consequence the relations (0.4) have passed into oblivion.
For a more complete account of this early period (see Hilgevoord 2005).
The asymmetry between space and time that seems to be implied by Pauli’s
statement, apparently contradicting the principles of relativity, has bothered
physicists for a long time. Many proposals for circumventing the difficulty have
been put forward, in particular a generalization of the axiom that observables
must correspond to self-adjoint operators on Hilbert space. One can weaken the
axiom to the postulate that “each observable is associated to a Positive Operator
Valued Measure (POVM)” (Egusquiza, Gonzalo Muga and Baute 2002: 283; see
also Busch 1989). POVMs are interesting in their own right, having many practical applications, but we shall not discuss them here, since we believe their use
as a way of nullifying Pauli’s objection to be fundamentally misdirected. Again
much attention has been given to finding an analogue of the uncertainty relation
(0.2) in the case of energy and time. If time is not an operator, a relation of this
type cannot exist. Nevertheless there do exist ‘uncertainty’ relations between
energy and time of a different kind, for example the relation between the energy
spread and the lifetime of a quantum state. The existence of such relations, in
which t is an ordinary number, might suggest a certain similarity with momen-
The problem dissolved
3
tum and position, but by the same token there appears to be a fundamental
difference between position and time in quantum mechanics.
Notwithstanding all these considerations, we shall show in the next section
that quantum mechanics does not involve a special problem for time, and that
there is no fundamental asymmetry between space and time in quantum mechanics over and above the asymmetry that already exists in classical physics.
In Sect. 3 we study time operators in detail, and in Sect. 4 various uncertainty
relations involving time are discussed.
0.2
The problem dissolved
To see that time poses no problem for quantum mechanics, one must distinguish
between two ways that it can appear in physics. First, time may figure as a
general parameter of the theory, and secondly it may appear as a dynamical
internal variable of some particular physical system described by the theory.
In its first guise, time t is on a par with the spatial coordinates x, y, z. This
is explicit in relativity theory, where t is added as a fourth coordinate to the
space coordinates to form a relativistic four-vector (x, y, z, ct). Like the space
coordinates, the time coordinate is independent of the physical systems the theory describes. In quantum mechanics the space and time coordinates remain
c-numbers; neither x, y, z, nor t become operators. The space-time coordinates
appear as parameters in the definition of well-known space-time symmetries, for
example rotation invariance, spatial and temporal translation invariance, and
Lorentz-invariance.
In its second guise, time is a dynamical variable belonging to a particular
physical system, and more than one such time variable may be present in the
system. Like the other dynamical variables, dynamical time variables become
operators in quantum mechanics. Examples of time operators are discussed in
the following section, where we will see that time operators, although they are less
common than position operators, are not fundamentally problematic in quantum
theory.
The work of Heisenberg that we discussed in the previous section clearly
betrays a confusion between the two ways in which time occurs in physics. On
the one hand, t in the first of Eqs.(0.4) is considered to be the analogue of w,
an internal dynamical variable of the system; but on the other hand t is looked
upon as the unique external time parameter. Also, Pauli seems to understand by
t the general time parameter, the c-number character of which we have seen to
be in fact unproblematic. The apparent problem of time arises when this time
parameter is put on a par with dynamical position variables rather than with
the coordinates of space. The confusion has proved to be quite persistent in the
quantum mechanics literature, and in the remainder of this section we will try
to see how this has come about.
Ironically, the origin of the problem is to be found in space rather than
in time, and its roots lie in classical mechanics. Much of fundamental physics
deals with ‘point’ particles. A point particle is a material object that can have
4
a position, a momentum, a mass, an energy, a charge, etc. At any moment
the particle is located at a point of space. Evidently a point particle and a
point of space are very different things. Nevertheless they are not always clearly
distinguished. Quite often the coordinates of space and the position variables of
a point particle are denoted by the same symbols x, y, z (e.g. when one writes
ψ(x, y, z, t) for the wave function of a particle). To avoid this confusion we shall
denote the dynamical position variables of a particle by q = (qx , qy , qz ), reserving
the symbols x, y, z for the coordinates of a point of space.
The same confusion has led to the erroneous view that the position q of a
particle and the time coordinate t form a relativistic four-vector. Though this
may be true numerically, since the numbers qx , qy , qz can coincide with the numbers x, y, z, the variable q and the coordinate t are conceptually quite distinct.
This becomes evident when there are several particles, which must share one
and the same t. Conflating position variables of particles and space coordinates
is responsible for the view that space is quantized whereas time is not, creating
the false impression of an asymmetry between the treatment of space and time in
quantum mechanics. We have seen that such an asymmetry does not exist, since
neither the space coordinates x, y, z, nor the time coordinate t are quantized. If
t is not the relativistic partner of q, what is the true partner of the latter? The
answer is simply that such a partner does not exist; the position variable of a
point particle is a non-covariant concept. It is an interesting fact that, whereas
in classical physics the non-covariance of q may easily remain hidden because the
pair q, t behaves as a four-vector, in quantum mechanics the non-covariance of q
is very clear. Newton and Wigner were the first to show that an operator that
represents the position of a particle must necessarily be non-covariant (Newton
and Wigner 1949; Schweber 1962: 60-62).
It is sometimes said that a worldline xµ (τ ) does provide a covariant description of a point particle, where xµ = (x, y, z, ct) and τ is the proper time along the
curve. However, just as xµ is only a point of spacetime, conceptually unrelated
to a particle, xµ (τ ) is just a curve in spacetime. And, just as the position q of a
point particle at time t may coincide with the spatial components of the point
xµ , so the orbit q(t) can coincide with the curve xµ (τ ) (though only if the tangent vector to the curve is timelike at every point). But the dynamical position
variable q of the particle remains without a relativistic partner. Likewise, the
dynamical time operators, to be discussed in the next section, are non-covariant
quantities. We conclude that time gives rise to no special problem in quantum
mechanics. For a fuller discussion see Hilgevoord (2005).
0.3 Time operators
An ideal clock is a device or system which produces a reading that mimics
coordinate time in much the same way that the position of a point particle
mimics coordinate space. Let us consider this analogy in detail for a system of
n free particles. The canonical commutation relations are
[q x,i , px,j ] = i I δij
[q x,i , q x,j ] = 0 = [px,i , px,j ] ,
(0.5)
Time operators
5
for i, j = 1, 2, . . . , n, where q x,i , px,i are the linear operators representing the
x-components of the positions and momenta, respectively, of the n particles.
Similar relations apply of course for the y and z components. Units are such
that h̄ = 1, and I is a unit operator on a suitable subspace of Hilbert space. In
this section and the following one we shall consistently use the convention that
operators are designated by bold letters, ordinary or c-numbers by normal fonts.
A clock reading is represented by an eigenvalue of a clock variable operator,
which may be either any real number between −∞ and ∞ (we then speak of
the linear clock), or it may be limited to the interval [0, 2π] (this is the cyclic or
periodic clock). There are interesting differences between the mathematics of the
linear and of the cyclic clocks, and we will concentrate first exclusively on the
linear clock, since the mathematics parallels that of the point particles, returning
to the cyclic clock in Subsect. 3.2.
0.3.1
The linear clock
Suppose that the clock variables associated with n clocks are represented by the
linear operators τi , and let the canonically conjugate variables be called η i . The
canonical commutation relations are
[τi , η j ] = i I δij
[τi , τj ] = 0 = [η i , η j ] .
(0.6)
Evidently these commutation relations are analogous to Eq.(0.5).
~,
The generator of a translation in space is the total momentum operator P
whereas the generator of a translation in time is the Hamiltonian H. Consider
first an infinitesimal translation, dx, in the x-direction. This is generated by Px ,
so any canonical coordinate, Ω, changes under the transformation to Ω + dΩ,
where
d Ω = [Ω, P x ]dx/i .
The y and z components of ~q i , and all the Cartesian components of ~pi , should
remain unchanged under this transformation. For the system of n point particles, only the position operators q x,i should be changed under the translation
generated by P x . Evidently the x-component of
~ =
P
n
X
~pi
i=1
generates precisely the required transformation. Thus
∂ q x,i
= [ q x,i , Px ]/i = I ,
∂x
and so the operators q x,i are linear functions of coordinate space, indeed
q x,i = q 0x,i + I x ,
(0.7)
6
where the q 0x,i are (operator) constants of integration (i.e. they are independent
of x). The eigenvalues of the shifted operators
Qx,i = q x,i − q 0x,i
are all equal to x, and in this sense the particle positions mimic coordinate
space. Indeed, it is from this fact that the pernicious error arises of identifying
the shifted operators with coordinate space, x. These operators are equal to
I x, not x, and the parallel distinction in the case of the clock is of cardinal
importance in resolving much confusion about time in quantum mechanics.
An infinitesimal time-translation, dt, is generated by H, changing any canonical coordinate Ω to Ω + d Ω, where
dΩ = [Ω, H]dt/i .
For a system of n ideal clocks, the clock variables τi should all be augmented
under this transformation by dt,
τ i → τ i + Idt ,
(0.8)
the variables ηi being unchanged. In this way the ideal clocks are close analogues
of the point masses: the position operators of the point masses are boosted
in space by the total momentum operator, while the ideal clock variables are
boosted in time by the Hamiltonian. The analogue of Eq.(0.7) is
H =
n
X
ηi .
(0.9)
i=1
This ensures that the clock variables transform as in Eq.(0.8), and that the ηi
remain unchanged. The clock variables in fact satisfy
dτi
= [τi , H]/i = I ,
dt
and so they are linear functions of time,
τi = τi0 + I t .
It is crucial to distinguish here coordinate time t from the clock readings that
are the eigenvalues of the clock variables τi . In an eigenbasis |τ i = |τ1 , . . . , τn i
of these operators,
τi |τ i = (τi0 + t)|τ i .
These n eigenvalues serve as indicators of coordinate time; but they are no more
conceptually identical to it than the positions of n point particles are identical
to coordinate space. The clock variables can be reset by defining
T i = τ i − τ 0i ,
so that Ti |τ i = t|τ i. The eigenvalues of the reset clock variables, T i , are all equal
to coordinate time t, much as the eigenvalues of the shifted position operators
Time operators
7
representing the particle positions were all equal to x. The temptation to identify
the reset clock variables with coordinate time is great, but it must be resisted:
these variables are equal to It, not to t.
It should be noted that the Hamiltonian (0.9) is not bounded from below,
indeed its eigenvalues extend over the entire real line (cf. Subsect. 3.3). We use
a Fourier integral to express the eigenvector |τ i as
1
|τ i =
n
(2π) 2
Z
∞
n
X
dη1 . . . dηn exp − i
ηi τi |ηi ,
∞
Z
...
−∞
−∞
(0.10)
i=1
where |ηi = |η1 , . . . , ηn i is an eigenvector of the variables η i . The inverse is
|ηi =
1
n
(2π) 2
Z
∞
Z
∞
...
−∞
n
X
dτ1 . . . dτn exp i
ηi τi |τ i .
−∞
(0.11)
i=1
In the analogue of configuration space, a state vector |ψi is assigned a wave
function ψ(τ ). This takes the form
1
ψ(τ ) ≡ hτ |ψi =
n
(2π) 2
Z
∞
Z
∞
...
−∞
n
X
ηi τi hη|ψi ,
dη1 . . . dηn exp i
−∞
i=1
and, in this space, τi is represented by the number τi , while η i is represented by
the differential operator −i∂/∂τi :
hτ |η i |ψi = −i
∂
1
ψ(τ ) =
n
∂τi
(2π) 2
Z
∞
Z
∞
...
−∞
n
X
ηi τi ηi hη|ψi .
dη1 . . . dηn exp i
−∞
i=1
Note that the Hamiltonian itself has the following representation as a differential
operator on the space spanned by the eigenvectors of the clock variables:
H ∼ −i
n
X
∂
.
∂τi
i=1
This is perfectly well defined, whereas the occasionally suggested equivalence
H = −id/dt is nonsense.
It follows from Eqs.(0.9)-(0.10) that the unitary time evolution operator,
U (t) = exp(−itH), induces the transformation
U (t)|τ1 , . . . , τn i = |τ1 + t, . . . , τn + ti ,
(0.12)
i.e. an eigenstate of the time operators simply evolves into an eigenstate of the
same operators at a later time.
8
An intuitively appealing realization of the clock algebra (0.6),(0.9) is afforded
by the following canonical transformation:
q̂i =
g τi2
2
−
ηi
mi g
p̂i = mi g τi .
These new canonical coordinates satisfy the standard commutation relations,
and the Hamiltonian (0.9) takes on the form
n 2
X
p̂i
− mi g q̂ i ,
(0.13)
H =
2mi
i=1
which we recognise as that of n freely falling point masses mi in a uniform gravitational field, g being the acceleration due to gravity. In this realization, the
clock times are provided by the canonical momenta p̂i . This Hamiltonian is not
bounded from below, of course, being simply (0.9) expressed in different variables. It is true that ideal clocks are ‘unphysical’ in a sense, but then so are
point particles. Both are consistent with the formalism of quantum mechanics.
Together they illustrate the similarity between the quantum mechanical treatment of indicators of position in space and indicators of position in time.
0.3.2
The cyclic clock
An alternative realization of the commutation relations (0.6) is afforded by the
cyclic clock, to which we alluded above. The eigenvalues of these clock variables,
τi , are readings, τi , that are limited to [0, 2π]. Such clock variables resemble
angle rather than position variables, and their conjugate variables, η i , resemble
angular momenta rather than linear momenta.
For notational simplicity we restrict our attention to just one cyclic clock,
but any number could be treated. The eigenvectors can be expanded in a Fourier
series:
∞
X
1
|τ i = √
e−imτ |ηm i ,
(0.14)
2π m=−∞
where |ηm i designates a discrete set of vectors, and |τ i refers to a continuous set
of vectors in the same space, the two sets being related by the discrete Fourier
transformation (0.14). The inverse of Eq.(0.14) is the finite Fourier integral
Z 2π
1
|ηm i = √
dτ eimτ |τ i .
(0.15)
2π 0
The eigenvector |ηm i of η evidently belongs to the eigenvalue ηm ≡ m, where
m = 0, ±1, ±2, . . ..
In the analogue of configuration space, a state vector |ψi is assigned a wave
function ψ(τ ):
∞
X
1
eimτ hηm |ψi .
(0.16)
ψ(τ ) ≡ hτ |ψi = √
2π m=−∞
Time operators
9
In this space, τ is represented by the number τ , and η by the differential operator
d
−i dτ
:
∞
X
d
1
hτ |η|ψi = −i ψ(τ ) = √
m eimτ hηm |ψi .
dτ
2π m=−∞
The Hamiltonian of the cyclic clock is η, and the unitary time evolution operator
is accordingly U (t) = exp(−itH) = exp(−itη). From Eq.(0.14), and recalling
that ηm = m, we find
U (t)|τ i = |τ + ti
(mod 2π) .
An operator on a Hilbert space of vectors is fully defined only if one gives its
domain, which is a subset of vectors in the space that are mapped by the operator
on to vectors in the space. It turns out that a careful definition of the relevant
domains is of crucial importance in resolving certain conceptual difficulties with
respect to the cyclic clock (and for the correct treatment of angular momentum),
and we will now give the necessary attention to these matters. In particular, it
will prove important to specify the domain of the unit operator appearing on
the right of Eq.(0.6).
It may be helpful first to explain some basic properties of unbounded linear
operators on Hilbert space. We shall then briefly consider their relevance to the
linear clock variables of the previous subsection, and then in more detail to those
of the cyclic clock, where their application is indispensable. An operator Ω is
defined by specifying a subset of the space called its domain, D(Ω), and a linear
mapping of a vector in that domain to another vector in the space. The domain
will not be the whole space if Ω is an unbounded operator; but if, for any |φi in
the whole space, one can nevertheless find a |ψi in D(Ω) that is arbitrarily close
to |φi in the sense of the norm, then the domain is said to be dense in the space.
If, for every |ψ1 i and |ψ2 i in the domain, which is presumed dense, it is the
case that
hψ1 |Ωψ2 i = hΩψ1 |ψ2 i ,
then Ω is said to be Hermitian (or symmetric). This property is however not
sufficient to guarantee the self-adjointness of Ω. For a fixed ψ2 in D(Ω), consider
the subset of vectors in the Hilbert space comprising all ψ1 such that there is a
φ in the space for which
hψ1 |Ωψ2 i = hφ|ψ2 i .
The set of all these ψ1 is called the domain of the operator adjoint to Ω . If Ω
is Hermitian, and this adjoint domain is equal to D(Ω), then Ω is said to be
self-adjoint. It is a basic assumption of quantum mechanics that any physical
observable is represented by a self-adjoint operator on Hilbert space, for then
a complete orthogonal set of eigenvectors exists (not admittedly always in the
Hilbert space itself, but always in the extended space of the associated rigged
Hilbert space, see Atkinson and Johnson 2002: 3). It should be noted that selfadjointness is a stronger constraint than mere Hermiticity, a fact that is ignored
in elementary introductions to quantum mechanics.
10
For the linear clock, the algebra of the operators τ and η is isomorphic to
that of the position and momentum operators of point particles, as we saw, and
the following statements can be demonstrated by adapting standard proofs, for
example those given in Yosida (1970: 198). In the space spanned by the eigend
vectors of τ , the operators τ and η are represented by τ and −i dτ
respectively.
2
Consider these as unbounded operators on the Hilbert space L (−∞, ∞), which
is defined to be the
R ∞ set of all square-integrable functions, i.e. all complex functions
ψ(τ ) for which −∞ dτ |ψ(τ )|2 < ∞ , equipped with the inner product
Z
∞
hφ|ψi =
dτ φ∗ (τ )ψ(τ ) .
−∞
From Yosida’s results we read off
1. τ is self-adjoint on the (dense) domain defined by the requirements that
both ψ(τ ) and τ ψ(τ ) lie in L2 (−∞, ∞).
2. η is self-adjoint on the (dense) domain defined by the requirements that
both ψ(τ ) and ψ 0 (τ ) lie in L2 (−∞, ∞), and moreover that ψ(τ ) be absolutely continuous, i.e. that ψ(τ ) be expressible in the form
Z τ
ψ(c) +
dx g(σ)
c
where g(σ) is a locally integrable function.
Returning to the cyclic clock, one defines the Hilbert space L2 (0, 2π) as the
R 2π
space of all complex functions ψ(τ ) for which 0 dτ |ψ(τ )|2 < ∞ , equipped with
an inner product like that in L2 (−∞, ∞), except that the integration domain is
limited to (0, 2π). Whereas τ is self-adjoint on L2 (0, 2π), η is not self-adjoint on
the whole of the space. The most useful self-adjoint extension is defined by the
requirements that both ψ(τ ) and ψ 0 (τ ) lie in L2 (0, 2π), that ψ(τ ) be absolutely
continuous, and that ψ(0) = ψ(2π). This periodicity is in accord with the Fourier
series representation Eq.(0.16).
Consider however the commutation relation,
[τ , η] = iI .
(0.17)
On what subspace of L2 (0, 2π) is I the unit operator? The difficulty is that,
whereas ψ(τ ) must satisfy ψ(0) = ψ(2π) to qualify for inclusion in a domain on
which η is self-adjoint, the function φ(τ ) = τ ψ(τ ) satisfies this condition only if
ψ(0) = ψ(2π) = 0. Hence the domain on which the commutator (0.17) is valid
is specified by the requirements that both ψ(τ ) and ψ 0 (τ ) lie in L2 (0, 2π), that
ψ(τ ) be absolutely continuous, and that ψ(0) = ψ(2π) = 0. In Eq.(0.17), I is
to be understood as the unit operator on this subspace alone. While there is no
objection to this limitation, the restricted subspace being also dense in L2 (0, 2π),
there are consequences for some of the uncertainty relations, as we will see in
Sect. 4.
Time operators
11
It may be noted that the above discussion of the cyclic clock variable and its
conjugate is equally applicable to the angle, θ, and the orbital angular momentum, L, for the motion of a point particle in a central force field. The algebra of
the operators τ and η for the cyclic clock is isomorphic to that of θ and L.
In the physics literature it is sometimes asserted that if two operators satisfy
the canonical commutation relation, then both must have continuous eigenvalues
on the whole real axis. In fact, Pauli’s negative conclusion about the existence
of a time operator mentioned at the beginning of this article was based on this
belief. For a proof, Pauli referred to the first edition of Dirac’s famous book on
quantum mechanics (Dirac 1930: 56). Our adaptation of his proof goes as follows:
by repeated application of relation (0.17), one obtains the equality
exp(icτ ) η exp(−icτ ) = η − cI ,
(0.18)
where c is a real number. Let |ηi be some eigenvector of η belonging to the
eigenvalue η. So
η exp(−icτ )|ηi = exp(−icτ )(η − c)|ηi = (η − c) exp(−icτ )|ηi ,
(0.19)
thus η − c is an eigenvalue of η, and since c may have any real value, so may
the eigenvalues of η. Note that the eigenvector exp(−icτ )|ηi is guaranteed not
to be the nullvector, since its norm is the same as that of |ηi, and that is surely
nonvanishing.1 On interchanging the roles of η and τ we find a similar result
for the eigenvalues of τ .
Now we have seen that, in the case of the cyclic clock, η has discrete eigenvalues. In view of Dirac’s result, how can this be? To understand that we need
to consider more carefully the conditions under which the above derivation is
valid. In fact, in order to use the commutator (0.17) repeatedly, so as to obtain
Eq.(0.18), we must find a domain on which the products τ η and ητ are welldefined, and on which both operators are self-adjoint. For the linear clock, this
common domain is dense in the whole of the Hilbert space L2 (−∞, ∞), so there
is no difficulty; but for the cyclic clock the matter is quite different.
As we have noted, for the cyclic clock η is self-adjoint on a subspace of
L2 (0, 2π) for which ψ(0) = ψ(2π), but the problem is that Eq.(0.17) does not
hold on the whole of this space. In fact it can be shown that, if |ψi and |φi are
any two vectors in the domain of η,
hψ|[τ , η] − iI|φi = −2πiψ ∗ (2π)φ(2π) ,
(0.20)
which means that Eq.(0.17) is simply not true on this space. It is only true on
the subspace of the domain specified by ψ(0) = ψ(2π) = 0, but the difficulty is
1 In the second edition (1935: 94) of the book of Dirac just cited, the author adds that
complex values of c are not allowed for physical reasons, since the putative eigenvector would
blow up exponentially at infinity. The mathematical version of this objection is that such a
complex value leads to a vector that is not contained in the extended Hilbert space.
12
that η is not self-adjoint on this restricted domain, and Eq.(0.18) does not hold,
so the argument of Dirac does not go through. There is no space on which τ
and η are self-adjoint, and on which Eq.(0.19) is valid. For further details, (see
Kraus 1965 and Uffink 1990).
0.3.3
Discussion
We have seen that the formalism of quantum mechanics allows for the existence
of ideal clocks, and that the energy of such systems is unbounded. While the
energy eigenvalues extend from −∞ to ∞, it is particularly the lack of a lower
bound that is sometimes thought to be a serious defect, since it is feared that
such a system could act as an infinite source of energy. However, as long as the
system is isolated or coupled to a system that cannot take up infinite amounts of
energy, nothing untoward will happen. In a sense, an ideal clock is better behaved
than a point particle, for an eigenstate of such a particle’s position spreads with
infinite velocity. By contrast, the eigenstates of time operators do not spread at
all, but rather transform into eigenstates belonging to a different eigenvalue, cf.
Eq.(0.12). We conclude that the existence of ideal clocks is perfectly consistent
with the formalism of quantum mechanics.
0.4
Uncertainty Relations
The commutation relation (0.1) and the inequality (0.3) are generally considered
to embody the essential content of elementary quantum mechanics. They express
two fundamental aspects of the theory: non-commutativity and irreducible uncertainty. In Sect. 3 we have seen that energy and time operators also exist
satisfying the relation [τ, η] = iI on suitably defined domains. Concerning (0.3)
we remark that although this inequality is traditionally considered to be the
mathematical expression of the quantum mechanical uncertainty principle, it is
actually quite unsatisfactory in this respect. The standard deviation is not the
most obvious and certainly not the most adequate measure of uncertainty in
quantum mechanics. For many perfectly normal quantum states the standard
deviation diverges, and even when the wave-function approximates a δ-function
the standard deviation may remain arbitrarily large. A consequence of this fact
is that inequality (0.3) permits probability distributions of p and q to be simultaneously arbitrarily narrow, contrary to what might be expected from an
uncertainty relation (Uffink and Hilgevoord 1985, Hilgevoord 2002). A more adequate measure of the spread of a probability distribution is the length Wα of the
smallest interval on which a sizeable fraction α of the distribution is situated.
An inequality of type (0.2) also holds for this measure:
Wα (p)Wα (q) ≥ ca h̄ ,
if
α ≥ 12 ,
(0.21)
where cα is of order 1 (Landau and Pollak 1961, Uffink 1990). This relation
expresses the intuitive content of the uncertainty principle in a much more satisfactory manner than does Eq.(0.3).
Uncertainty Relations
13
As to the time and energy variables, since the linear clock is mathematically
isomorphic to the case of p and q, we simply have
∆η∆τ ≥ 12 h̄
(0.22)
and
Wα (η)Wα (τ ) ≥ ca h̄ ,
if
α ≥ 12 .
(0.23)
In the case of the cyclic clock the eigenvalues of η are discrete and proper eigenstates exist. For these states ∆η = 0, and this contradicts Eq.(0.22)! The problem
is in fact well known, being analogous to the much discussed and mathematically identical case of angle and angular momentum. It turns out, however, that
inequalities using more appropriate measures of uncertainty of the type (0.23)
exist in this case too, as might be expected, since they are simply consequences
of the Fourier transformation that relates the conjugate variables of the cyclic
clock (Eqs. (0.14)-(0.15)). We refer to Uffink (1990) for a full discussion.
In Sect. 1 we mentioned that the apparent lack of a counterpart to inequality
(0.3) for a time operator stimulated people to look for uncertainty relations
containing time as a parameter. Many instances of such relations in quantum
mechanics are known, for example the familiar relation between the lifetime and
energy-spread of a decaying state. An extensive discussion of such energy-time
uncertainty relations is given in Busch (1989, 2000). Here we will briefly discuss
a powerful and general class of relations which brings out the full symmetry
between the four coordinates t, x, y, z. For a detailed discussion see Hilgevoord
(1998). Basic to these relations is a notion of uncertainty that differs from the
one in Eq.(0.3) and Eq.(0.21). It is an uncertainty concerning the state in which
a quantum system finds itself (Hilgevoord and Uffink 1991).
If a system is in quantum state |Ψi, the probability of finding it in a different
state |Φi is generally non-zero; in fact this probability is given by the number
|hΦ|Ψi|2 . The closer |hΦ|Ψi|2 is to 1 the harder it will be to distinguish between
the two states by measurements. Accordingly, it is useful to define a number ρ
as follows: |hΦ|Ψi| = 1 − ρ, with 0 ≤ ρ ≤ 1, and call it the reliability with which
the states |Ψi and |Φi can be distinguished. If the states coincide this reliability
is 0, whereas it has its maximal value 1 when the states are orthogonal.
Let us apply these ideas to states that are translated with respect to each
other in time and in space, respectively. For simplicity we consider only one space
coordinate x, and we once more put h̄ = 1. The unitary operators U t (τ ) and
U x (ξ) of translations in time and space are given by
U t (τ ) = exp(−iHτ )
and
U x (ξ) = exp(iPx ξ),
(0.24)
where H is the operator of the total energy and Px the x-component of the
operator of the total momentum of the system, as in Sect. 3. Let the system be
initially in state |Ψi. Then we define τρ to be the smallest time for which the
following equality is valid:
14
|hΨ|U t (τρ )|Ψi| = 1 − ρ .
(0.25)
Similarly, ξρ is the smallest distance for which
|hΨ|U x (ξρ )|Ψi| = 1 − ρ .
(0.26)
That is, the state |Ψi must be translated over at least an interval τρ in time to
become distinguishable from the original state with reliability ρ, or it must be
translated over at least a distance ξρ in space to become distinguishable with
reliability ρ.
We shall call τρ and ξρ the temporal and spatial translation widths of the
state |Ψi. Both quantities have well-known physical meanings. If (1 − ρ)2 = 12 ,
τρ is the so-called half-life of the state; and relation (0.26) is closely related to
Rayleigh’s criterion for the distinguishability of two spatially translated states,
1/ξρ being a generalization of the notion of the resolving power of an optical
instrument.
These translation widths are subject to very general uncertainty relations,
which connect them to the width of the energy and momentum spectrum of the
state |Ψi, respectively. Let |Ei and |Px i denote complete sets of eigenstates of
Hand Px , where a possible degeneracy is ignored for simplicity. Using Eq.(0.24),
we have
Z
Z
hΨ|U t (τρ )|Ψi = |hE|Ψi|2 e−iτ E dE and hΨ|U x (ξρ )|Ψi = |hPx |Ψi|2 eiξPx dPx ,
(0.27)
where the integrals may include summation over discrete eigenvalues. Define
the overall widths Wα (E) and Wα (Px ) ofR the energy and momentum distributions as the smallest intervals such that Wα (E) |hE|Ψi|2 dE = α and similarly
R
|hPx |Ψi|2 dPx = α. It can then be shown (Uffink and Hilgevoord 1985,
Wα (Px )
Uffink 1993) that
τρ Wα (E) ≥ C(α, ρ)h̄ and ξρ Wα (Px ) ≥ C(α, ρ)h̄
(0.28)
for ρ ≥ 2(1 − α). For sensible values of the parameters, say α = 0.9 or 0.8, and
0.5 ≤ ρ ≤ 1, the constant C(α, ρ) is of order 1. Inequalities (0.28) hold for all
states |Ψi, and the only assumptions needed for their validity are the existence
of the translation operators Eq.(0.24) and the completeness of the energy and
momentum eigenstates. Furthermore, since in relativity theory t and ~x on the one
hand, and E and P~ on the other, are united to form a 4-vector, the inequalities
(0.28) lead to a relativistically covariant set of uncertainty relations.
Let us expand a little on the physical meaning of inequalities (0.28). The first
of these is a useful general expression of the relation between the lifetime and
the energy spread of a state. Usually such an inequality is obtained in the approximation in which the decay is exponential, but here it is completely general.
Though these inequalities deal with coordinate time and coordinate space, they
also have relevance to time operators and position operators. This may be seen
Uncertainty Relations
15
as follows. Taking position q as an example, let us see how the second inequality
(0.28) relates to inequality (0.21). If the main part of the probability distribution
|hq|Ψi|2 of q in state |Ψi is concentrated on an interval of length a, the width
Wα (q) is of order a. In this case the translation width ξρ will be of the same
order a. If, on the other hand, the distribution consists of a number of narrow
peaks of width b, as in an interference pattern, the translation width ξρ will be
of order b, whereas Wα (q) remains of the order of the overall width a of the distribution. Thus, inequalities (0.28) are stronger than (0.21). Next consider the
case where a number of position operators q i are present in the system. Suppose
the spread Wα (P ) of the total momentum spectrum is small. From Eq.(0.28)
it follows then that the translation width ξρ must be large. This implies that
the spread in all position variables of the system must be large. Conversely, if
the spread in only one position variable is small, then ξρ is small and Eq.(0.28)
implies that the spread in the total momentum must be large. Thus it is the total
momentum that determines whether or not the position variables of a system
can be sharply determined. Similar conclusions follow for the relation between
the spread Wα (E) of the total energy and the spread of all time operators. It is
the total energy that decrees whether or not the time variables of a system can
be sharply determined.
REFERENCES
Atkinson, D., and Johnson, P.W. (2002), Quantum Field Theory. A Self-Contained
Course (Rinton Press, Princeton).
Busch, P. (1989), ‘On the Energy-Time Uncertainty Relations, I and II’, in
Foundations of Physics 20: 1-43.
Busch, P. (2002), ‘The Time-Energy Uncertainty Relation’, in J.G. Muga, R.
Sala Mayato and I.L. Egusquiza (eds.), Time in Quantum Mechanics (Springer,
Berlin), 69-98.
Dirac, P.A.M. (1930), The Principles of Quantum Mechanics (Clarendon Press,
Oxford).
Egusquiza, I.L., Gonzalo Muga, J. and Baute, A. (2002), ‘“Standard” Quantum
Mechanical Approach to Times of Arrival’, in J.G. Muga, R. Sala Mayato and
I.L. Egusquiza (eds.), Time in Quantum Mechanics (Springer, Berlin), 279304.
Hilgevoord, J. and Uffink, J. (1990), ‘A new view on the uncertainty principle’,
in A.I. Miller (ed.), 62 Years of Uncertainty, Historical and Physical Inquiries
into the Foundations of Quantum Mechanics (Plenum, New York), 121-139.
Hilgevoord, J. and Uffink, J. (1991), ‘Uncertainty in Prediction and in Inference’, in Foundations of Physics 21: 323-341.
Hilgevoord, J. (1996), ‘The uncertainty principle for energy and time’, in American Journal of Physics 64: 1451-1456.
Hilgevoord, J. (1998), ‘The uncertainty principle for energy and time II’, in
American Journal of Physics 66: 396-402.
Hilgevoord, J. (2002a), ‘The standard deviation is not an adequate measure of
quantum uncertainty’, in American Journal of Physics 70: 983.
Hilgevoord, J. (2002b), ‘Time in quantum mechanics’, in American Journal of
Physics 70/3: 301-306.
Hilgevoord, J. (2005), ‘Time in quantum mechanics: a story of confusion’, in
Studies in History and Philosophy of Modern Physics 36: 29-60.
Kraus, K. (1965), ‘Remark on the uncertainty between angle and angular momentum’, in Zeitschrift für Physik 188: 374-377.
Newton, T.D. and Wigner, E.P. (1949), ‘Localized states for elementary systems’, in Rev. Mod. Phys. 21: 400-406.
Pauli, W. (1933), ‘Die allgemeine Prinzipien der Wellenmechanik’, in Handbuch
der Physik, 2. Auflage, Band 24, 1. Teil (Springer-Verlag, Berlin), 83-272. An
English translation of the first nine chapters exists under the title: General
Principles of Quantum Mechanics (Springer-Verlag, Berlin, 1980).
Schweber, S.S. (1962), An introduction to relativistic quantum field theory (Harper
and Row, New York).
References
17
Uffink, J. and Hilgevoord, J. (1985), ‘Uncertainty Principle and Uncertainty
Relations’, in Foundations of Physics , 15: 925-944.
Uffink, J. (1990), Measures of uncertainty and the uncertainty principle, (Ph.D.
thesis, Utrecht University).
Uffink, J. (1993), ‘The rate of evolution of a quantum state’, in American Journal of Physics 61: 935-936.
Yosida, K. (1970), Functional Analysis (Springer, Berlin).