Muga G. Time in Quantum Mechanics

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10 J.M. S

anchez-Ron

Ψ (q, −t) is not, because the equation it satisﬁes is

Ψ (q, −t) =−c ·[∂Ψ(q, −t)/∂t], (1.2)

which differs from the original in the minus sign on the right-hand side.

And here

Reichenbach [35, pp. 209–210] wrote,

There remains the problem of distinguishing between Ψ (q, t)andΨ (q, −t). In order to

discriminate between these two functions, we would ﬁrst have to know whether (1.1) or

(1.2) is the correct equation. But the sign of the term on the right in Schr

odinger’s equation

can be tested observationally only if a direction of time has been previously deﬁned. We

use here the time direction of the macrocosmic systems by the help of which we compare

the mathematical consequences of Schr

odinger’s equation with observations. Therefore, to

attempt a deﬁnition of time direction through Schr

odinger’s equation would be reasoning

on a circle; this equation merely presents us with the time direction which we introduced

previously in terms of macrocosmic processes.

And he added,

It may be recalled that even in classical physics the time direction of a molecule is not ascer-

tainable from observations of the molecule, even if such direct observations could be made,

but is determined only by comparison with macroprocesses, for which statistics deﬁne a

time direction. In the same way, the time direction of a quantum-mechanical elementary

process, like the movement of an electron, is determined only with reference to the time of

macroprocesses.

This consideration shows that the fundamental quantum-mechanical law governing the time

development of physical systems does not distinguish one time direction from its opposite.

Since the laws governing the observables of quantum physics are not causal laws, but prob-

ability laws, the reversibility of elementary processes assumes here the form of a symmetry

of the relations connecting probability distributions. These connecting relations are strict

laws expressible through a differential equation, namely, Schr

odinger’s equation.

“Time direction of a quantum-mechanical elementary process,” he wrote, “is

determined only with reference to the time of macroprocesses.” Not a stimulating

comment for anyone who would have thought that so radical physics revolution as

quantum mechanics should affect also our ideas of what time is.

1.5 Reichenbach on Feynman’s Theory of the Positron

One thing that strikes one when reading Reichenbach’s book is how scarce the

references to works of physicists who dealt with the quantum are, whether with

quantum mechanics or with quantum electrodynamics. It seems as if it was more a

philosophical inner discussion, illuminated mainly by quantum mechanics (mainly

N. of E.: The standard “time-reversal invariance” argument is based on the commutation of the

Hamiltonian with the antiunitary time-reversal operator, see, e.g., Chap. 12 in this volume. The

time-reversed state of Ψ (q, t)isΨ (q, t)

∗

.IfΨ (q, t)

∗

evolves for a time t, the resulting state is

Ψ (x, 0)

∗

, which is the time-reversed state of Ψ (x, 0). For a critical analysis see A.T. Holster, New

J. Phys. 5, 130 (2003).

1 Schlick and Reichenbach on Time in Quantum Mechanics 11

Schr

odinger’s version), statistical mechanics, and the mathematical theory of proba-

bility. There is, however, an important exception: Reichenbach’s reference to works

of the Lausanne professor E. C. G. St

uckelberg and the American Richard Feynman

on the theory of positrons, where they considered positrons as electrons moving

backward in time [46, 47, 13, 14].

“Surprisingly enough,” he wrote [35, pp. 263–264], “recent developments have

demonstrated that the genidentity [that is, the physical identity of a thing] of mate-

rial particles can be questioned more seriously than is done in Bose statistics. The

difference between one and two, or even three, material particles can be shown to be

a matter of interpretation; that is, this difference is not an objective fact, but depends

on the language used for the description. The number of material particles, therefore,

is contingent upon the extension rules of language. However, the interpretations thus

admitted for the language of physics differ in one essential point from all others:

they require an abandonment of the order of time.”

Those “recent developments” were the “conceptions... developed by E. C. G.

uckelberg and R. P. Feynman. Their investigations showed that a positron – that is,

a particle of the mass of an electron, but carrying a positive charge – can be regarded

as an electron moving backward in time.” To Reichenbach [35, pp. 266, 268] such

interpretation “does not merely signify a reversal of time direction; it represents an

abandonment of time order...This is the most serious blow the concept of time

has ever received in physics. Classical mechanics cannot account for the direction

of time; but it can at least deﬁne a temporal order. Statistical mechanics can deﬁne

a temporal direction in terms of probabilities; but this deﬁnition presupposes time

order for those atomic occurrences the statistical behavior of which supplies time

direction. Quantum physics, it appears, cannot even speak of a unique time order of

the processes, if further investigations conﬁrm Feynman’s interpretation, which is at

present still under discussion.”

Even without addressing the fundamental problem of putting in a sound theoret-

ical basis for the time–energy uncertainty relation and deriving its consequences for

the concept of time, Reichenbach had found that the realm of the quantum was a

dangerous territory for the conception of time that classical physics had favored.

1.6 Epilogue

In his book Philosophie der Raum-Zeit-Lehre, and thinking in the case of the rel-

ativity theories, Reichenbach [32] [36, p. 109] stated that “philosophy of science

has examined the problems of time much less than the problems of space. Time

has generally been considered as an ordering schema similar to, but simpler than,

that of space, simpler because it has only one dimension. Some philosophers have

Feynman’s work here was inﬂuenced by his previous collaboration with John Wheeler on an

action-at-distance electrodynamics, in which they used retarded as well as advanced potentials;

that is, electromagnetic waves moving forward and backward in time [49, 50].

12 J.M. S

anchez-Ron

believed that a philosophical clariﬁcation of space also provided a solution of the

problem of time.” It seems to me, after having reviewed here what Moritz Schlick

and Hans Reichenbach – just two, certainly, German-speaking philosophers of sci-

ence, but, nevertheless, very prominent ones – had to say about time in quantum

mechanics, that the same comment can be applied to the analysis of time in the

quantum domain. How different, and much less frequent, were the comments that

the two uncertainty relations aroused apropos the time concept is a good example

of such assertion. Of course, it is true – obviously true – that scientiﬁcally time is

a much more problematic and difﬁcult concept to deﬁne and study than space, but

it is so fundamental! Without it, there would be nothing, just “something” (I resist

calling it “world”) unknowledgeable. With it, we have science, but also mystery, the

mystery of a concept perhaps too difﬁcult for us to fully understand.

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Correspondence, May–December 1920, and Supplementary Correspondence, 1909–1920)

(Princeton University Press, Princeton, 2006) 2, 3

1 Schlick and Reichenbach on Time in Quantum Mechanics 13

23. D. Kormos Buchwald, R. Schulmann, J. Illy, D.J. Kenneﬁck, T. Sauer (eds.), The Collected

Papers of Albert Einstein, vol. 9 (The Berlin Years: Correspondence, January 1919–April

1920) (Princeton University Press, Princeton, 2004) 2

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Papers of Albert Einstein, vol. 10 (The Berlin Years: Correspondence, May–December 1920,

and Supplementary Correspondence, 1919–1920) (Princeton University Press, Princeton,

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Heinemann, Oxford, 1977) 5, 6

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ur Physik 69, 56 (1931) 5

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ur Physik 69, 56 (1931), in Quan-

tum Theory and Measurement, J.A. Wheeler, W.H. Zurck (eds.) (Princeton University Press,

Princeton, 1983), p. 465 5

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(1945), English translation: J. Phys. URSS 9, 249 6

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vol. II (Time in a Quantized Universe) (Reidel, Dordrecht, 1980), p. 203 8

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Press, Berkeley, 1944) 8

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e 13, part 2, 109 (1952/1953), English

translation in Moritz Schlick: Philosophical Papers (1925–1936), vol. II, H. Mulder, B.F.B.

van de Velde-Schlick (eds.) (Reidel, Dordrecht, 1979), p. 237 9

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tion of Philosophie der Raum-Zeit-Lehre (Walter de Gruyter, Berlin, 1928) 11

37. H. Reichenbach, Philosophical Foundations of Quantum Mechanics (University of California

Press, Berkeley, 1964), ﬁrst California paperback printing 8

38. H. Reichenbach, in Selected Writings, 1909–1953, vol. I, M. Reichenbach, R.S. Cohen (eds.)

(Reidel, Dordrecht, 1978) 2, 3

39. M. Schlick, Zeitschrift f

ur Philosophie und philosophische Kritik 159, 129 (1915), English

translation in Moritz Schlick: Philosophical Papers (1909–1922), vol. I, H. Mulder, B.F.B.

van de Velde-Schlick (eds.) (Reidel, Dordrecht, 1979), p. 153 2

40. M. Schlick, Die Naturwissenschaften 5, 161, 177 (1917) 2

41. M. Schlick, Raum und Zeit in der gegenw

artigen Physik (Julius Springer Verlag, Berlin, 1917) 2

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43. M. Schlick, University of California Publications in Philosophy XV, 99 (1932), reprinted in

Moritz Schlick: Philosophical Papers (1925–1936), vol. II, H. Mulder, B.F.B. van de Velde-

Schlick (eds.) (Reidel, Dordrecht, 1979), p. 238 7, 8

44. M. Schlick, in Moritz Schlick: Philosophical Papers (1925–1936), vol. II, H. Mulder, B.F.B.

van de Velde-Schlick (eds.) (Reidel, Dordrecht, 1979), p. 176 7

45. D. Sciama, in The Nature of Time, R. Flood, M. Lockwood (eds.) (Basil Blackwell, Oxford,

1986), p. 6 9

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uckelberg, Helvetica Physica Acta 14, 588 (1941) 11

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uckelberg, Helvetica Physica Acta 15, 23 (1942) 11

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Chapter 2

The Time-Dependent Schr

odinger Equation

Revisited: Quantum Optical and Classical

Maxwell Routes to Schr

odinger’s Wave

Equation

Marlan O. Scully

2.1 Introduction

In a previous paper [1, 2] we presented quantum ﬁeld theoretical and classical

(Hamilton–Jacobi) routes to the time-dependent Schr

odinger equation (TDSE) in

which the time t and position r are regarded as parameters, not operators. From this

perspective, the time in quantum mechanics is argued as being the same as the time

in Newtonian mechanics. We here provide a parallel argument, based on the photon

wave function, showing that the time in quantum mechanics is the same as the time

in Maxwell equations.

The next section is devoted to a review of the photon wave function which is

based on the premise that a photon is what a photodetector detects. In particular, we

show that the time-dependent Maxwell equations for the photon are to be viewed

in the same way we look at the time-dependent Dirac–Schr

odinger equation for the

(massive) π meson particle or (massless) neutrino.

In Sect. 2.3 we then recall previous work which casts the classical Maxwell

equations into a form which is very similar to the Dirac equation for the neutrino.

Thus, we are following de Broglie more closely than did Schr

odinger, who followed

a Hamilton–Jacobi approach to the quantum mechanical wave equation. In this

way, with nearly a century of hindsight, we arrive naturally at the time-dependent

Schr

odinger equation without operator baggage. Figures 2.1 and 2.2 summarize the

physics of the present chapter.

M.O. Scully (B)

Texas A&M University, College Station, TX 77843; Princeton University, Princeton,

NJ 08544, USA, mscully@Princeton.edu

It is a pleasure to dedicate this chapter to David Woodling who has enriched our lives through his

engineering and mechanical gifts and his insightful and gentle ways.

Scully, M.O.: The Time-Dependent Schr

odinger Equation Revisited: Quantum Optical and

Classical Maxwell Routes to Schr

odinger’s Wave Equation. Lect. Notes Phys. 789, 15–24 (2009)

DOI 10.1007/978-3-642-03174-8

 Springer-Verlag Berlin Heidelberg 2009

16 M.O. Scully

Fig. 2.1 Comparison of the quantum ﬁeld, wave mechanical, and classical descriptions of the spin

1 photon, spin

neutrino, and spin 0 meson; adapted from Scully and Zubairy “Quantum Optics”

[3]

Fig. 2.2 Top Down: The time-dependent Schr

odinger wave equation follows from the quantum

optical “a photon is what a photodetector detects” deﬁnition. This is in accord with the usual wave

function deﬁnition Ψ (r, t) =r|Ψ (t) since |r=

(r)|0. Bottom Up: The time-dependent

Schr

odinger wave follows nicely from the classical Maxwell equations by, for example, working

with a combination of electric and magnetic ﬁelds

2.2 The Quantum Optical Route to the Time-Dependent

Schr

odinger Equation

Quantum optics is an offshoot of quantum ﬁeld theory in which we are often inter-

ested in intense light beams such as provided by the laser. However the issue of the

photon concept, and how we should think of the “photon,” is a topic of current and

reoccurring discussion.

Perhaps the most logical, at least the most operational, approach is to say that the

photon is what a photodetector detects. In this spirit we consider the excitation of a

2 Time Dependent Schr

odinger Equation 17

single atom at point r at time t to be our photodetector and, following [3], write the

probability of exciting the atom as

(r, t) = ηΨ



†

(r, t)

E(r, t)



Ψ  . (2.1)

Several points should be made:

1. We consider the state

Ψ  to be a single photon state. For example, the state

generated by the emission of a single photon (see [3], Eq. 6.3.18)



=



k , (2.2)

where the state |k is expressed in terms of the radiation creation operator

†

k=

†

0 and in the simple case of a scalar photon, we ﬁnd

= g

−ik·r

(ν

−ω

) + iΓ/2

, (2.3)

where g

is the atom-ﬁeld coupling constant, r

is the atomic position vector, ν

and ω

are the photon and atomic frequencies, and Γ is the atomic decay rate.

2. The uninteresting photodetection efﬁciency constant η will be ignored in the

following.

†

(r, t) and

E(r, t) are the creation and annihilation operators deﬁned by

†

(r, t) =



k,λ

(λ)

†

k,λ

−iν

t+ik·r

, (2.4)

where ε

is the unit vector for light having polarization λ and wave vector k,

= ck = c|k| and the electric ﬁeld “per photon” E

√

ν

/2ε

V , where we

use MKS units so that ε

= 1/c

and V is the quantization volume.

Next we insert a sum over a complete set of states,



|nn|=1 in Eq. (2.1) and

note that since there is only one photon in ψ

(and

E(r, t) annihilates it), only the

vacuum term |00| will contribute. Hence we have

(r, t) =ψ

†

(r, t)|00|

E(r, t)|ψ

, (2.5)

and we are therefore led to deﬁne the single photon detection amplitude as

(r, t) =0|

E(r, t)|ψ

. (2.6)

As shown in detail in Sect. 2.4, the one photon state |ψ

 yields

(r, t) =

Δr



t −

Δr



−i(t−Δr/c)(ω−iΓ/2)

, (2.7)

18 M.O. Scully

where E is a constant, Δr is the distance from the atom to the detector, and Θ(x)is

the usual step function. More generally we have

(r, t) =0|

E(r, t)|ψ

=







k,λ

ˆε



ν

2ε

k,λ

−iν

t+ik·r





. (2.8)

The ﬁeld is sharply peaked about the frequency ω so that we may replace the

frequency ν

as it appears in the square root factor by ω and write

(r, t) =



ω

2ε

(r, t) , (2.9)

where

(r, t) =



k,λ

ˆε

(λ)





k,λ

−iν

t+ik·r

√





. (2.10)

The complete “photon wave function” also involves the magnetic analog of the

proceeding. To that end we write

(r, t) =0|

H(r, t)|ψ

 , (2.11)

where

H(r, t) is the annihilation operator for the magnetic ﬁeld which is given by

H(r, t) =



r,λ

× ˆε

(λ)



ν

2μ

k,λ

−iν

t+ik·r

√

, (2.12)

and we introduce the notation

(r, t) =



ω

2μ

(r, t) , (2.13)

where

(r, t) =







k,λ

× ˆε

(λ)

k,λ

−iν

t+ik·r

√





. (2.14)

Finally, we write ϕ

(r, t) and χ

(r, t)inmatrixformas

⎡

⎣

⎤

⎦

,χ

⎡

⎣

⎤

⎦

, (2.15)

2 Time Dependent Schr

odinger Equation 19

in terms of which Maxwell equations may be written as

i

∂

∂t







0 −cs ·p

cs ·p 0





, (2.16)

where p =



∇ and

⎡

⎣

00 0

00−1

01 0

⎤

⎦

, s

⎡

⎣

001

000

−100

⎤

⎦

, s

⎡

⎣

0 −10

100

000

⎤

⎦

(2.17)

are the 3 × 3 matrices for the (spin 1) photon.

Finally, we note the close correspondence with the two-component (spin

)

neutrino,



photon



←→



neutrino

neutirno



, (2.18)

and the Dirac equation for the neutrino

i

∂

∂t







0 −cσ ·p

cσ ·p 0





, (2.19)

where σ is given in terms of the 2 ×2 Pauli matrices and p =



∇.

We conclude by noting that, just as in the quantum ﬁeld theory [4, 5] route to

the Schr

odinger equation, the appearance of

∂

∂t

and ∇ in Eq. (2.16) has not arisen

from operator arguments. In the next sections, we follow a de Broglie wave–particle

duality path to the Schr

odinger equation.

2.3 The Classical Maxwell Route to the Schr

odinger Equation

In the previous section, we followed a top-down quantum ﬁeld route to the

Schr

odinger equation, see Fig. 2.2. In particular, we saw that the quantum opti-

cal analysis of the single photon wave equation provided an interesting connection

between the Schr

odinger (Dirac) equations for photons and neutrinos.

In the present section, we start with the classical Maxwell equations and obtain

a Schr

odinger equation for the combination E + iH which previous workers [6, 7]

call the photon wave function. It is then natural to follow de Broglie and associate a

wave function with matter waves. This provides another (operator-free) route to the

Schr

odinger equation.

20 M.O. Scully

Thus, we deﬁne the “classical” photon wave function as

(r, t) = E(r, t) + iH(r, t) =

⎡

⎣

(r, t)

⎤

⎦

, (2.20)

where the subscript m stands for Maxwell. Along the lines of the discussion in

Sect. 2.2, we may write the Maxwell equations as

i

(r, t) =−cs ·pΨ

(r, t) , (2.21)

where p =



∇, as before, but now

⎡

⎣

00 0

00−i

0 i 0

⎤

⎦

, s

⎡

⎣

00i

000

−i 00

⎤

⎦

, s

⎡

⎣

0 −i 0

i 00

000

⎤

⎦

. (2.22)

The present s matrix is related to the s of Sect. 2.2 by the factor i. It also should be

noted that the present photon wave function ψ

is a 1×3 matrix whereas that of 2.2

is a 1 × 6 matrix. That is, the quantum optical analysis involves a two-component

wave function in Ψ

and Ψ

; in the present analysis we ﬁnd it convenient to combine

the electric and magnetic contributions at the outset.

Since the energy per photon is ω = ck = cp, we write

i

(r, t) = HΨ

(r, t) , (2.23)

where the Hamiltonian is given by

H =−cs · p . (2.24)

The natural extension of this Schr

odinger equation for the spin one massless

photon to the case of a spin zero particle of mass m is clear. That is, since E =



+ p

is the ﬁnite mass extension of E = pc, we follow the lead of de

Broglie and write

i

Ψ (r, t) =



+ p

Ψ (r, t) , (2.25)

where p =



∇, just as it is for the photon.

Hence when m

 pc we may write



+ p

∼

+ m

, and we

have

i

Ψ (r, t) =

−

∇

Ψ (r, t) , (2.26)

Muga G. Time in Quantum Mechanics - Vol. 2

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