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Is quantum theory intrinsically nonlinear?

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2013 Phys. Scr. 87 038117

(http://iopscience.iop.org/1402-4896/87/3/038117)

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Phys. Scr. 87 (2013) 038117 (10pp) doi:10.1088/0031-8949/87/03/038117

Is quantum theory intrinsicallynonlinear?Dieter Schuch

Institut fur Theoretische Physik, J W Goethe-Universitat Frankfurt am Main, Max-von-Laue-Straße 1,D-60438 Frankfurt am Main, Germany

E-mail: Schuch@em.uni-frankfurt.de

Received 2 November 2012Accepted for publication 4 December 2012Published 11 February 2013Online at stacks.iop.org/PhysScr/87/038117

AbstractIn contrast with classical physics, complex quantities have a fundamental physical meaning inquantum physics and action, being essentially the quantized entity, should be given moreattention instead of focusing mainly on Hamiltonians or Lagrangians that have the dimensionof energy. Phase and amplitude of the complex quantities in (time-dependent andtime-independent) quantum mechanics are not independent of each other but coupled via someconservation law. This coupling can be understood if the systems are described in terms ofcomplex nonlinear Riccati equations. These equations not only enable a connection to thePythagorean triples, probably the oldest and most abstract ‘quantization’ problem, but alsolead to dynamical invariants with the dimension of action. Factorization of the correspondingoperator provides generalized creation and annihilation operators, which is also possible fordissipative systems where no conventional Hamiltonian formalism exists. Formal similaritieswith other fields, particularly with nonlinear dynamics, are shown.

PACS numbers: 03.65.Ta, 03.65.−w, 03.65.Yz

1. Introduction

On 4 July 2012, CERN announced the discovery of a newelementary particle that is probably the Higgs boson whichis needed to complete the standard model of elementaryparticles. If it turns out to be true, is this the completionof theoretical physics? Definitely not! Not only is theunification of the electromagnetic, weak and strong forceswith gravity still an unsolved puzzle, there are even moreserious shortcomings of the present state of theoretical physicsas, e.g. stated by Rothman in a recent paper [1]. He claimsthat the building of theoretical physics resembles a dilapidatedTower of Babel with many cracks. He correctly points outthat in the so-called fundamental equations of motion there isno distinction between past and future, no direction of timeand phenomena like friction producing heat and increasingentropy in irreversible processes do not occur. Besides, healso criticizes the large number of adjustable parameters andthus the lack of ‘beauty’ in the standard model and thefact that even rather simple systems like a double-pendulumobey nonlinear (NL) equations of motion that may lead tochaotic time-evolution. Also many other phenomena in oursurrounding world (like the weather) obey NL evolutionequations. On the other hand, the most fundamental and

successful theory, quantum theory, is supposedly a lineartheory. Is it possible to unite the missing aspects of non-linearity, irreversibility and dissipation with the existingfundamental theories, particularly with quantum mechanics?

In this paper, it shall be shown that a positive answer canbe given to this question if certain ‘prejudices’ are abandoned.This will lead to a NL formulation of quantum mechanicsthat still allows for a superposition principle and is basedon a complex, quadratically-nonlinear Riccati equation andthe replacement of energy by action as the fundamentalquantity in physics. This formulation is advantageous becauseit is also to be found in the same mathematical formin other fields of physics, like optics, quantum optics,Bose–Einstein condensates (BEC), supersymmetric (SUSY)quantum mechanics, NL dynamics, cosmology and others.

2. The roots of ‘quantization’

Considering quantum mechanics, properties come to mindlike linear wave equation (and thus superposition principle),wave-particle duality, uncertainty principle, quantized energylevels etc. But what are really the essential differencesbetween classical and quantum physics?

0031-8949/13/038117+10$33.00 Printed in the UK & the USA 1 © 2013 The Royal Swedish Academy of Sciences

Phys. Scr. 87 (2013) 038117 D Schuch

The era of quantum physics began around 1900 [2] withMax Planck’s introduction of h, the quantum of action, thathe needed to quantize the energy of the harmonic oscillator(HO). So the actual quantized quantity is action, not energy;the quantized energy of the HO is only a consequence of theconstant frequency ω0 of the oscillator; for a time-dependent(TD) frequency ω(t) it looks quite different, as will bediscussed later.

In classical as well as in quantum mechanics, theHamiltonian, representing the energy of the system andcontaining the potential from which the force can be derived,plays the dominant role. For the HO with TD frequency, theHamiltonian is no longer a constant of motion, but an invariantstill exists with the dimension of an action in this case. In theRiccati equations that occur in SUSY quantum mechanics andin our NL version of quantum mechanics, the potential playsonly the role of an inhomogeneity that essentially enters in aparticular solution of the corresponding differential equation.

Another fundamental difference between classical andquantum physics, as stated by Yang in his talk ‘square rootof minus one, complex phase and Erwin Schrodinger’ [3], isthat with quantum mechanics, for the first time, the imaginaryunit enters physics in a fundamental way and ‘complexnumbers became a conceptual element of the very foundationsof physics’. He continues that the very meaning of thefundamental equations of matrix mechanics (Heisenberg’scommutation relation) and of wave mechanics (the TDSchrodinger equation (SE)) ‘would be totally destroyed if onetries to get rid of i by writing them in terms of real andimaginary parts’.

I totally agree with this statement but want to go a stepfurther. The reason it is not sufficient to simply write thesecomplex equations in terms of two real ones is due to thefact that real and imaginary parts, or phase and amplitude,respectively, are not independent of each other but coupledvia some kind of conservation law, as will be shown below.

Where does the idea of quantization actually originate?And what is it that should be quantized? In the western cultureone might first think of the Greek philosopher Democritus(about 460–about 371 BC) and his idea of dividing our worldinto minute components that are not further divisible, leadingto the term ‘atom’ that is still used for the building blocksof our physical world which, for some time, seemed furtherindivisible.

A similar idea was formulated in a more abstract wayby another equally famous Greek philosopher living aroundthe same time. In his work Timaios, Plato (428/27–348/47BC) gives his view of how the world is built up in termsof right-angled triangles, essentially his ‘quanta’. WernerHeisenberg, equally fascinated with, and puzzled by, this text,summarizes this idea [4] in the general sense as follows:‘Matter is made up of right-angled triangles which, afterbeing paired to form isosceles triangles or squares, are joinedtogether to build the regular bodies of stereometry: cube,tetrahedron, octahedron and icosahedron. These four solidsare then supposed to be the basic units of the four elementsearth, fire, air and water’.

So, geometric objects like right-angled triangles shouldbe the quanta of nature. But one could go even further andask if particular right-angled triangles might play a special

role in quantization. Yet another Greek philosopher enters thescene. We remember Pythagoras’ theorem from school, i.e.a2 + b2

= c2 where a and b are the two shorter sides (catheti)and c is the longest side (hypotenuse) of a right-angledtriangle. Pythagoras (around 570–500 BC) and his discipleswere well-known for their dogma ‘everything is number’,with number meaning integer. So let us look for right-angledtriangles where the lengths of all three sides are integers(a kind of ‘second quantization’) fulfilling the Pythagoreantheorem. The most common example for such a Pythagoreantriple is (3, 4, 5) with 9 + 16 = 25. But, asked for a few moreexamples of the kind, even mathematically-affiliated personshave difficulties finding any—although an infinite number oftriples exists! Moreover, there is even a rather simple rule offinding these triples. This rule was probably already knownto the Babylonians more than 3500 years ago [5] but, atleast, Diophantus of Alexandria (around 250 AD) knew of it.Why do I mention this here? What does this have to do withquantum mechanics? In the following it will be shown thatfrom the dynamics of Gaussian wave packet (WP) solutionsof the TDSE a complex NL evolution equation (Riccatiequation) can be obtained that also contains the key for theanswer to the above question of obtaining the Pythagoreantriple. This complex Riccati equation will lead to a dynamicalinvariant that has the dimension of an action and still existsfor cases where the Hamiltonian is no longer invariant; e.g.for the HO with TD frequency or certain dissipative systems.This will also allow for the construction of generalizedcreation/annihilation operators and corresponding coherentstates (CS).

The same type of NL Riccati equations also occur intime-independent (TI) SE, as well as in BECs, cosmology andother fields mentioned before. More details will be given insections 4 and 6.

In section 3, the WP solutions of the TDSE and thecorresponding complex Riccati equation will be discussed.A generalization to include irreversibility and dissipationleading to new qualitative properties like bifurcations willbe outlined in section 5 and a comparison with systemsshowing relaxation, scale-invariance or both, as they occurin NL dynamics, will be given in section 6. A summary andperspectives conclude the paper.

3. Time-dependent Schrodinger equation (SE),complex Riccati equation and dynamical invariant

In the following, one-dimensional problems with exactanalytic solutions of the TDSE in the form of Gaussian WPswill be considered, particularly the free motion (potentialV (x) = 0) and the HO (V =

m2 ω2x2) with constant frequency,

ω = ω0, or TD frequency, ω = ω(t). In these cases, thesolution of the TDSE (here for the HO, the case V = 0, inthe following, is always obtained in the limit ω → 0)

ih∂

∂t9(x, t) =

{−

h2

2m

∂2

∂x2+

m

2ω2x2

}9(x, t), (1)

(where h =h

2π) can be written as

9(x, t) = N (t) exp

{i

[y(t) x2 +

〈p〉

hx + K (t)

]}(2)

2

Phys. Scr. 87 (2013) 038117 D Schuch

with the shifted coordinate x = x − 〈x〉 = x − η(t), wherethe mean value 〈x〉 =

∫ +∞

−∞9∗x9 dx = η(t) corresponds to

the classical trajectory, 〈p〉 = mη represents the classicalmomentum and the coefficient of the quadratic term in theexponent, y(t) = yR(t) + iyI(t), is a complex function of time.The (possibly TD) normalization factor N (t) and the purelyTD function K(t) in the exponent are not relevant to thefollowing discussion in this section.

The equations of motion for η(t) and y(t), or(

2hm y

),

that are obtained by inserting WP (2) into equation (1), areimportant for our purpose and given by

η + ω2η = 0, (3)

and (2h

my

)+

(2h

my

)2

+ ω2= 0, (4)

where overdots denote derivatives with respect to time. TheNewtonian equation (3) simply means that the maximumof the WP, located at x = 〈x〉 = η(t), follows the classicaltrajectory. The equation for the quantity 2h

m y(t) has theform a of a complex NL Riccati equation and describesthe time-dependence of the WP width that is related withthe position uncertainty via yI =

14〈x2〉

with 〈x2〉 = 〈x2

〉 −

〈x〉2 being the mean square deviation of position. This

quadratically NL complex equation will be the link to our‘Pythagorean quantization’ as well as to an invariant with thedimension of an action.

To show this, a new (real) variable α(t) is introducedvia

(2hm yI

)=

1α2 . Inserting this into the imaginary part of

equation (4) allows one to determine the real part of thevariable as

(2hm yR

)=

αα

, which, when inserted into the realpart of (4) together with the above definition of

(2hm yI

), finally

turns the complex Riccati equation into the real NL so-calledErmakov equation1 for α(t),

α + ω2α =1

α3. (5)

It had been shown by Ermakov [9] in 1880, i.e. 45 yearsbefore quantum mechanics was formulated by Schrodingerand Heisenberg, that from the pair of equations (3) and (5),coupled via ω2, by eliminating ω2 from the equations, adynamical invariant, the Ermakov-invariant

IL =1

2

[(ηα − ηα)2 +

(η

α

)2]

= const (6)

can be obtained (this invariant was rediscovered by severalauthors, also in a quantum mechanical context; see,e.g. [10–12]).

This invariant has (at least) two remarkable properties:(i) it is also a constant of motion for ω = ω(t), in thecase where the corresponding Hamiltonian does not havethis property; (ii) apart from a missing constant factor m,

1 The author is grateful to the referee for bringing to his attention that thisequation had been studied already in 1874 by Steen [6]. However, Steen’swork was ignored by mathematicians and physicists for more than a century,because it was published in Danish in a journal usually not containing manyarticles on mathematics. An English translation of the original paper [7] andgeneralizations can be found in [8].

i.e. mass of the system, it has the dimension of an action,not of an energy. The missing factor m is due to the factthat Ermakov used the mathematical equation (3), whereasin a physical context, Newton’s equation of motion, i.e.equation (3) multiplied by m, is relevant.

Furthermore, as will be shown below, an invariant of thistype also exists for certain dissipative systems, i.e. systemsfor which a conventional Hamiltonian does not even exist.Also, factorization of the corresponding operator leads togeneralized creation and annihilation operators (see end ofthis section). In this context, the complex Riccati equation (4)again plays the central role.

There are different ways of treating the (inhomogeneous)Riccati equation. Instead of transforming it into the (real)NL Ermakov equation (5), it can be solved directly bytransforming it into a homogeneous NL (complex) Bernoulliequation if a particular solution

(2hm y

)of the Riccati equation

is known. The general solution of equation (4) is then givenby 2h

m y =2hm y + 2h

m v(t) where 2hm v(t) fulfils the Bernoulli

equation (2h

mv

)+

(4h

my

) (2h

mv

)+

(2h

mv

)2

= 0. (7)

The coefficient A = 2( 2hm y) of the linear term depends

on the particular solution. Equation (7) can be linearized via2hm v =

1w(t) to yield

w − Aw = 1, (8)

which can be solved straightforwardly. For constant A, w(t)can be expressed in terms of exponential or hyperbolicfunctions (for real A) or trigonometric functions (forimaginary A). For A being TD, w(t) and hence 2h

m v(t) can be

expressed in terms of I(t) =∫ t dt ′e−

∫ t ′ dt ′′ A(t ′′). So the generalsolution of equation (4) can be written as

2h

my(t) =

2h

my +

d

dtln [w0 + I(t)], (9)

defining a one-parameter family of solutions depending on the

(complex) initial value of w0 =(

2hm v0

)−1as parameter.

Comparison with SUSY quantum mechanics [13, 14]shows that this solution is formally identical to the mostgeneral superpotential W (x), fulfilling a real Riccati equationand leading to a one-parameter family of isospectral potentialsthat have the same SUSY partner potential (see, e.g. [15–17]).A major difference between the SUSY situation and the onein our TD case (apart from replacing the spatial variableby a temporal one) is the fact that the variables of the NLequations (4) and (7) are complex, whereas W (x) is real. Also,

the parameter w0 =(

2hm v0

)−1in our case is generally complex

and determines the initial conditions.Another property of the Riccati equation, particularly

interesting in a quantum mechanical context, is the existenceof a superposition principle for this NL differentialequation [18–20]. This is related to the fact that the Riccatiequation can always be linearized. In our case, this can beachieved using the ansatz(

2h

my

)=

λ

λ, (10)

3

Phys. Scr. 87 (2013) 038117 D Schuch

with complex λ(t), leading to

λ + ω2(t)λ = 0, (11)

which has the form of the Newton-type equation (3) of thecorresponding problem, but now for a complex variable.

First, a kind of geometric interpretation of the motionof λ in the complex plane shall be given. Expressed inCartesian coordinates, λ can be written as λ = u + iz, or inpolar coordinates as λ = α eiϕ . Inserting the polar form intoequation (10) leads to(

2h

my

)=

α

α+ iϕ, (12)

where the real part is already identical to 2hm yR, as defined

above.The quantity α defined above in 2h

m yI as beingproportional to the position uncertainty is identical to theabsolute value of λ if it can be shown that

ϕ =1

α2. (13)

This, however, can be proven by simply inserting real andimaginary parts of (12) into the imaginary part of the Riccatiequation (4). Comparing relation (13), that can also be writtenin the form

zu − uz = α2ϕ = 1, (14)

with the motion of a particle under the influence of acentral force in two-dimensional physical space, it showsthat this relation corresponds to the ‘conservation of angularmomentum’, but here for the motion in the complex plane!

Relation (14) also shows, that real and imaginary parts,or phase and amplitude, respectively, of the complex quantityare not independent of each other but uniquely coupled. Thiscoupling, which is, as mentioned in the introduction, typicalfor quantum systems (but not only for these) is due to thequadratic nonlinearity in the Riccati equation. We will find ananalogous situation also in the TI case, discussed in the nextsection.

It should also be mentioned that the real part of thevariable

(2hm y

), as given in equation (12), does not depend on

the actual ‘size’ of α since only the relative change in timematters; so this quantity is invariant on different scales.

The variables α, α and ϕ allow one to express theuncertainties of position and momentum in a way that thecorresponding contribution to the energy of the WP can bewritten in Lagrangian or Hamiltonian form. The uncertaintiesthen read

〈x2〉L =

h

2mα2

L, 〈 p2〉L =

hm

2

(α2

L + ϕ2α2L

),

〈[x, p]+〉L = hαLαL, UL =h2

4

[1 + (αLαL)2

],

(15)

where UL = 〈x2〉L〈 p2

〉L, p = p − 〈p〉 and [. . . , . . .]+ denotesthe anti-commutator (the subscript L has been added todistinguish between quantities corresponding to the linearSE and those related to its NL modification, described insection 5).

The energy of the WP can be written as

E = 〈HL〉 = Ecl + E (16)

with the classical energy Ecl =m2 η2 + m

2 ω2η2=

12m 〈p〉

2 +m2 ω2

〈x〉2 and the contribution from the uncertainties

(corresponding to the ground-state energy E = E0 =h2 ω) in

the form

E =〈 p2

〉

2m+

m

2ω2

〈x2〉 =

h

4{α2 + ϕ2α2 + ω2α2

}. (17)

In search of a Hamiltonian formalism for the uncertaintiessimilar to the classical Hamiltonian formalism, one can tryusing the difference L=

12m 〈 p2

〉 −m2 ω2

〈x2〉, expressed in

terms of α, α and ϕ, as a kind of Lagrangian to determinethe corresponding canonical momenta, leading to

∂L∂α

= pα =h

2α and

∂L∂ϕ

= pϕ =h

2α2ϕ =

h

2. (18)

So, E can finally be written as a Hamiltonian in the form

H(α, pα, ϕ, pϕ) =p2

α

h+

p2ϕ

hα2+

h

4ω2α2, (19)

which yields the correct Hamiltonian equation of motionequivalent to the Ermakov equation (5) for α(t).

Let us now try to establish the relation to the Pythagoreantriples, introduced in section 2. For this purpose we considerthe complex quantity C =R+ iI =

2hm y =

αα

+ iϕ, where R=

αα

and I = ϕ =1α2 can be considered the catheti and the

absolute value |2hm y| =

√(αα

)2+ ϕ2 the hypothenuse of a

right-angled triangle in the complex plane.Allow for a short intermezzo before proceeding to the

triples. If we multiply all three sides by h2 α2 we obtain as new

catheti a =h2 αα, b =

h2 = pϕ and thus Pythagoras’ theorem

gives us the uncertainty relation expressed as products ofthe canonical variables we introduced above, i.e. a2 + b2

= c2

yields

p2ϕ + (αpα)2

=h2

4

[1 + (αα)2

]= UL. (20)

Now let us return to the time evolution of our complexquantity C =

2hm y. In the following, only the case V = 0 (i.e.

ω = 0) will be considered explicitly. As shown above, with aparticular solution of the Riccati equation its inhomogeneitycan always be removed. The resulting additional linear term(at least for constant coefficient A) can also be removed. Sowe are dealing with a complex equation of the form

d

dtC + C2

= 0. (21)

Then −ddt C is also a complex quantity, C2, where its real

and imaginary parts as well as its absolute value again definea right-angled triangle (in the complex plane) and each sidecontains contributions from R and I, i.e. <{C2

} =R2− I2,

={C2} = 2 RI and |C|2 =R2 + I2.If we now assume that R and I are integers (with

R > I), all three sides of the right-angled triangle createdby C2 in the complex plane are also integers. As examples, we

4

Phys. Scr. 87 (2013) 038117 D Schuch

choose: (a) R= 2, I = 1: R2− I2

= 3, 2 RI = 4, R2 + I2=

5 with 9 + 16 = 25; (b) R= 3, I = 2: R2− I2

= 5, 2 RI =

12 and R2 + I2= 13 with 25 + 144 = 169.

All possible Pythagorean triples can be obtained in thisway2 just by applying all integers R and I with R> I. In aphysical context this means that whenever a physical quantityobeys a complex Riccati equation and this quantity can be‘quantized’ in the sense that its real and imaginary parts canbe expressed as multiples of some units, its evolution (in timeor space, depending on the respective derivative) can alsobe expressed in terms of the same units. An example of aRiccati equation where the complex variable depends on space(instead of time) will be given in the next section.

To conclude the discussion based on WP solutions ofthe TDSE which (particularly in the context of quantumoptics) can also be considered as CSs, it shall also be shownhow the complex Riccati variable can be used to definegeneralized creation/annihilation operators. These can beused to construct CSs with TD width that are no minimumuncertainty WPs but fulfil the Schrodinger–Robertsonuncertainty relation [22, 23].

WPs with TD width are not only known from the TDSEfor the free motion. They also occur for the HO with constantfrequency ω0 if the initial state is not the ground state (leadingto oscillating width), for the HO with TD frequency ω(t)and for effective Hamiltonians describing open dissipativequantum systems.

The standard creation/annihilation operators can beobtained by factorizing the Hamiltonian Hop of the HO[24, 25] or an operator related to it via

Hop =Hop

hω0= (a+a + 1

2 ), (22)

where a+a is the so-called number operator and the creationand annihilation operators are defined by

a+=−i

√m

2hω0

(pop

m+i ω0 x

)=

1

2hω0

(−

h√

m

∂

∂x+√

mω0 x

),

(23)

a = i

√m

2hω0

(pop

m− i ω0 x

)=

1

2hω0

(h

√m

∂

∂x+

√mω0 x

),

(24)

where pop =hi

∂∂x and a is the adjoint operator of a+.

The number that is the eigenvalue of a+a is the numberof quanta of the action h since Hop

ω0has the dimension of an

action! With the help of a, the ground state wave functioncan be obtained and from this, by successive application ofa+, the excited states can be created. Via superposition ofall these states, Schrodinger obtained a stable Gaussian WP(with constant width) [26]. Generalizations of Schrodinger’sapproach were achieved for the description of coherent

2 The Pythagorean triples can also be understood in terms of rationalnumber points on the unit circle. This can be seen dividing both sides of

the Pythagorean theorem by c2 leading to( a

c

)2+

( bc

)2= 1. Therefore, a

correspondence between points on the unit circle with rational coordinatesand Pythagorean triples exists. The abovementioned algorithm can then bederived by trigonometric methods or by stereographic projection. For furtherdetails, see e.g. [21].

light beams emitted by lasers in terms of what is nowcalled CS.

One of at least three different definitions of CSs isthat these are eigenstates of the annihilation operator a with(complex) eigenvalue z, a|z〉 = z|z〉. Comparing the CS |z〉for the HO with the minimum uncertainty WP solution inthe form of equation (2), it shows that ω0 =

2hm yI =

1α2

0. So,

in definitions (23) and (24), iω0 can be replaced by i 2hm yI.

Therefore, for the more general case of WPs or CSs with TDwidth,

(2hm yR 6= 0

), iω0 must be replaced by the full complex

quantity(

2hm y

)in a and by

(2hm y∗

)in the adjoint operator a+.

If one then substitutes 1√

ω0= α0 in front of the brackets with

α(t), the generalized creation and annihilation operators takethe form

a+(t) = −i

√m

2hα(t)

(pop

m−

(2h

my∗

)x

)(25)

a(t) = i

√m

2hα(t)

(pop

m−

(2h

my

)x

). (26)

These operators can even be turned into constants ofmotion if an additional phase factor is taken into account.But in the case of CSs, as discussed here, this factor canbe absorbed into the phase of the CS and will therefore beomitted in the following (for further details see [27]).

Employing the above definition of the CS, but nowwith our generalized annihilation operator, i.e. a(t)|z〉 = z|z〉,the CS (in position representation, 〈x |z〉 = 9z(x, t)) can beobtained as

9z(x, t) =

(m

π h

)1/4

λ−1/2 exp

{i

[yx2 +

〈p〉x

h+

〈p〉〈x〉

2h

]},

(27)

which is in complete agreement with our WP definition (2)(here, also N (t) and K (t) are now specified).

From the mean values of position and momentum, 〈x〉z =

η and 〈p〉z = mη, calculated with these CSs, the real andimaginary parts of the complex eigenvalue z = zR + izI can bedetermined to be

zR =

√m

2h

(η

α

), zI =

√m

2h(ηα − ηα) , (28)

which looks familiar when compared with the Ermakovinvariant (6). Indeed, the absolute square of z is, up to aconstant factor, identical to IL,

IL =h

m

(z2

I + z2R

)=

h

mzz∗

=h

m|z|2. (29)

An operator, corresponding to IL can then be written inanalogy to H op =

Hop

hω0as

m

hIL ,op = [a+(t) a(t) + 1

2 ]. (30)

Factorization of this operator was also used [28] to findgeneralized creation and annihilation operators for the HOwith TD frequency but these operators were expressed interms of α and α or in terms of a complex variable [29]

5

Phys. Scr. 87 (2013) 038117 D Schuch

corresponding to λ(t) which fulfills the linear equation (11)instead of 2h

m y fulfilling the NL equation (4). The Riccatiequation (4) for 2h

m y can be solved directly, as shown above.The sensitivity of the solutions to the initial conditionsbecomes obvious immediately and the Riccati form is stillvalid in cases where a dissipative environment is taken intoaccount (see below); advantages the other approaches aremissing.

To conclude this section dealing with the TDSE, itshould be mentioned that a corresponding Riccati equationfor the WP width also exists in momentum space. There thevariable is just the inverse of the one in position space, i.e.(

2hm y

)−1=

λ

λ. Furthermore, the Ermakov invariant is directly

proportional to the exponent of the Wigner function whichis closest to the classical phase-space picture of thecorresponding problem (for details, see [30]).

4. Complex Riccati equations and thetime-independent SE

We have seen in the TD case that the real and imaginaryparts, or phase ϕ and amplitude α of the complex variableλ(t) = α eiϕ which fulfils the linear equation (11), obtainedvia equation (10) from the Riccati equation (4), are notindependent of each other but coupled via the conservationlaw (14). A similar situation exists when considering theTISE, but now in the space-dependent case.

This can be shown using Madelung’s hydrodynamicformulation of quantum mechanics [31] where the wavefunction is written in polar form as

9(r, t) = %1/2(r, t) exp

(i

hS(r, t)

)(31)

with the square root of the probability density % = 9∗9 asamplitude and 1

h S as phase (r is the position vector in threedimensions).

Inserting this form into the TDSE (1) (now in threedimensions), and replacing ∂

∂x by the nabla operator ∇), leadsto a modified Hamilton–Jacobi equation for the phase,

∂

∂tS +

1

2m(∇S)2 + V −

h2

2m

1%1/2

%1/2= 0, (32)

and a continuity equation for the amplitude,

∂

∂t% +

1

m∇(%∇S) = 0. (33)

Already here, the coupling of phase and amplitude can beseen clearly since the Hamilton–Jacobi equation for the phaseS contains a term (misleadingly called ‘quantum potential’,Vqu) depending on %, and the continuity equation for thedensity % contains ∇S. It can be shown that also in the TIcase this coupling is not arbitrary but related to a conservationlaw.

In 1994, Reinisch [32] presented a NL formulation ofTI quantum mechanics. Since in this case ∂

∂t % = 0 and∂∂t S = −E are valid, the continuity equation (33) (we now usethe notation %1/2

= |9| = a) turns into

∇(a2∇S) = 0 (34)

and the modified Hamilton–Jacobi equation into

−h2

2m1a + (V − E) a = −

1

2m(∇S)2a. (35)

Equation (34) is definitely fulfilled for ∇S = 0, turning(35) into the usual TISE for the real wave function a = |9|

with position-independent phase S. (NB: the kinetic energyterm divided by a is just identical to Vqu!)

However, equation (34) can also be fulfilled for ∇S 6= 0if only the conservation law

∇S =C

a2(36)

is fulfilled with constant (or, at least, position-independent) C .This relation now shows explicitly the coupling between

phase and amplitude of the wave function and is equivalentto equation (13) in the TD case. Inserting (36) into the rhs ofequation (35) changes this into the Ermakov equation

1a +2m

h2 (E − V ) a =

(1

h∇S

)2

a =

(C

h

)2 1

a3, (37)

equivalent to equation (5) in the TD case.A similar formulation of the TISE in terms of this

equation, but within a different context and differentapplications has also been given in [33]. In another paper [34],the relation between the Ermakov equation (37) and theTISE has been extended to also include magnetic fieldeffects. The NL differential equation (37) has also been usedto obtain numerical solutions of the TISE for single anddouble-minimum potentials as well as for complex energyresonance states; for details see [35, 36].

Returning to the method described in [32], so far theenergy E occurring in equation (37) is still a free parameterthat can take any value. However, solving this equationnumerically for arbitrary values of E leads, in general, tosolutions a that diverge for increasing x . Only if the energyE is appropriately tuned to any eigenvalue En of the TISE(see equation (39), below) this divergence disappears andnormalizable solutions can be found. So, the quantizationcondition that is usually obtained from the requirement of thetruncation of an infinite series in order to avoid divergenceof the wave function is, in this case, obtained from therequirement of nondiverging solutions of the NL Ermakovequation (37) by variation of the parameter E . This has beennumerically verified in the case of the one-dimensional HOand the Coulomb problem and there is the conjecture that thisproperty is ‘universal’ in the sense that it does not depend onthe potential V (see [32, 37]).

The corresponding complex Riccati equation is nowgiven by

∇

(∇9

9

)+

(∇9

9

)2

+2m

h2 (E − V ) = 0 (38)

with the complex variable(

∇99

)=

∇aa + i 1

h ∇S which corres-

ponds to(

2hm y

)=

λλ

=αα

+ iϕ in the TD problem.It is possible to show straightforwardly that equation (38)

can be linearized to yield the usual TISE

−h2

2m19 + V 9 = E9, (39)

6

Phys. Scr. 87 (2013) 038117 D Schuch

but in this form, the information on the coupling of phase andamplitude, expressed by equation (36) and originating fromthe quadratic NL term in equation (38), gets lost.

5. Dissipative systems with effective Hamiltonians

The conventional way of treating open dissipative systemsuses the system-plus reservoir approach, i.e. the systemof interest is coupled to some (in the limit infinitelymany) environmental degrees of freedom (often HOs) andsystem plus reservoir together are considered a closedHamiltonian system. Taking certain limits and applyingaveraging processes finally leads to an irreversible dissipativeequation of motion for the system of interest. One of themost often quoted approaches of that kind is the one ofCaldeira and Leggett [38, 39]. A similar idea, but with themost minimalistic environment, namely only one additionalposition variable (plus the corresponding momentum) isbehind the Bateman Hamiltonian [40] that represents aconstant of motion and provides an irreversible equation ofmotion for the system,

x + γ x +1

m

∂

∂xV = 0, (40)

i.e. a Newtonian equation with an additional linear velocity (ormomentum) dependent friction force (with friction coefficientγ ); actually the Langevin equation without stochastic force.

Since the environmental degrees of freedom are anywayeliminated or ignored in the end, several approachesexist where only the effect of the environment on theobservable system is taken into account without consideringthe individual environmental degrees of freedom. This canlead to modifications of the classical Lagrange/Hamiltonformalism where the corresponding modified (linear) SEis obtained via subsequent canonical quantization. Thecanonical variables of these approaches are related with thephysical position and momentum variables via non-canonicaltransformations in the classical case, corresponding tonon-unitary transformations in the quantum mechanical case(for further details see [41, 42]). The most frequently appliedapproach of that kind is the one of Caldirola [43] andKanai [44] which is uniquely related to one using anexponentially expanding coordinate system [45], leading toa Hamiltonian that is still a constant of motion. Theseapproaches can be directly linked to the aforementionedones. Using standard methods to eliminate the environmentaldegrees of freedom, Yu and Sun [46, 47] have shown howthe conventional approach of Caldeira–Leggett leads directlyto the Hamiltonian operator of Caldirola–Kanai. It is alsopossible to eliminate the second set of variables of theBateman approach by imposing TD constraints [48] to getto the Hamiltonian of the expanding system. Furthermore,this approach and the one of Caldirola–Kanai are connectedvia an explicitly TD canonical transformation. In our contextit is interesting that for these two approaches also an exactErmakov invariant exists. In the quantized version, GaussianWP solutions can be obtained in the same cases as inthe conventional reversible theory, but now the maximumfollows a damped motion according to equation (40) and thetime-dependence of the width is determined by a modified

complex Riccati equation that can again be transformed into a(real) Ermakov-type equation.

Another type of effective approaches starts already on thequantum level by adding some friction terms W (x, pop, t; 9)

to the Hamiltonian operator. This usually leads to NLHamiltonians, HNL = HL + W , where quite different forms ofnonlinearities are considered in the literature (some are NLonly because some mean-value 〈· · · 〉 occurs in W ) [49–60]3.Of these, an exact invariant was found [61, 62] for only twoapproaches [58, 59].

In the following, only those NLSEs possessing anErmakov invariant shall be discussed explicitly since it canbe shown that the canonical approaches are unambiguouslyrelated to these by a non-unitary transformation [41, 42].In particular, the equations of motion for the WP maximumand width can be uniquely transformed into each other [42].The approach of Hasse [58] uses a combination of productsof position and momentum operators and their meanvalues. The other one [59] is based on an irreversibleFokker–Planck-type equation for the probability density thatis obtained from the usual continuity equation by adding atime-symmetry-breaking diffusion term. Following a methodby Madelung and Mrowka [63, 64] this (real) so-calledSmoluchowski equation can be separated into two complexequations: namely a modified SE for the wave function 9 andits complex conjugate 9∗, provided the separation condition

− D∂2

∂x2 %

%= γ (ln % − 〈ln %〉) (41)

with diffusion coefficient D is fulfilled (for details see,e.g. [59, 66]).

This leads to the NLSE

ih∂

∂t9NL(x, t) =

{HL + γ

h

i(ln 9NL − 〈ln 9NL〉)

}9NL(x, t)

(42)with a complex logarithmic nonlinearity.

The additional NL term (WSCH) can be written as real andimaginary contributions in the form

WSCH = WR + iWI =γ

2

h

i

(ln

9NL

9∗

NL

−

⟨ln

9NL

9∗

NL

⟩)+

γ

2

h

i(ln %NL − 〈ln %NL〉) , (43)

where the real part only depends on the phase of the wavefunction and provides the friction force in the averagedequation of motion. The imaginary part does not contribute todissipation but introduces irreversibility into the evolution ofthe wave function. It corresponds to the diffusion term in theSmoluchowski equation, but still allows for normalizabilitydue to the subtraction of the mean value of ln %. Comparisonwith the afore-mentioned approaches shows that the real partis just identical to Kostin’s term [54] and the imaginary partcorresponds to Beretta’s term [51–53] introduced to describenon-equilibrium systems (without dissipation).

The imaginary part breaks the time-reversal symmetry onthe level of the probability density, introduces a non-unitarytime evolution and turns the Hamiltonian into a non-Hermitian

3 Sussmann [57] is quoted in [58].

7

Phys. Scr. 87 (2013) 038117 D Schuch

one while still guaranteeing normalizable wave functions andreal energy mean values since its mean value vanishes.

An interesting interpretation of WI can be found if oneidentifies, according to Grossing et al [65], the Einsteindiffusion coefficient with the quantum mechanical one (ifthe SE is considered a diffusion equation with imaginarydiffusion coefficient), i.e. D =

kTmγ

=h

2m with temperature Tand Boltzmann’s constant k. Then, WI turns into

WI = −iTk (ln %NL − 〈ln %NL〉) , (44)

where −k〈ln %〉 = −k∫ +∞

−∞% ln % dx has a form like the

definition of entropy, S. So, the mean value of the linearHamiltonian that still represents the energy of the system,〈HL〉 = E , together with the second term of (44) would looklike E − iTS, i.e. it has similarity with an expression forthe free energy only that here, again, the imaginary unit iturns up in the quantum mechanical context. This point stillneeds further investigation. In the context of our logarithmicNLSE (42), %NL is the probability density corresponding tothe Gaussian WP, which can be considered as a pure state,so a direct comparison with the von Neumann entropy in theusual way seems problematic. However, in Beretta’s work WI

is discussed in the context of density operators and direct linksto an interpretation in terms of entropy are given [51–53].

Also from the real part of WSCH no additional termto the energy mean value occurs, so this is still given bythe mean value of the operators of kinetic and potentialenergies. This real part is however not arbitrary but is uniquelydetermined by the separation condition and provides thecorrect dissipative friction forces in the equation of motion forthe mean values. Besides, the ratio of energy dissipation (forthe classical contribution) is in agreement with the classicalcounterpart and arises because the mean values are calculatedwith 9NL (the solution of equation (42)) instead of 9L.

The real part, by itself, would provide dissipationbut retain a unitary time-evolution of the wave function,whereas the imaginary part, on its own, would provideirreversibility via a non-unitary time-evolution but nodissipation. Consequently only the combination of real andimaginary parts provides all the desired properties of thequantum system under consideration. The reason for this is thecoupling of phase and amplitude of the wave function sinceWR depends on the phase and WI on the amplitude.

The relation between the two NL approaches is discussedin detail in [42] and can be traced back to a modification ofthe Riccati equation (38) by adding a linear term. The twoNLSEs have the same WP solutions where, in both cases, themaximum η(t) follows an equation of motion, like (40), witha linear friction force and the WP width is determined by themodified Riccati equation(

2h

my

)NL

+ γ

(2h

my

)NL

+

(2h

my

)2

NL

+ ω2(t) = 0 (45)

with an additional linear term depending on γ .As in the conservative case,

(2hm yI

)NL

=h

2 m〈x2〉NL=

1α2

NLis

valid but the real part of the complex Riccati variable nowtakes the modified form(

2h

myR

)NL

=αNL

αNL−

γ

2. (46)

The corresponding Ermakov equation and invariant aregiven now by

αNL +

(ω2

−γ 2

4

)αNL =

1

α3NL

(47)

and

INL =1

2α2

NL

[(η −

(αNL

αNL−

γ

2

)η

)2

+

(1

α2NLη

)2]eγ t

= const. (48)

From this it is obvious that, apart from the factor eγ t , INL

can be written in exactly the same form as in the conservativecase if expressed in terms of η and

(2hm y

)instead of α and

α, i.e.

INL =1

2α2

NL

[(η −

(2h

myR

)NL

η

)2

+

((2h

myI

)NL

η

)2]

eγ t ,

(49)which again shows the more universal validity of relationswhen expressed in terms of the Riccati variable. Also, inthis dissipative case, the invariant (without the exponentialfactor) can be factorized to yield generalized creation andannihilation operators where the CSs obtained as eigenstatesof the annihilation operator are identical to the WP solutionsof the NLSEs (for details, see [27]).

6. Similarities with nonlinear dynamics and otherfields of physics

In NL dynamics, an important phenomenon is the Hopfbifurcation as it can be the first step on a route to turbulenceand chaos [67]. A system displaying this property can bedescribed by the NL evolution equation

r + 0r + r3= 0, (50)

which has the solution

r2(t) =0r2

0 e−20t

r20 (1 − e−20t ) + 0

. (51)

For 0 > 0, the trajectory approaches a fixed point (theorigin); however, for 0 < 0, it spirals towards a limit cyclewith radius r∞ = |0|

1/2 [67]. The same type of differentialequation is also discussed by Großmann with respect toself-similarity and scale-invariance (see [68]).

The relation to our Riccati equations (4) or (45) is easilyseen by multiplying equation (50) by 4r and introducing a newvariable R = 2r2, leading to

R + 20R + R2= 0. (52)

This is exactly the form of the Bernoulli equation (7) thatcan be obtained if a particular solution

(2hm y

)of the Riccati

equation is known. The coefficient 20 of the linear term inequation (52), corresponds to A = 2

(2hm y

)in equation (7) and,

in the dissipative case, is simply replaced by A = 2(

2hm y

)+ γ .

It has indeed been shown that, in the dissipative case, this

8

Phys. Scr. 87 (2013) 038117 D Schuch

bifurcation occurs and one obtains two different WP solutionswith different spreading behaviour of the WP width, differentuncertainties and different energies (for details, see [69]).

Another system with cubic nonlinearity is the Gross–Pitaevskii equation

ih∂

∂t9 =

{−

h2

2m1 + V (r, t) + g|9|

2

}9, (53)

which is used in a mean field approximation to describethe macroscopic WP of a BEC where V (r, t) can begiven by V (r, t) =

m2 ω2(t)r2, i.e. a HO with TD frequency;

g parameterizes the strength of the atomic interaction.Although equation (53) cannot be solved analytically, the

dynamics of the BEC characterized by this equation can bedescribed in terms of so-called moments Mn (n = 1 − 4) (fordetails, see e.g. [70]), where M1 represents the norm, M2 thewidth, M3 the radial momentum and M4 the energy of the WP.It can be shown that these moments satisfy a set of coupledfirst-order differential equations (where d

dt M1 corresponds tothe conservation of probability or particle number). This setcan be reduced to a single equation for M2 which can beexpressed, using a new variable X =

√M2, in the form of an

Ermakov equation,

X + ω2(t)X =k

X3, (54)

which, as shown in section 3, is equivalent to a complexRiccati equation.

To include dissipative effects, one could add anotherNL term like the logarithmic one from equation (42) tothe Gross–Pitaevskii equation which would correspond toadding a linear term to the Riccati equation. So, onesimply has to solve this modified Riccati equation (or thecorresponding Ermakov equation) to obtain all moments Mn

for the dissipative BEC [71].This treatment of the BEC is also interesting for

another reason. It has been shown by Lidsey [72] that acorrespondence can be established between positively-curvedisotropic, perfect fluid cosmologies and the two-dimensionalharmonically-trapped BEC by mapping the equations ofmotion for both systems onto a one-dimensional Ermakovequation. The moments Mn defined above can be identifiedin the cosmological context with M2 = scale factor, M3 =

Hubble expansion parameter and M4 = energy density of theUniverse. So the expanding Universe can be represented as anErmakov or complex Riccati system.

Without going into details, it should be noted that acomplex Riccati equation occurs also in the context ofproblems related to quantum gravity (see, e.g. [73, 74]).

More examples could be mentioned from fields like elec-trodynamics, optics, quantum optics, NL dynamics, super-symmetry and others, but further details would go beyond thescope of this paper.

7. Conclusions and perspectives

In classical physics only real quantities have any physicalsignificance and energy in the form of Hamiltonians orLagrangians plays the dominant role. In quantum physics

however, action, i.e. the product of energy and time (orposition and momentum), is essentially the quantized entity.The appearance of i =

√−1, and hence the use of complex

quantities in quantum mechanics, is not just a matter ofmathematical convenience but has fundamental physicalmeaning. In the TISE, the wave function is such a complexquantity that fulfils a linear differential equation. We haveseen that this TISE is actually a linearized form of acomplex NL Riccati equation. Why should one bother witha more complicated NL equation if there is a simplerlinear version at hand for which such nice properties likea superposition principle exist? Because, in the linear form,it is not obvious that real and imaginary parts, or phaseand amplitude, respectively, of the complex wave functionare not independent of each other but uniquely coupled viaa kind of conservation law. This coupling can be tracedback to the quadratic nonlinearity in the Riccati equationand always occurs in systems that can be described bycomplex Riccati equations. This complex quadratic term isactually also the key to the ‘quantization’ problem that goesback to Plato and Pythagoras, namely, the search for analgorithm that supplies the so-called Pythagorean triples. Thismost abstract quantization problem in terms of numbers,particularly integers, can therefore be related to physicalquantization problems whenever the evolution (in time orspace) of the physical system can be described by a complexRiccati equation.

We found another example of that kind in the TDSEwhere the time-evolution of the quantum uncertainties obeyssuch an equation. The linearized version of this Riccatiequation is just a complex Newtonian equation of motion fora quantity λ(t) where the coupling of phase and amplitudeof this quantity corresponds to the conservation of angularmomentum for the motion of λ in the complex plane!

The complex TD Riccati equation (or its transformedversion, the real NL Ermakov equation) together with theclassical Newtonian equation for the system, lead to adynamical invariant with the dimension of action. ThisErmakov invariant not only essentially determines the Wignerfunction of the system but, when the corresponding operator isfactorized, one obtains generalized creation and annihilationoperators that also apply in cases where the correspondingHamiltonian is no longer invariant. Specifically, this is alsovalid for certain dissipative systems when the Ermakovinvariant is expressed in terms of the complex Riccati variable.This has been shown using some effective models for thedescription of such open systems.

Finally, an initial link to NL dynamics was madewhere properties like scale-invariance, bifurcations as aroute to chaos and other similar properties already emergewhen real Riccati equations occur. The relations to our NLversion of quantum mechanics, in particular the effect of‘complexification’ will be further investigated. In addition,formal similarities to fields like SUSY quantum mechanics,quantum optics and cosmology shall be explored.

Acknowledgments

The author would like to dedicate this paper to ProfessorFrancesco Iachello on the occasion of his 70th birthday in2012.

9

Phys. Scr. 87 (2013) 038117 D Schuch

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