Riccati and Ermakov Equations in Time-Dependent and Time-Independent Quantum Systems
aa r X i v : . [ qu a n t - ph ] M a y Symmetry, Integrability and Geometry: Methods and Applications SIGMA (2008), 043, 16 pages Riccati and Ermakov Equations in Time-Dependentand Time-Independent Quantum Systems ⋆ Dieter SCHUCHInstitut f¨ur Theoretische Physik, J.W. Goethe–Universit¨at,Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany
E-mail: [email protected]
Received December 28, 2007, in final form May 07, 2008; Published online May 12, 2008Original article is available at
Abstract.
The time-evolution of the maximum and the width of exact analytic wave packet(WP) solutions of the time-dependent Schr¨odinger equation (SE) represents the particle andwave aspects, respectively, of the quantum system. The dynamics of the maximum, locatedat the mean value of position, is governed by the Newtonian equation of the correspondingclassical problem. The width, which is directly proportional to the position uncertainty,obeys a complex nonlinear Riccati equation which can be transformed into a real nonlinearErmakov equation. The coupled pair of these equations yields a dynamical invariant whichplays a key role in our investigation. It can be expressed in terms of a complex variable thatlinearizes the Riccati equation. This variable also provides the time-dependent parametersthat characterize the Green’s function, or Feynman kernel, of the corresponding problem.From there, also the relation between the classical and quantum dynamics of the systemscan be obtained. Furthermore, the close connection between the Ermakov invariant and theWigner function will be shown. Factorization of the dynamical invariant allows for compari-son with creation/annihilation operators and supersymmetry where the partner potentialsfulfil (real) Riccati equations. This provides the link to a nonlinear formulation of time-independent quantum mechanics in terms of an Ermakov equation for the amplitude of thestationary state wave functions combined with a conservation law. Comparison with SUSYand the time-dependent problems concludes our analysis.
Key words:
Riccati equation; Ermakov invariant; wave packet dynamics; nonlinear quantummechanics
Participating in a conference on nonlinear mathematical physics taking place in Kiev, one canfind more colleagues than usual who are familiar with the name Ermakov; some are even able totell where one can find traces of him from the time he used to work at the university in Kiev. Thisperiod was, however, before quantum mechanics was established and Vasili Petrovich Ermakov(1845–1922) died already four years before Schr¨odinger published his celebrated equation [1]. So,what can be his relation with quantum theory? The second name in the title of this paper alsoseems to be surprising in this context because Jacopo Riccati (1676–1754) was born actually 250years prior to Schr¨odinger’s publication! The surprising answer to our question is that Ermakov’sand Riccati’s contributions are not only closely connected with quantum theory in the formestablished by Schr¨odinger and Heisenberg, but also with later formulations due to Feynmanand Wigner. Therefore, their work plays a central role in finding underlying connections amongstall these different formulations. ⋆ D. SchuchOne way of showing how Riccati and Ermakov enter the formalism of quantum theory is thestudy of cases where exact analytic Gaussian wave packet (WP) solutions of the time-dependentSchr¨odinger equation (SE) exist, in particular, the harmonic oscillator (HO) and the free mo-tion. Therefore, in the following, we will restrict the discussion of the time-dependent problemsto the (one-dimensional) HO (where the frequency ω may also be time-dependent) and the freemotion is obtained in the limit ω →
0. In Section 2, it will be shown how a (complex) nonlinearRiccati equation that governs the dynamics of a typical quantum effect, namely, the positionuncertainty, emerges from the WP solution of the time-dependent SE. This Riccati equationcan be transformed into a (real) nonlinear Ermakov equation that, together with the classicalNewtonian equation of the system, provides a corresponding dynamical invariant, the so-calledErmakov invariant, that will play an important role in our discussion. As shown in Section 3,the linearization of the Riccati equation can be accomplished by introducing new complex vari-ables. These variables are connected with the time-dependent parameters that characterize thetime-dependent Green’s function, also called Feynman kernel, of the corresponding problem.They also provide the key to understanding the relations between the dynamics of the classicaland quantum aspects of the system; in particular, the importance of the initial position un-certainty will become obvious. Furthermore, in Section 4, the close relationship between theErmakov invariant and the Wigner function will be shown. The link to the occurrence of Ric-cati and Ermakov equations in time-independent quantum mechanics is obtained in Section 5.Factorization of the Ermakov invariant allows for a comparison with creation and annihilationoperators as well as their generalizations used in supersymmetry (SUSY). In the latter case, thesupersymmetric partner potentials fulfil real
Riccati equations. This shows similarities witha nonlinear formulation of time-independent quantum mechanics by Reinisch [2], where he ob-tains an Ermakov equation (with spatial derivatives instead of temporal ones) for the amplitudeof the stationary state wave functions. This Ermakov equation is related with a kind of com-plexification of the Riccati equation appearing in SUSY. Finally, in Section 6, the main resultswill be summarized and possible generalizations mentioned.
Starting point of our investigation is the time-dependent SE for the HO with frequency ω ( t ), i ~ ∂∂t Ψ WP = (cid:26) − ~ m ∂ ∂x + m ω ( t ) x (cid:27) Ψ WP (1)that possesses Gaussian WP solutions of the typeΨ WP ( x, t ) = N ( t ) exp (cid:26) i (cid:20) y ( t )˜ x + h p i ~ ˜ x + K ( t ) (cid:21)(cid:27) , (2)where y ( t ) = y R ( t ) + iy I ( t ), ˜ x = x − h x i = x − η ( t ) (i.e., h x i = η ( t ) is the classical trajectorycalculated as mean value R Ψ ∗ WP x Ψ WP dx = h x i ), h p i = m ˙ η is the classical momentum and N ( t )and K ( t ) are purely time-dependent terms that are not relevant for the following discussion.Inserting the WP (2) into equation (1) yields the equations of motion for η ( t ) and y ( t ). Theequation for the WP maximum, located at x = η ( t ), is just the classical equation of motion¨ η + ω ( t ) η = 0 , (3)where overdots denote time derivatives.The equation of motion for the complex quantity y ( t ) that is connected with the WP widthand, thus, the position uncertainty, is given by the complex Riccati-type equation2 ~ m ˙ y + (cid:18) ~ m y (cid:19) + ω ( t ) = 0 . (4)iccati and Ermakov Equations in Quantum Theory 3Equation (4) can be separated into real and imaginary parts,Im : 2 ~ m ˙ y I + 2 (cid:18) ~ m y I (cid:19) (cid:18) ~ m y R (cid:19) = 0 , (5)Re : 2 ~ m ˙ y R + (cid:18) ~ m y R (cid:19) − (cid:18) ~ m y I (cid:19) + ω ( t ) = 0 . (6)The real part y R ( t ) can be eliminated from equation (6) by solving equation (5) for y R andinserting the result into equation (6).It is useful to introduce a new variable α ( t ) that is connected with y I ( t ) via2 ~ m y I = 1 α ( t ) , (7)where α ( t ) is directly proportional to the WP width, or position uncertainty, i.e., α = (cid:16) m ~ y I (cid:17) / = (cid:16) m h ˜ x ( t ) i ~ (cid:17) / with h ˜ x i = h x i − h x i . Inserting this definition (7) into equation (5) shows thatthe real part of y ( t ) just describes the relative change in time of the WP width,2 ~ m y R = ˙ αα = 12 ddt h ˜ x ih ˜ x i . (8)Together with definition (7), this finally turns equation (6) into¨ α + ω ( t ) α = 1 α , (9)the so-called Ermakov equation.It has been shown by Ermakov in 1880 [3] that the system of differential equations (3) and (9),coupled via the possibly time-dependent frequency ω , leads to a dynamical invariant that hasbeen rediscovered by several authors in the 20th century [4], I L = 12 (cid:20) ( ˙ ηα − η ˙ α ) + (cid:16) ηα (cid:17) (cid:21) = const . (10)It is straightforward to show that ddt I L = 0; a proof following Ermakov’s method can be foundin [5]. The Ermakov invariant not only depends on the classical variables η ( t ) and ˙ η ( t ), but alsoon the quantum uncertainty connected with α ( t ) and ˙ α ( t ). Additional interesting insight intothe relation between the variables η and α can be obtained from a different treatment of theRiccati equation (4). Introducing a new complex variable λ ( t ), the complex variable in the Riccati equation (4) canbe replaced by the logarithmic time-derivative of λ , i.e., (cid:18) ~ m y (cid:19) = ˙ λλ , (11)thus turning the nonlinear Riccati equation into the complex linear equation¨ λ + ω ( t ) λ = 0 , looking exactly like equation (3) for η ( t ), which is not just by accident, as will be shown later. D. SchuchThe complex variable λ ( t ) can be written in polar coordinates as well as in cartesian coordi-nates, i.e., λ = αe iϕ = u + iz. The choice of the symbol α for the absolute value of λ is also not coincidental, as will now beshown. Writing relation (11) in polar coordinates yields˙ λλ = ˙ αα + i ˙ ϕ = (cid:18) ~ m y (cid:19) . (12)The real part already looks identical to the one given in equation (8). So, the absolute value of λ would be (up to a constant factor) identical with the square root of the position uncertainty h ˜ x i ,if ˙ ϕ = 1 α (13)is fulfilled. This can easily be verified by inserting (cid:0) ~ m y R (cid:1) and (cid:0) ~ m y I (cid:1) , as given in equation (12),into the imaginary part of the Riccati equation, thus turning equation (5) into¨ ϕ + 2 ˙ αα ˙ ϕ = 0 , in agreement with equation (13). From equation (6) for the real part then, again, the Ermakovequation is obtained as¨ α + ω ( t ) α = α ˙ ϕ = 1 α . After the physical meaning of the absolute value of λ in polar coordinates and its relation tothe phase angle via (13) have been clarified, the interpretation of the cartesian coordinates u and z needs to be ascertained. For this purpose, it can be utilized that the WP solution (2) attime t can also be obtained with the help of an initial WP at, e.g., t ′ = 0 and a time-dependentGreen’s function, also called time-propagator or Feynman kernel, viaΨ WP ( x, t ) = Z dx ′ G ( x, x ′ , t, t ′ = 0)Ψ WP (cid:0) x ′ , (cid:1) . (14)For the considered Gaussian WP with initial distributionΨ WP (cid:0) x ′ , (cid:1) = (cid:18) mβ π ~ (cid:19) / exp (cid:26) im ~ h iβ x ′ + 2 p m x ′ i(cid:27) , (15)where β = ~ m h ˜ x i = α and p = h p i ( t = 0), the Green’s function can be written as G (cid:0) x, x ′ , t, (cid:1) = (cid:18) m πi ~ α z (cid:19) / exp ( im ~ " ˙ zz x − xz (cid:18) x ′ α (cid:19) + uz (cid:18) x ′ α (cid:19) . (16)iccati and Ermakov Equations in Quantum Theory 5Since in the definition of Ψ WP ( x, t ), according to equation (14), only G actually depends on x and t , the kernel G , as defined in equation (16), must also fulfil the time-dependent SE. Inser-ting (16) into the SE (1) shows that z ( t ) and u ( t ) not only fulfil the same equation of motionas η ( t ) and λ ( t ) but, in addition, are also uniquely connected via the relation˙ zu − ˙ uz = 1 . (17)Expressing u and z in polar coordinates according to u = α cos ϕ and z = α sin ϕ shows that thiscoupling is identical to relation (13) that connects the amplitude and phase of λ . From equa-tion (17) it also follows that, e.g., with the knowledge of z , the quantity u can be calculated as u = − z Z t z dt ′ . The last necessary step is to explicitly perform the integration in (14), using (15) and (16), toyield the WP solution in the formΨ WP ( x, t ) = (cid:16) mπ ~ (cid:17) / (cid:18) u + iz (cid:19) / exp (cid:26) im ~ (cid:20) ˙ zz x − ( x − p α m z ) z ( u + iz ) (cid:21)(cid:27) . Comparison with the same WP, written in the form given in equation (2), shows that therelations z = mα p η ( t ) (18)and 2 ~ m y = ˙ zz − zλ = ˙ λλ are valid, where λ = u + iz is identical to our linearization variable and equation (17) has beenapplied. So, from equation (18), it then follows that the imaginary part of λ is, apart froma constant factor, just the particle trajectory.The equivalence between deriving the time-dependent Green’s function via a Gaussian ansatzor via Feynman’s path integral method has been mentioned in [6], where also the relation to theErmakov invariant is considered.In conclusion, one can say that the complex quantity λ contains the particle as well as thewave aspects of the system. In polar coordinates, the absolute value α of λ is directly connectedwith the quantum mechanical position uncertainty; in cartesian coordinates, the imaginary partof λ is directly proportional to the classical particle trajectory η . Absolute value and phase, orreal and imaginary parts, of λ are not independent of each other but uniquely connected via theconservation laws (13) and (17), respectively. Further insight into the relation between the classical and quantum dynamics of the system canbe gained by rewriting the invariant (10), with the help of equation (18), in terms of z and ˙ z instead of η and ˙ η , I L = 12 (cid:16) α p m (cid:17) (cid:20) ( ˙ zα − z ˙ α ) + (cid:16) zα (cid:17) (cid:21) = const . D. SchuchSince zα = sin ϕ , for I L to be constant it is necessary that ( ˙ zα − z ˙ α ) = cos ϕ = (cid:0) uα (cid:1) is valid.So, up to a ± sign, one obtains u = ˙ zα − z ˙ αα = α (cid:18) ˙ z − ˙ αα z (cid:19) . (19)With this form of u , a certain asymmetry in the exponent of the Feynman kernel (16), namely ˙ zz as coefficient of x compared to α uz as coefficient of the initial quantity x ′ , can be explained.According to (19), ˙ zz can be written as˙ zz = 1 α ( t ) uz + ˙ αα . (20)This shows that, in the case when α is time-dependent, not only α must be replaced by α ( t ),but also an additional term ˙ αα must be taken into account. For constant α = α or at t = 0,relation (20) reduces to˙ zz = 1 α uz or u = α ˙ z = mα p ˙ η, (21)i.e., u is simply proportional to ˙ η . Note the explicit occurrence of α , the initial positionuncertainty, because it has important consequences for the time-dependence of α ( t ). Inserting u ,as given in (21), into α ( t ) = u + z yields (with v = p m and β = α ) α ( t ) = α v (cid:2) ˙ η + β η (cid:3) = 2 m ~ h ˜ x i . (22)This shows that the quantum mechanical uncertainty of position (at any time t ) can be expressedsolely in terms of the classical trajectory η ( t ) and the corresponding velocity ˙ η ( t ), if the initialvelocity v and the initial position uncertainty , expressed by α are known.This explains why Feynman’s procedure [7] of deriving his kernel based only on the classical Lagrangian provides the correct time-evolution of the system since the time-dependence entersonly via the classical variables η ( t ) and ˙ η ( t ). However, the importance of the initial positionuncertainty α for the quantum dynamics should not be underestimated.The influence of the initial uncertainty becomes clear if one inserts the expressions for η ( t )and ˙ η ( t ) into equation (22).a) For the free motion, one obtains with η = v t , ˙ η = v : α = α (cid:2) β t ) (cid:3) ;b) for the HO with η = v ω sin ωt and ˙ η = v cos ωt , equation (22) yields α = α ( cos ωt + (cid:18) β ω sin ωt (cid:19) ) . Only if the initial state is the ground state, is β = ~ m h ˜ x i = α = ω valid and, hence, α = α , i.e., the WP width remains constant; in all other cases it oscillates. This oscillatingWP width corresponds to the general solution of the Riccati equation (4) and yields in the limit ω → ω → α ( t ) = α (cid:2) β t ) (cid:3) = α ( t ) , iccati and Ermakov Equations in Quantum Theory 7whereas the WP usually presented, with constant width α = (cid:0) ω (cid:1) / , provides in this limit onlya plane wave, no WP!The time-derivative of α , or,˙ αα = α v ˙ η (cid:2) β η + ¨ η (cid:3) = α v ˙ η (cid:20) β η − ∂∂η V ( η ) (cid:21) , respectively, only vanishes if the term in square brackets is equal to zero, which dependson ∂∂η V ( η ). Therefore, for V = 0 (¨ η = 0), ˙ α = 0 is always valid; for the HO (¨ η = − ω η ),˙ α = 0 is only valid for β = α = ω , otherwise, and in particular for ω = ω ( t ) (which describes,e.g., the motion of an ion in a Paul trap [8]), ˙ α = 0 always holds. So, obviously the initialvalue α of the position uncertainty plays an important role in the qualitative behaviour of thequantum aspect of the dynamics. A more detailed discussion of this problem can be found in [9],also including dissipative effects. In the previous section, it has been shown how the real and imaginary parts (in cartesiancoordinates) of the complex variable λ ( t ), that allows the linearization of the complex Riccatiequation (4), enter the Feynman kernel that describes the transformation of an initial quantumstate into a state at a later time t as time-dependent parameters. But, also in polar coordinates α and ϕ (which are not independent of each other but related via ˙ ϕ = α ), λ ( t ) is related, via theErmakov invariant (10), with another description of quantum systems that shows close similaritywith the classical phase space description of dynamical systems, namely the Wigner function.In order to show this connection the invariant I L shall be written explicitly in the form I L = 12 (cid:20)(cid:18) ˙ α + 1 α (cid:19) η − α ˙ αη ˙ η + α ˙ η (cid:21) with terms bilinear in η and ˙ η . How the coefficients of these terms are related with the quantumuncertainties, and thus with α , ϕ and λ , follows from λλ ∗ = α = m ~ y I = 2 m ~ h ˜ x i L , (23)˙ λ ˙ λ ∗ = ˙ α + α ˙ ϕ = ~ m y + y y I = ~ m h ˜ p i L , (24) ∂∂t ( λλ ∗ ) = 2 ˙ αα = 2 (cid:18) y R y I (cid:19) = 2 ~ h [˜ x, ˜ p ] + i L = 2 ~ h ˜ x ˜ p + ˜ p ˜ x i L . (25)With the help of these relations, the invariant takes the form I L = 12 (cid:20) ˙ λ ˙ λ ∗ η − ∂∂t ( λλ ∗ ) η ˙ η + λλ ∗ ˙ η (cid:21) = 1 m ~ (cid:2) h ˜ p i L η − h [˜ x, ˜ p ] + i L η ( m ˙ η ) + h ˜ x i L ( m ˙ η ) (cid:3) . (26)The connection with the Wigner function becomes obvious if one performs the Wigner trans-formation [10] of our WP (2) according to W ( x, p, t ) = 12 π ~ Z + ∞−∞ dq e ipq/ ~ Ψ ∗ WP (cid:16) x + q , t (cid:17) Ψ WP (cid:16) x − q , t (cid:17) D. Schuchyielding the Wigner function in the form W ( x, p, t ) = 1 π ~ exp (cid:26) − (cid:18) y + y y I (cid:19) ˜ x − ˜ p ~ y I + 2 ~ (cid:18) y R y I (cid:19) ˜ x ˜ p (cid:27) . Using relations (23)–(25), this can be expressed as W ( x, p, t ) = 1 π ~ exp (cid:26) − ~ (cid:2) h ˜ p i L ˜ x − h [˜ x, ˜ p ] + i L ˜ x ˜ p + h ˜ x i L ˜ p (cid:3)(cid:27) . (27)In particular, at the origin of the phase space, i.e., for x = 0 and p = 0, where ˜ x → η ,˜ p → ( m ˙ η ) and ˜ x ˜ p → mη ˙ η , one obtains W (0 , , t ) = 1 π ~ exp (cid:26) − m ~ I L (cid:27) = constwith I L as given in (26) which fulfils ∂∂t W = 0 since I L is an invariant. For x = 0 and p = 0, W ( x, p, t ) takes the form (27) and the equation of motion is given, as expected, by ∂∂t W ( x, p, t ) = − pm ∂W∂x + ∂V∂x ∂W∂p , i.e., a continuity equation for an incompressible medium, also called Liouville equation in phasespace.Further details, also concerning the different forms to express the Ermakov invariant inthe exponent of the Wigner function and the physical interpretation of these forms is givenin [11]. Other attempts to construct connections between the Ermakov invariant and the Wignerfunction can be found in [6]. Formal similarities between the Ermakov invariant and the algebraic treatment of the HO usingcreation and annihilation operators (or complex normal modes, in the classical case) and itsgeneralization in the formalism of SUSY can be found if I L is written in a form that allows forfactorization I L = 12 α "(cid:18) ˙ η − ˙ αα η (cid:19) + (cid:16) ηα (cid:17) = 12 α AA ∗ with A = (cid:18) ˙ η − ˙ αα η (cid:19) − i α η = ˙ η − (cid:18) ~ m y (cid:19) η (28)and A ∗ = (cid:18) ˙ η − ˙ αα η (cid:19) + i α η = ˙ η − (cid:18) ~ m y ∗ (cid:19) η. (29)For the HO with time-independent frequency ω and constant WP width α , the real partof (cid:0) ~ m y (cid:1) vanishes ( ˙ α = 0) and the imaginary part is simply α = ˙ ϕ = ω , so equations (28)and (29) turn into A = ˙ η − iω η, A ∗ = ˙ η + iω η. (30)iccati and Ermakov Equations in Quantum Theory 9These expressions are, up to constant factors, identical with the complex normal modes or, inthe quantized form, with the creation and annihilation operators of the HO. In this context,it should be mentioned that the Ermakov invariant has also been quantized, where α and ˙ α remain c-numbers, whereas position η and momentum m ˙ η are, following the rules of canonicalquantization, replaced by the corresponding operators, i.e., η → q op = q , m ˙ η → p op = ~ i ∂∂q (fordetails see, e.g., [12]). Expressions (30) would then turn into the operators A , op = 1 m p op − iω q, A ∗ , op = 1 m p op + iω q. For comparison, the Hamiltonian operator of the HO can be written as H op , HO = 12 m p + m ω q = ~ ω (cid:18) ˆ b + ˆ b − + 12 (cid:19) with the creation/annihilation operatorsˆ b ± = ∓ i r m ~ ω (cid:16) p op m ± iω q (cid:17) = r m ~ ω (cid:16) ω q ∓ i p op m (cid:17) . (31)Comparison shows that α A , op = − i r ~ m ˆ b − , α A ∗ , op = + i r ~ m ˆ b + . So, A and A ∗ (or A op and A ∗ op ) are generalizations where the constant factor ± iω , in frontof q , is replaced by the complex time-dependent functions (cid:0) ~ m y (cid:1) or (cid:0) ~ m y ∗ (cid:1) , respectively. Thefactorization of the dynamical invariant and the relation to creation/annihilation operators isalso discussed in [6]. A different generalization of the creation/annihilation operators is found in SUSY where, essen-tially, the term linear in the coordinate q is replaced by a function of q , the so-called “superpo-tential” W ( q ), leading to the operators B ± = 1 √ (cid:20) W ( q ) ∓ i p op √ m (cid:21) . In this case, the term ω q with constant ω is replaced by a real, position-dependent func-tion W ( q ). The operators B ± fulfil the commutator and anti-commutator relations[ B − , B + ] − = ~ √ m dWdq , { B − , B + } + = W + p m . The supersymmetric Hamiltonian H SUSY = (cid:18) H H (cid:19) can be expressed with the help of B ± in the form H = B + B − = − ~ m d dq + V ( q ) , H = B − B + = − ~ m d dq + V ( q ) . V ( q ) and V ( q ) fulfil real Riccati equations, whichfollows directly from the definition of B ± , i.e., V = 12 (cid:20) W − ~ √ m dWdq (cid:21) (32) V = 12 (cid:20) W + ~ √ m dWdq (cid:21) . (33)The energy spectra of H and H are identical apart from the ground state. H has theground state E (1)0 = 0, whereas the ground state E (2)0 of H is identical with the first excitedstate E (1)1 of H . The ground state wave function of H , Ψ (1)0 , has no node and determines thesuperpotential via W = − ~ √ m ddq Ψ (1)0 Ψ (1)0 . From equations (32) and (33), then, the partner potentials V and V follow. On the otherhand, Ψ (1)0 is connected with V via the solution of the equation H Ψ (1)0 = 0, i.e., H Ψ (1)0 = − ~ m d dq Ψ (1)0 + 12 (cid:20) W − ~ √ m dWdq (cid:21) Ψ (1)0 = E (1)0 Ψ (1)0 = 0 . The connection between the spectra of H and H , i.e. E (1) n and E (2) n , and the correspondingwave functions, Ψ (1) n and Ψ (2) n , is determined via the generalized creation/annihilation opera-tors B ± according toΨ (1) n +1 = 1 q E (2) n B + Ψ (2) n and Ψ (2) n = 1 q E (1) n +1 B − Ψ (1) n +1 , (34)where B + creates a node and B − annihilates a node in the wave function. So, e.g., the firstexcited state Ψ (1)1 of H (which has one node) can be obtained from the ground state Ψ (2)0 of H (which has no node) by applying B + onto it as described in (34).In order to obtain the higher eigenvalues and eigenfunctions, Ψ (1)0 in the definition of W ≡ W must be replaced by Ψ (2)0 , leading to W = − ~ √ m ddq Ψ (2)0 Ψ (2)0 etc., i.e., W s = − ~ √ m ddq Ψ ( s )0 Ψ ( s )0 with the corresponding operators B ± s = 1 √ (cid:20) W s ∓ ~ √ m ddq (cid:21) , thus creating a hierarchy that provides all eigenvalues and eigenfunctions of the Hamiltonians H and H .In the context of this paper, only two systems with exact analytic solutions shall be consideredexplicitly, namely, the one-dimensional HO (with constant frequency ω = ω ) and the Coulombproblem. The latter case, a three-dimensional system with spherical symmetry ( V ( ~r ) = V ( r ) =iccati and Ermakov Equations in Quantum Theory 11 − e r ), can be reduced to an essentially one-dimensional problem via separation of radial andangular parts. Using the ansatz Φ nlm ( ~r ) = r Ψ nl ( r ) Y lm ( ϑ, ϕ ) = R ( r ) Y lm ( ϑ, ϕ ) for the wavefunction (with n = total quantum number, l = azimuthal quantum number, m = magneticquantum number, r, ϑ, ϕ = polar coordinates), the energy eigenvalues E n of the system can beobtained from the radial SE (cid:26) − ~ m d dr + V eff (cid:27) Ψ nl ( r ) = E n Ψ nl ( r )with the effective potential V eff = V ( r ) + l ( l + 1) ~ mr = − e r + l ( l + 1) ~ mr . (35)The superpotential W , the energy eigenvalues E n and the supersymmetric potential V forthe systems under consideration are given by:a) HO: V ( q ) = m ω q (eigenfunctions Ψ n ( q ): Hermite polynomials) W = ωq,E n = ~ ω (cid:18) n + 12 (cid:19) ,V = m ω q − ~ ω = V ( q ) − E . (36)b) Coulomb potential: V ( r ) = − e r (eigenfunctions Ψ nl ( r ): Laguerre polynomials) W = √ me ( l + 1) ~ − ( l + 1) ~ √ mr ,E n ′ = − mc (cid:18) e ~ c (cid:19) n ′ + l + 1) ,V = − e r + l ( l + 1) ~ mr + mc (cid:18) e ~ c (cid:19) l + 1) = V eff − E . (37)In the second case, the radial quantum number n ′ occurs which indicates the number ofnodes in the wave function and is connected with the total quantum number n , that actuallycharacterizes the energy eigenvalue, via n = n ′ + l + 1.Particularly the quantities V , given in equations (36) and (37), shall be compared with similarexpressions obtained in the next subsection where a nonlinear formulation of time-independentquantum mechanics is presented. In the following discussion of a nonlinear formulation of quantum mechanics, that is essentiallybased on the work of Reinisch [2], formal similarities with SUSY (in the time-independent case)and the complex Riccati formalism (in the time-dependent case) shall be pointed out.Starting point is Madelung’s hydrodynamic formulation of quantum mechanics [14] that usesthe polar ansatzΨ( ~r, t ) = a ( ~r ) exp (cid:26) − i ~ S ( ~r, t ) (cid:27) ~r, t ), turning the linear SE (1) into two coupled equations for theamplitude a ( ~r ) and the phase S ( ~r, t ), i.e., the Hamilton–Jacobi-type equation ∂∂t S + 12 m ( ∇ S ) + V − ~ m ∆ aa = 0 (38)(with ∇ = Nabla operator, ∆ = Laplace operator) and the continuity equation ∂∂t a + 1 m ∇ ( a ∇ S ) = 0 , (39)where a = Ψ ∗ Ψ = ̺ ( ~r, t ) is the usual probability density. For stationary states, the energy of thesystem is related with the action S via ∂∂t S = − E = const and the density is time-independent,i.e., ∂∂t a = 0 , where it subsequently follows that the second term on the lhs of equation (39) must also vanish.In the usual textbook treatment, this is achieved by taking ∇ S = 0, thus turning equation (38)into the conventional time-independent linear SE − ~ m ∆ a + V a = Ea. (40)This is, however, not the only possibility of fulfilling equation (39) since, also for ∇ S = 0,this can be accomplished if only ∇ S = Ca (41)with constant C . In this case, equation (38) takes the form of a nonlinear Ermakov equation∆ a + 2 m ~ ( E − V ) a = (cid:18) C ~ (cid:19) a . (42)The corresponding complex Riccati equation, equivalent to equation (4) in the time-depen-dent case, is given here by ∇ (cid:18) ∇ ΨΨ (cid:19) + (cid:18) ∇ ΨΨ (cid:19) + 2 m ~ ( E − V ) = 0 , (43)where the following substitutions must be made ∂∂t ↔ ∇ , (cid:18) ~ m y (cid:19) = ˙ λλ ↔ ∇ ΨΨ , λ = αe iϕ ↔ Ψ = ae i S ~ . Considering first the one-dimensional HO, and introducing the dimensionless variable ζ via ζ = | k | q with ~ k = p = ±√ mE , ˜ V ( ζ ) = V [ q ( ζ )] and ¨ a = d dζ a , equation (42) acquires thefamiliar form¨ a + − ˜ VE ! a = 1 a . (44)A similar formulation of the time-independent SE in terms of this equation, but withina different context and different applications has also been given in [15]. In another paper [16]the relation between the Ermakov equation (44) and the time-independent SE has been extendediccati and Ermakov Equations in Quantum Theory 13to also include magnetic field effects. The nonlinear differential equation (44) has also beenused to obtain numerical solutions of the time-independent SE for single and double-minimumpotentials as well as for complex energy resonance states; for details see [17]. Here we want toconcentrate on the similarities between the time-dependent and time-independent situation, inparticular with respect to SUSY, as mentioned in Sections 5.1 and 5.2.Following the method described in [2], from the real solution a NL ( ζ ) of this nonlinear Ermakovequation (44) the complex solution a L ( ζ ) of the linear SE (40) can be obtained via a L ( ζ ) = a NL ( ζ ) exp (cid:26) − i ~ S (cid:27) = a NL ( ζ ) exp (cid:26) − i Z ζζ dζ ′ a ( ζ ′ ) (cid:27) , (45)from which a real (not normalized) solution of the time-independent SE can be constructedaccording to˜ a L ( ζ ) = ℜ [ a L ( ζ )] = 12 h a NL e i ~ S + a NL e − i ~ S i = a NL cos (cid:18)Z ζζ dζ ′ a ( ζ ′ ) (cid:19) . (46)So far, the energy E occurring in equation (44) is still a free parameter that can take anyvalue. However, solving equation (44) numerically for arbitrary values of E leads, in general,to solutions a NL that diverge for increasing ζ . Only if the energy E is appropriately tuned toany eigenvalue E n of equation (40) does this divergence disappear and the integral in the cosineof equation (46) takes for ζ → ∞ exactly the value π , i.e., the cosine vanishes at infinity. So,the quantization condition that is usually obtained from the requirement of the truncation of aninfinite series in order to avoid divergence of the wave function is, in this case, obtained fromthe requirement of nondiverging solutions of the nonlinear Ermakov equation (44) by variationof the parameter E . This has been numerically verified in the case of the one-dimensional HOand the Coulomb problem and there is the conjecture that this property is “universal” in thesense that it does not depend on the potential V (see [2]).For comparison with the situation in SUSY, the HO and the Coulomb problem can be writtenin the form:a) HO: with µ = (cid:0) ~ ω E (cid:1) , E = E n = ( n + ) ~ ω → µ n = n +1) and µζ = m ω q E = VE follows:¨ a + (cid:0) − µζ (cid:1) a = ¨ a + − ˜ VE ! a = ¨ a − U n E a = 1 a , (47)where U n = m ω q − ~ ω (cid:18) n + 12 (cid:19) = V ( q ) − E n . b) Coulomb problem: with a ( r, ϑ, ϕ ) = R ( r ) Y lm ( ϑ, ϕ ) the radial part can be separated and,with the dimensionless variable ζ = | k | r with now ~ k = p = ± p m ( − E ) ( E < X ( ζ ) = r ( ζ ) X [ r ( ζ )], which corresponds to Ψ nl ( r ) in SUSY.This function obeys, again, an Ermakov equation, namely¨ X + ˜ WE − ! X = ¨ X + U n ′ E X = 1 X , (48)where ˜ W ( ζ ) = ˜ V [ r ( ζ )] + l ( l + 1) ~ mr ( ζ ) ˆ= V eff V eff from SUSY (see equations (35) and (37)) and E = E n = − me ~ n with n = n ′ + l + 1. The coefficient of the term linear in X can again be expressed with thehelp of the potential-like expression U n ′ as U n ′ = − e r + l ( l + 1) ~ mr + mc (cid:18) e ~ c (cid:19) n ′ + l + 1) = V eff − E n ′ . In both cases, the ground state ( n = 0) wave functions are real, nodeless ( n ′ = 0) and thephase does not depend on spatial variables (i.e., ∇ S = 0). Therefore, the rhs of equations (47)and (48) vanishes since a ∝ ( ∇ S ) a (similar for the Coulomb problem), i.e., the nonlinearErmakov equations turn into the usual time-independent SEs. In this case, comparison showsthat for the HO and the Coulomb problem, the potential-like terms U are identical with thecorresponding V of SUSY. For n > n ′ >
0, however, U n and U n ′ are different from V anddescribe higher excited states. In SUSY, these states can only be obtained from the hierarchydescribed in the previous subsection. Here, the price that must be paid to include these excitedstates is the nonlinearity on the rhs of equations (47) and (48).Comparing the situation in this nonlinear formulation of time-independent quantum mechan-ics with SUSY and the time-dependent systems discussed in the first part of this paper, one cansee the following similarities:i) Comparison with SUSY :The real superpotential W = − ~ √ m (cid:16) ∇ Ψ Ψ (cid:17) is replaced by the complex “superpotential” C ( q ) = − ~ √ m h(cid:16) ∇| Ψ || Ψ | (cid:17) + i ∇ S ~ i , i.e., the ground state Ψ is replaced by the absolute value | Ψ | of anyeigenstate and an additional imaginary part depending on the phase S ~ of the wave functionoccurs, being responsible for the non-vanishing rhs of the Ermakov equations (47) and (48).ii) Comparison with time-dependent SE :As mentioned before, the complex quantity C ( q ) can be compared with the time-dependentquantity (cid:0) ~ m y (cid:1) fulfilling the complex Riccati equation (4), C ( q ) = − ~ √ m ∇ ΨΨ ↔ (cid:0) ~ m y ( t ) (cid:1) = ˙ λλ , or,in terms of real and imaginary parts, ∇| Ψ || Ψ | ↔ ˙ αα and ∇ S ~ ∝ | Ψ | ↔ ˙ ϕ ∝ α . As a result of this investigation, one can state that the Ermakov invariant is a quantity of centralimportance that connects different forms for the description of the dynamics of quantum sys-tems, such as the time-dependent SE, the time-dependent Green’s function (or Feynman kernel)and the time-dependent Wigner function. Unlike the classical Hamiltonian or Lagrangian, thisinvariant not only depends on the classical variables such as position and momentum, but also onthe quantum uncertainties of these quantities contained in α and ˙ α . Therefore, the initial valuesof these quantities also play an important role for the time-evolution of the quantum system, ashas been demonstrated in the discussion of the time-dependence of the WP width or positionuncertainty. So, the time-evolution of a typical quantum mechanical property can be totallydescribed if one only knows the classical trajectory η and the classical velocity ˙ η (including theirinitial conditions) plus the initial position uncertainty . This traces quantum dynamics entirelyback to the classical one plus the existence of an uncertainty principle.So far, the discussion of time-dependent systems included only systems where the potentialis at most quadratic in its variables (a similar treatment of the motion in a magnetic field isalso possible; see, e.g., [18]). This might not be as restrictive as it seems at first sight since onemay sometimes perform canonical transformations to reduce a given Hamiltonian to a quadraticform [19] which has been shown explicitly by Sarlet for some polynomial Hamiltonians. To whatextent this method can also be applied in our case requires more detailed studies. Another wayiccati and Ermakov Equations in Quantum Theory 15to extend considerably the class of Hamiltonians for which an exact invariant can be found is bymaking use of generalized canonical transformations [20]. Further generalizations of applyingthe Ermakov-type invariants also in the context of field-atom interactions, systems of coupledoscillators including damping and/or time-dependent masses and attempts of obtaining thecorresponding Wigner functions can be found in [21]. For a further survey of two-dimensionalproblems in this context, see also [22].Similar to the factorization of the Hamiltonian of the HO in terms of (classical) complexnormal modes, or, (quantum mechanical) in terms of creation and annihilation operators b ± ,a factorization of the Ermakov invariant is possible that looks like a complex time-dependentgeneralization of this formalism. The major difference compared with the conventional case isthe replacement of ± iω in front of the spatial variable q in equation (31) by the complex time-dependent quantity (cid:0) ~ m y (cid:1) , which fulfils the Riccati equation (4), or its conjugate complex (cid:0) ~ m y ∗ (cid:1) ,respectively.Another generalization of the creation/annihilation operator formalism of the HO, concerningthe space-dependence instead of the time-dependence, is found in SUSY where the term linearin the spatial coordinate q is replaced by a function of q , the so-called “superpotential” W ( q ).The generalized creation and annihilation operators B ± are obtained by replacing the term ω q in b ± by (up to constants) W ( q ), where W ( q ) fulfils the real Riccati equations (32) and (33).In a nonlinear formulation of time-independent quantum mechanics proposed by Reinisch [2],the amplitude of the wave function fulfils a real nonlinear (space-dependent) Ermakov equation,that can be connected with the time-independent SE written as a complex space-dependentRiccati equation (see (43)) together with a kind of conservation law (see (41)), similar to theconservation of “angular momentum” in the complex plane (see (13)) in the time-dependentcase. In SUSY, the superpotential is initially determined by the amplitude of the ground statewave function (i.e., without any contribution from the phase) and the excited states can beobtained by some hierarchy based thereupon, in the nonlinear formulation of time-independentquantum mechanics,the ground state wave function is replaced by the absolute value of anyeigenstate plus an imaginary part depending on the gradient of the phase of this state. Similarto the time-dependent situation, this looks like a complex generalization where, now, the realsuperpotential W ( q ) is replaced by a complex term that not only depends on the amplitude ofthe wave function, but, due to an additional imaginary contribution, also on (the gradient of)its phase. Further clarification of these facts will be subject of forthcoming studies. Acknowledgements
The author wishes to thank G. Reinisch for numerous encouraging and inspiring discussions.
References [1] Schr¨odinger E., Quantisierung als Eigenwertproblem,
Ann. Phys. (1926), 361–376.[2] Reinisch G., Nonlinear quantum mechanics, Phys. A (1994), 229–252.Reinisch G., Classical position probability distribution in stationary and separable quantum systems,
Phys.Rev. A (1997), 3409–3416.[3] Ermakov V.P., Second-order differential equations, Conditions of complete integrability, Univ. Izv. Kiev (1880), no. 9, 1–25.[4] Milne W.E., The numerical determination of characteristic numbers, Phys. Rev. (1930), 863–867.Pinney E., The nonlinear differential equation y ′′ + p ( x ) y + cy − = 0, Proc. Amer. Math. Soc. (1950), 681.Lewis H.R., Classical and quantum systems with time-dependent harmonic-oscillator-type Hamiltonians, Phys. Rev. Lett. (1967), 510–512. [5] Lutzky M., Noether’s theorem and the time-dependent harmonic oscillator, Phys. Lett. A (1979), 3–4.Ray J.R., Reid J.L., More exact invariants for the time-dependent harmonic oscillator, Phys. Lett. A (1979), 317–318.[6] Malkin I.A., Man’ko V.I., Trifonov D.A., Linear adiabatic invariants and coherent states, J. Math. Phys. (1973), 576–582.Markov M.A. (Editor), Invariants and evolution of nonstationary quantum systems, Proceedings of theLebedev Physical Institute , Vol. 183, Nova Science, New York, 1989.[7] Feynman R.P., Hibbs A.R., Quantum mechanics and path integrals, McGraw-Hill, New York, 1965.[8] Schleich W.P., Quantum optics in phase space, Wiley-VCh, Berlin, Chapter 17, 2001.[9] Schuch D., Moshinsky M., Connection between quantum-mechanical and classical time evolution via a dy-namical invariant,
Phys. Rev. A (2006), 062111, 10 pages.Schuch D., Connection between quantum-mechanical and classical time evolution of certain dissipative sys-tems via a dynamical invariant, J. Math. Phys. (2007), 122701, 19 pages.[10] Wigner E.P., On the quantum correction for thermodynamical equilibrium, Phys. Rev. (1932), 749–759.Hillery M. O’Connell R.F., Scully M.O., Wigner E.P., Distribution functions in physics: fundamentals, Phys.Rep. (1984), 121–167.[11] Schuch D., On the relation between the Wigner function and an exact dynamical invariant,
Phys. Lett. A (2005), 225–231.[12] Lewis H.R., Riesenfeld W.B., An exact quantum theory of the time-dependent harmonic oscillator and ofa charged particle in a time-dependent electromagnetic field,
J. Math. Phys. (1969), 1458–1473.Hartley J.G., Ray J.R., Ermakov systems and quantum-mechanical superposition law, Phys. Rev. A (1981), 2873–2876.[13] Cooper F., Khare A., Sukhatme U., Supersymmetry in Quantum Mechanics, World Scientific, Singapore,2001.Kalka H., Soff G., Supersymmetrie, Teubner, Stuttgart, 1997.[14] Madelung E., Quantentheorie in hydrodynamischer Form, Z. Phys. (1926), 322–326.[15] Lee R.A., Quantum ray equations, J. Phys. A: Math. Gen. (1982), 2761–2774.[16] Kaushal R.S., Quantum analogue of Ermakov systems and the phase of the quantum wave function, Intern.J. Theoret. Phys. (2001), 835–847.[17] Korsch H.J., Laurent H., Milne’s differential equation and numerical solutions of the Schr¨odinger equation.I. Bound-state energies for single- and double minimum potentials, J. Phys. B: At. Mol. Phys. (1981),4213–4230.Korsch H.J., Laurent H. and Mohlenkamp, Milne’s differential equation and numerical solutions of theSchr¨odinger equation. II. Complex energy resonance states, J. Phys. B: At. Mol. Phys. (1982), 1–15.[18] Schuch D., Relations between wave and particle aspects for motion in a magnetic field, in New Challenges inComputational Quantum Chemistry, Editors R. Broer, P.J.C. Aerts and P.S. Bagus, University of Groningen,1994, 255–269.Maamache M. Bounames A., Ferkous N., Comment on “Wave function of a time-dependent harmonicoscillator in a static magnetic field”, Phys. Rev. A (2006), 016101, 3 pages.[19] Ray J.R., Time-dependent invariants with applications in physics, Lett. Nuovo Cim. (1980), 424–428.Sarlet W., Class of Hamiltonians with one degree-of-freedom allowing applications of Kruskal’s asymptotictheory in closed form. II, Ann. Phys. (N.Y.) (1975), 248–261.[20] Lewis H.R., Leach P.G.L., Exact invariants for a class of time-dependent nonlinear Hamiltonian systems, J. Math. Phys. (1982), 165–175.[21] Sebawa Abdalla M., Leach P.G.L., Linear and quadratic invariants for the transformed Tavis–Cummingsmodel, J. Phys. A: Math. Gen. (2003), 12205–12221.Sebawa Abdalla M., Leach P.G.L., Wigner functions for time-dependent coupled linear oscillators via linearand quadratic invariant processes, J. Phys. A: Math. Gen.38