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Evolution of quantum systems with a scaling type of time-dependent Hamiltonians


Evolution of quantum systems with a scaling type
arXiv:quant-ph/0212074v1 12 Dec 2002

of time-dependent Hamiltonians
ˇ L. SAMAJ

Institute of Physics, Slovak Academy of Sciences, D? ubravsk? a cesta 9, 842 28 Bratislava, Slovakia; tel. +421 2 5941 0522, fax +421 2 5477 6085 e-mail: fyzimaes@savba.sk February 1, 2008

Abstract We introduce a new class of quantum models with time-dependent Hamiltonians of a special scaling form. By using a couple of time-dependent unitary transformations, the time evolution of these models is expressed in terms of related systems with time-independent Hamiltonians. The mapping of dynamics can be performed in any dimension, for an arbitrary number of interacting particles and for any type of the scaling interaction potential. The exact solvability of a “dual” time-independent Hamiltonian automatically means the exact solvability of the original problem with model time-dependence.

PACS numbers: 03.65.-w, 03.65.Fd, 03.65.Ge, 02.30.Tb Keywords: Quantum motion, time-dependent Hamiltonian, wave functions, unitary transformations, exactly solvable models. 1

Dynamics of quantum systems, governed by time-dependent Hamiltonians, has attracted much of attention for a long time. A relevant progress has been made mainly in the study of one-dimensional time-dependent harmonic oscillators1?10 which have many applications in various areas of physics (see e.g. Refs. 11-13). Lewis1 and Lewis and Riesenfeld2 (LR) have introduced for these systems a quantum-mechanical LR invariant, and derived a relation between the invariant eigenstates and exact solutions of the corresponding time-dependent Schr¨ odinger equation. Using the LR invariant method, exact wavefunctions have been obtained for harmonic oscillators with time-dependent frequency,3?5 time-dependent mass and frequency,6,7 and linear driving terms.8 At present, the exact solution can be, in principle, constructed for a general one-dimensional timedependent Hamiltonian of N coupled quantum oscillators.9,10 The addition of a singular inverse quadratic potential does not break the exact solvability of the time-dependent harmonic oscillator, as has been shown by combining the LR invariant method with a unitary transformation in Refs. 14-17. In this paper, we present a new vast array of quantum models with time-dependent Hamiltonians of a special scaling form. By using a couple of time-dependent unitary transformations, time evolution of these models is expressed in terms of related systems with time-independent Hamiltonians. The mapping of dynamics can be performed in any dimension, for an arbitrary number of interacting particles and for any type of the interaction potential. The exact solvability of the time-independent Hamiltonian automatically means the exact solvability of the original problem with model time-dependence; otherwise, dynamics induced by the time-dependent Hamiltonian can be studied by simpler techniques known for time-independent Hamiltonians. We ?rst restrict ourselves to the case of one particle situated in a d-dimensional space of points r = (x1 , x2 , . . . , xd ). The proposed time-dependent Hamiltonian has the following scaling form:
′ f (0) ?2 ? (t) = f (t) f (t) p + V H f ′ (0) f (0) 2m f (t)

f (0) r f (t)

,

(1)

where p ? = ?i? h?r is the momentum operator, V is an arbitrary one-body potential and 2

the prime denotes the di?erentiation with respect to the argument. The normalization of parameters was chosen such that at initial time t = 0 the Hamiltonian takes the standard form p ?2 ? H(t = 0) = + V (r ). 2m (2)

The function f (t) is assumed to be real continuous function of time in order to ensure the hermiticity of the Hamiltonian. To simplify the notation, without any lost of generality we shall assume that f (0) = 1, f ′ (0) = ? (? real and nonzero) (3)

and rewrite the Hamiltonian (1) as follows
′ ?2 1 ? (t) = f (t) f (t) p + V H ? 2m f (t)

r f (t)

.

(1′ )

Two choices of the function f (t) are of special interest. The ?rst one, f (t) = (1 + 2?t)1/2 (4)

with ? > 0 in order to avoid singularities in time, corresponds to the constant particle mass. The corresponding Hamiltonian (1) takes the form ?2 1 ? (t) = p H + V 2m 1 + 2?t r . (1 + 2?t)1/2 (5)

The classical version of this Hamiltonian has been introduced in Ref. 18 in connection with the study of adiabatic propagation of distributions, with ? taken as the adiabatic slowness parameter. The second choice is f (t) = exp(?t), and the corresponding Hamiltonian reads ?2 ? (t) = e2?t p + V e??t r . H 2m (7) (6)

In one dimension and when V (x) = mω 2 x2 /2, the model (7) is known as the CaldirolaKanai oscillator.19?21 Its exact quantum states are known, even in a more general case with the singular inverse square potential16 , V (x) = mω 2 x2 /2 + k/x2 . 3

The evolution of the system with Hamiltonian (1′ ) is governed by the time-dependent Schr¨ odinger equation i? h ? ? (t)ψ (r, t), ψ (r, t) = H ?t (8)

with the initial condition for the wavefunction ψ (r, t = 0) = ψ (r). (9)

In the next two paragraphs, we apply successively two time-dependent unitary transformations, ? (t), ψ (r, t)} =? {H ? 1 , ψ1 (r, t)} =? {H ? 2 , ψ2 (r, t)}, {H ? (t) to the new time-independent to go from the original time-dependent Hamiltonian H ? 2 , via an intermediate one H ? 1. Hamiltonian H We start with the unitary transformation t′ = x′j It is easy to verify that it holds f ′ (t) 1 ? ? = ? r′ · ?r′ , ?t f (t) ? ?t′ ?2 1 ?2 , j = 1, 2, . . . , d. = ?x2 [f (t)]2 ?x′j 2 j The transformation (10) converts the evolution equation (8) to the new one i? h ? ? 1 ψ1 (r, t), ψ1 (r, t) = H ?t ?2 ?1 = p ? ?r · p ? + V (r ). H 2m (12a) (12b) (11a) (11b) 1 ln f (t), ? xj = , j = 1, 2, . . . , d. f (t) (10a) (10b)

The original wavefunction ψ in (8) is expressible in terms of ψ1 as follows ψ (r, t) = ψ1 and the initial condition (9) now reads ψ1 (r, t = 0) = ψ (r). 4 (14) r 1 , ln f (t) , f (t) ? (13)

? 1 in (12) can be reexpressed as follows The time-independent Hamiltonian H ?1 = H 1 1 (p ? ? m?r)2 + ? (p ? ·? r ?? r·p ?) 2m 2 1 +V (r) ? m?2 r 2 . 2

(15)

Since p ? ·? r=? r·p ? ? i? hd, the evolution Eq. (12) can be written as i? h 1 d? ? ψ1 (r, t) = + (p ? ? m?r)2 ψ1 (r, t) ?t 2 2m 1 + V (r) ? m?2 r 2 ψ1 (r, t). 2 (16)

(17)

Using the unitary transformation ψ1 (r, t) = exp ? we ?nally arrive at i? h ? ? 2 ψ2 (r, t), ψ2 (r, t) = H ?t ?2 1 ?2 = p H + V (r) ? m?2 r 2 . 2m 2 (19a) (19b) d? im? 2 t+ r ψ2 (r, t) 2 2? h (18)

The initial condition for ψ2 is implied by Eqs. (14) and (18) as follows ψ2 (r, t = 0) = ψ (r) exp ? im? 2 r . 2? h (20)

The original wavefunction ψ is obtained in terms of ψ2 by combining relations (13) and (18), ψ (r, t) = r 1 im? r 1 ? ψ2 exp ? , ln f (t) . d/ 2 [f (t)] 2? h f (t) f (t) ?
?
2

?

(21)

Using the above mapping of dynamics, standard techniques able to deal with timeindependent Hamiltonians (see e.g. Ref. 22) can be directly applied to time-dependent Hamiltonians. The mapping becomes even more appealing when the resulting timeindependent problem with potential 1 V (r) ? m?2 r 2 2 5

is exactly solvable, because this automatically means the exact solvability of the original model time-dependence. It is clear that V (r) must be expressible as an integrable potential plus the harmonic-oscillator potential m?2 r 2 /2. In one dimension, there exists an in?nite chain of exactly solvable “re?ectionless potentials”.23,24 In higher dimensions, there are standard models which admit the exact solution.22 We brie?y discuss the simplest example – the isotropic d-dimensional harmonic oscillator with V (r) = mω 2 r 2 /2. For the special choices (4) and (6) of the function f (t), the respective time-dependent Hamiltonians ?2 1 ω ? (t) = p H + m 2m 2 1 + 2?t and 1 ?2 ? (t) = e2?t p + e?2?t mω 2 r 2 H 2m 2 are transformed to the same time-independent Hamiltonian 1 ?2 ?2 = p + m(ω 2 ? ?2 )r 2 . H 2m 2 (24) (23)
2

r2

(22)

The time evolution associated with this Hamiltonian can be obtained by using the method of the separation of variables. Without going into details, there are three regimes: if ω > ? one expands the wavefunction ψ2 (and, consequently, ψ ) in Hermite polynomials, if ω = ? in plane waves and if ω < ? in hypergeometric functions. The extension of the formalism to the case of an arbitrary number of particles is straightforward and we only write down ?nal formulae. For N particles, the Hamiltonian (1′ ) generalizes to
N ′ p ?2 1 ? (t) = f (t) ?f (t) + H V ? f (t) j =1 2mj

?

rj f (t)

?

?,

(25)

where f (0) = 1 and f ′ (0) = ?, and the interaction function V [with particle coordinates uniformly scaled by 1/f (t)] involves all possible one-, two-, . . ., N -body potentials. The solution of the time-dependent Schr¨ odinger equation i? h ? ? (t)ψ ({rj }, t) ψ ({rj }, t) = H ?t 6 (26)

with the initial condition ψ ({rj }, t = 0) = ψ ({rj }) reads 1 i? N rj ? ψ2 ? ψ ({rj }, t) = exp mj N d/ 2 [f (t)] 2? h j =1 f (t) i? h
?
2

(27)

?

1 rj , ln f (t) . f (t) ?

(28)

Here, the wavefunction ψ2 evolves under the action of a time-independent Hamiltonian, ? ? 2 ψ2 ({rj }, t), ψ2 ({rj }, t) = H ?t ? ? N N p ?2 1 j 2 2 ?2 = H + ? V (r ) ? mj ? rj ? , 2 m 2 j j =1 j =1 i? N 2? mj rj . ψ2 ({rj }, t = 0) = ψ ({rj }) exp ?? 2? h j =1
? ?

(29a) (29b)

with the initial condition

(30)

As the simplest exactly solvable example, we mention a system of N coupled d-dimensional harmonic oscillators, which is solvable in the transformed picture with the time-independent Hamiltonian by the ordinary normal-mode technique. The couple of time-dependent unitary transformations presented in this paper can be applied directly to the quantum Liouville equation for the density matrix, extending in this way the treatment also to the propagation of distributions. In conclusion, we believe that the results presented will enable one to answer some problems concerning the evolution of quantum systems under a time-dependent Hamiltonian, for instance, to predict a qualitative change in the analytic structure of leading corrections to the ideal adiabaticity when going from a few degrees of freedom to a dissipative system with in?nite degrees of freedom. For classical adiabatic processes in one dimension, the corresponding exact analysis has been done in Ref. 17.

Acknowledgements: I am grateful to Prof. J. K. Percus for stimulating discussions. This work was supported by Grant VEGA 2/7174/20. 7

References
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[19] R.W. Hasse, J. Math. Phys. 16, 2005 (1975). [20] V.V. Dodonov and V.J. Mon’Ko, Nuovo Cimento B44, 265 (1978). [21] A.D. Jannussis, G.N. Brodimas and A. Streclas, Phys. Lett. A74, 6 (1979). [22] C. Cohen-Tannoudji, B. Diu and F. Lalo¨ e, M? ecanique Quantique I, II (Hermann, Paris, 1980). [23] A. Shabat, Inverse Prob. 8, 303 (1992). [24] D.T. Barclay, R. Dutt, A. Gangopadhyaya, A. Khare, A. Pagnamenta and U. Sukhatme, Phys. Rev. A48 2786 (1993).

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