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Physica Scripta

Revival, Mirror Revival and Collapse may Occur even in a Harmonic Oscillator Wave Packet

Kalbe Razi Naqvi, S. Waldenstrøm

Published Jul 1, 2000• 11 citations
Publisher

Abstract

If the probability density P ( x , t ) = |ψ ( x , t )| 2 of a non-stationary state happens to meet the condition P ( x , t 0 ) = P ( x , t 0 + Υ), the wave packet is said to experience exact revivals at intervals of Υ. The definition implies that a harmonic oscillator wave packet will experience revivals with a period Υ = 2π/ω, where ω is the angular frequency of the corresponding classical system. A harmonic oscillator wave packet of Gaussian form, centered initially ( t = 0) at x = x 0 , is shown to spread almost everywhere, if its initial width is sufficiently narrow, in a time very much shorter than the period of the corresponding classical oscillator. However, the packet reassembles itself twice during a single period: (i) it experiences a mirror revival, defined by the relation P ( x , ½Υ) = P (- x , 0), and (ii) an exact revival, P ( x , Υ) = P ( x , 0). These results are used to counter a recent claim (which apparently gives voice to a commonly held belief) that a harmonic oscillator wave packet ψ( x , t ) does not collapse and has no revivals. Plots of the recurrence probability |⟨ψ( x , 0)|ψ( x , t )⟩| 2 are presented and the need for formulating unambiguous definitions of the two concepts, revival and collapse, is stressed.

DOI
10.1238/physica.regular.062a00012

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If the probability density P ( x , t ) = |ψ ( x , t )| 2 of a non-stationary state happens to meet the condition P ( x , t 0 ) = P ( x , t 0 + Υ), the wave packet is said to experience exact revivals at intervals of Υ. The definition implies that a harmonic oscillator wave packet will experience revivals with a period Υ = 2π/ω, where ω is the angular frequency of the corresponding classical system. A harmonic oscillator wave packet of Gaussian form, centered initially ( t = 0) at x = x 0 , is shown to spread almost everywhere, if its initial width is sufficiently narrow, in a time very much shorter than the period of the corresponding classical oscillator. However, the packet reassembles itself twice during a single period: (i) it experiences a mirror revival, defined by the relation P ( x , ½Υ) = P (- x , 0), and (ii) an exact revival, P ( x , Υ) = P ( x , 0). These results are used to counter a recent claim (which apparently gives voice to a commonly held belief) that a harmonic oscillator wave packet ψ( x , t ) does not collapse and has no revivals. Plots of the recurrence probability |⟨ψ( x , 0)|ψ( x , t )⟩| 2 are presented and the need for formulating unambiguous definitions of the two concepts, revival and collapse, is stressed.
We investigate the quantum and classical wave packet dynamics in an harmonic oscillator that is perturbed by a disorder potential. This perturbation causes the dispersion of a Gaussian wave packet, which is reflected in the coordinate-space and the momentum-space Shannon entropies, the latter being a measure for the amount of information available on a system. Regarding the sum of the two quantities, one arrives at an entropy that is related to the coordinate-momentum uncertainty. Whereas in the harmonic case, this entropy is strictly periodic and can be evaluated analytically, this behavior is lost if disorder is added. There, at selected times, the quantum mechanical probability density resembles that of a classical oscillator distribution function, and the entropy assumes larger values. However, at later times and dependent on the degree of disorder and the chosen initial conditions, quantum mechanical revivals occur. Then, the observed effects are reversed, and the entropy may decrease close to its initial value. This effect cannot be found classically.
The problem investigated refers to periodically driven 1D quantum inverted harmonic oscillator (IHO) with the Hamiltonian of . The model is used widely in huge quantum applications concerned unstable molecular complexes and ions under laser light affection. Non-stationary Schrödinger equation (NSE) was solved analytically and numerically by means of Maple 17 with initial wave function (w.f.) of generalized Gaussian type. This one described the converging 1D probability flux and fitted well the quantum operator of initial conditions (IC). For the IC one can observe, first, the collapse of w.f. packet into extremely narrow 1D space interval of length and, second, its spreading back up to its starting half width, and all that - at dimensionless times. At certain phases j defined by W and s0 the wave packet center displayed nonharmonic oscillating behavior near some slowly drifting space position within this time interval and after that leaved onto infinity while the unlimited packet spreading. And the phases themselves served as bifurcation points separating the NSE solutions with the outgoing to from those with. In “resonant” case of the values obeyed an inverted Fermi-Dirac formula of; for differing the asymptotic of obeyed well classical law.
The temporal evolution of a wave packet ψ( x,t ) confined to an infinite square well (occupying the region 0 ≤ x ≤ L ), as manifested by fractional revivals of the probability density P ( x,t ) ≡ | ψ( x,t ) | 2 and the time-dependence of the expectation values of position, momentum and force ( X ( t ) ≡ ⟨ x ⟩ t , Π( t ) ≡ ⟨ ⟩ t and Φ( t ) ≡ ⟨ ⟩ t , respectively) are studied, using wave packets for which Π(0) = 0. In the region accessible to the wave packet (0 < x < L ), and the Hamiltonian Ĥ commute; still, is, in general, not a constant of the motion, and the three expectation values are all time-dependent, except when the initial packet has a definite parity, that is ψ( x ,0) = ±ψ( L - x , 0), which implies X ( t ) = L /2 and Π( t ) = 0 = Φ( t ). For a packet with no definite parity, Y ( t ) ≡ X ( t ) - L /2 stays, if the packet is sufficiently narrow at t = 0 close to zero nearly all the time, but executes large excursions in the neighbourhood of instants at which the probability density experiences a mirror revival or a revival. This behaviour, with no classical counterpart, can be understood by paying attention to the presence of infinite potential barriers (at x = 0 and x = L ) and taking account of the net force felt by an initially off-centred wave packet. It is shown that though plots (against time) of expectation values catch a glimpse of the revival structure of a wave packet, such plots fail to reveal some fractional revivals.
By using a test-function method, we construct nnn exact solutions of a quantum harmonic oscillator with a time-dependent "spring constant". Any nnn-th solution describes a wave-packet train consisting of n+1n+1n+1 packets. Its center oscillates like the classically harmonic oscillator with variable frequency, and width and highness of each packet change simultaneously. When the deformation is small, it behaves like a soliton train, and the large deformation is identified with collapse and revival of the wave-packet train.
We investigate the short-, medium-, and long-term time dependence of wave packets in the infinite square well. In addition to emphasizing the appearance of wave packet revivals, i.e., situations where a spreading wave packet reforms with close to its initial shape and width, we also examine in detail the approach to the collapsed phase where the position-space probability density is almost uniformly spread over the well. We focus on visualizing these phenomena in both position- and momentum-space as well as by following the time-dependent expectation values of and uncertainties in position and momentum. We discuss the time scales for wave packet collapse, using both an autocorrelation function analysis as well as focusing on expectation values, and find two relevant time scales which describe different aspects of the decay phase. In an Appendix, we briefly discuss wave packet revival and collapse in a more general, one-dimensional power-law potential given by V(k)(x)=V0|x/a|k which interpolates between the case of the harmonic oscillator (k=2) and the infinite well (k=∞).
Fractional revivals of wave packets in an infinite square well are scrutinized with a viewpoint rooted in Fourier analysis, and a compact relation, expressing the wavefunction Ψ( x , t ) for certain values of t in terms of spatially displaced copies of Ψ( x ,0), is derived without appealing to a classical analogy ; conditions for the appearance of an interference pattern in a plot of the position probability density of a wave packet are deduced, along with related results obtained previously from different considerations, with minimal effort. The specific case of a packet of Gaussian shape is analysed in greater detail to provide concrete illustrations of fractional revivals with or without interference between overlapping copies of wave packets, depending on whether a finite value of the so-called classical time can or cannot be assigned to the wave packet.