We describe the time evolution of a wave function in the infinite square well using a fractional revival formalism, and show that at all times the wave function can be described as a superposition of translated copies of the initial wave function. Using the model of a wave form propagating on a dispersionless string from classical mechanics to describe these translations, we connect the reflection symmetry of the square-well potential to a reflection symmetry in the locations of these translated copies, and show that they occur in a ``parity-conserving'' form. The relative phases of the translated copies are shown to depend quadratically on the translation distance along the classical path. We conclude that the time-evolved wave function in the infinite square well can be described in terms of translations of the initial wave-function shape, without approximation and without any reference to its energy eigenstate expansion. That is, the set of translated initial wave functions forms a Hilbert space basis for the time-evolved wave functions.
We describe the time evolution of a wave function in the infinite square well using a fractional revival formalism, and show that at all times the wave function can be described as a superposition of translated copies of the initial wave function. Using the model of a wave form propagating on a dispersionless string from classical mechanics to describe these translations, we connect the reflection symmetry of the square-well potential to a reflection symmetry in the locations of these translated copies, and show that they occur in a ``parity-conserving'' form. The relative phases of the translated copies are shown to depend quadratically on the translation distance along the classical path. We conclude that the time-evolved wave function in the infinite square well can be described in terms of translations of the initial wave-function shape, without approximation and without any reference to its energy eigenstate expansion. That is, the set of translated initial wave functions forms a Hilbert space basis for the time-evolved wave functions.
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.
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.
A particle of mass M moves in an infinite square well of width L (the “particle in a box”). Classically, the motion has period L2M/E, which depends on the initial condition through the energy E. Quantum mechanically, any wave function repeats exactly with period 4ML2/πℏ, independent of the initial condition. Given this qualitative difference, how can the classical motion possibly be the limit of the quantal time development? The resolution of this paradox involves the difference between the exact revival (recurrence) of the wave function and the approximate periodicity of expectation values such as 〈x(t)〉. (The latter may recur an odd integral number of times before the full wave function recurs.) The period of the expectation values does depend on the initial condition and can possess the expected classical limit. [An Appendix demonstrates that, under suitably quasiclassical conditions, the quantal time evolution of 〈x(t)〉 passes over to the classical result not only in period, but also in its exact functional form. Another Appendix proves four theorems concerning state-dependent exact revival times.]
In the infinite square well, any wave function will return to its initial state at integer multiples of the revival time. Most quantum systems do not exhibit perfect revivals, but some exhibit partial revivals in which the wave function returns close to its initial state. Subsequent partial revivals usually deteriorate in quality. We discuss the reasons for the perfect revivals in the infinite square well and how a small change in the potential disrupts the revivals. As an example, we examine partial revivals of a Gaussian wave packet in an infinite square well with a step. First-order and second-order perturbation theory show that the rate at which revivals decay depends on the location of the step.
The time evolution of quantum mechanical states in a square well of infinite depth with a Dirac δ function at its center has been examined for cases where the initial state was localized in one of the wells. Let well A denote the well in which the initial state is localized, and let P(t) denote the integrated probability density Ψ*Ψ in well A. For opaque barriers the time-dependent system is adequately described by a two-state model in which only the pair of stationary states of even and odd parity are considered whose wave functions in well A, apart from an arbitrary phase factor, are largely identical with the wave function of the initial state. For P(t) a harmonic oscillation is observed whose frequency νt is well approximated by the well-known formula for the tunneling frequency νt≂ΔE/h, where ΔE represents the energy separation among the pair of states in the model. For the present model of a symmetric double well it has been shown that for highly transparent barriers a three-state model can describe the time evolution adequately. The three stationary states involved in this model are a state of odd parity whose wave function in well A is largely the same as the wave function of the initial state and a pair of stationary states of even parity which on the energy scale are immediately above and below the first state. In this three-state model the function P(t) is a superposition of two sinusoidal functions with nearly identical amplitudes and frequencies plus a constant. As a consequence, the amplitude of P(t) changes harmonically. In the present model a δ function has been used as a barrier in order to minimize the mathematical detail involved in the time-dependent treatment. It is to be expected that the beating in P(t) can be observed also in the time evolution of a state localized in one of the wells of a symmetric double-minimum potential with a more realistic ‘‘low’’ barrier if the density of energy levels near the energies of the levels to be considered in the corresponding three-state model varies slowly with energy.
We present an analytical investigation of revival phenomena in the finite square-well potential. The classical motion, revival, and super-revival time scales are derived exactly for wave packets excited in the finite well. These time scales exhibit a richer dependence on wave-packet energy and on potential-well depth than has been found in other quantum systems: They explain, for example, the difficulties in exciting wave packets with strong classical features at the bottom of a finite well, or with clearly resolved super-revivals in a shallow well. In the proper regions of validity, the time scales predict the instances of wave-packet reformation extremely accurately. Revivals at the bottom of the well are explored as a ``universal'' limit of the general theory, which offers the clearest connection with the series of fractional and full revivals seen in the dynamics of the infinite square-well potential.