Semiclassical wavefunctions and Schrodinger equation

Michael Berry

Francesco R. Ruggeri

The quantum oscillator ground state energy may be obtained by solving the time-independent Schrodinger equation and selecting the lowest energy eigenvalue. In (1) it is suggested that ground state oscillator energy appears in an semiclassical calculation of oscillator energy using the Moyal product (**) i.e. (p+iwx) ** (p-iwx) .5 = pp/2 + wwxx/2 + hbar w/2 for m=1. In this note, we try to show that the Moyal product: (p+iwx)/sqrt(2){ d/dx (left)d/dp (right) - d/dp (left) d/dx (right) } (p-iw)/sqrt(2) follows from the fact that Wo(x) and ao(p) where Wo is the ground state wavefunction and ao the ground state momentum wavefunction have the same functional form and that this form is in fact a Gaussian so that Wo(x)ao(p) = exp(-constant( pp/2 + wwxx/2). By using both, one may create a combined time-independent Schrodinger equation with both p and x present at the same time giving the appearance of a semiclassical approach. With respect to the ground state energy, d/dx d/dx acting on a Gaussian exp(-axx-bpp) first brings down 2ax followed by 2a from the second derivative (and similar results for d/dp d/dp). The other part of the second derivative is x dependent and cancels the x dependent potential. Thus one may think in terms of a single derivative d/dx acting on ax or a single derivative acting on bp. Combining in a symmetric form leads to the Moyal product acting on p+iwx and p-iwx to yield the ground state oscillator energy. We argue that this is essentially the same information as the time-independent Schrodinger equation with the special Gaussian form of the ground state wavefunction. Thus derivatives essentially act on pp/2 + wwxx/2 in a product of the x-wavefunction and p-wavefunction and this may be converted into a scheme of single derivative acting on the variable p+iwx and p-iwx because the relative piece of the d/dx d/dx of the Schrodinger equation is: d/dx d/dx exp(-axx) →d/dx 2ax exp(-axx) → 2a (linear in a) with the other term of the second derivative canceling with .5kxx exp(-axx).

Vasile Catrinel Grădinaru, Oliver Rietmann

The semiclassical (or Hagedorn) wavepackets depending on a fixed set of parameters are an orthonormal L 2 -basis of generalized coherent states. They have been used to solve numerically the time-dependent Schrödinger equation in its semiclassical formulation, yet their localization property makes them inefficient in case of non-local phenomena such as quantum tunneling. In order to overcome this difficulty, we use simultaneously several bases with different parameters. We propose an algorithm to expand a given wavefunction in terms of multiple families of Hagedorn wavepackets; each family can then be accurately and efficiently propagated using modern semiclassical time-splittings.

Bogdan M. Mihalcea

"We investigate quantum dynamics for an ion confined within an oscillating quadrupole field, starting from two well known and elegant approaches. It is established that the Hamilton equations of motion, in both Schrodinger and Heisenberg representations, are equivalent to the Hill equation. One searches for a linear independent solution associated to a harmonic oscillator (HO). An adiabatic invariant, which is also a constant of motion, is introduced based on the Heisenberg representation. Thus, the state of the non-autonomous system can be determined at any subsequent moment of time. The quantum states for trapped ions are demonstrated to be Fock (number) states, while the exact solutions of the Schrodinger equation for a trapped ion are exactly the quasienergy states. Semiclassical dynamics is also investigated for many-body systems of trapped ions, where the wavefunction associated to the Schrodinger equation is prepared as a Gauss package multiplied by a Hermite polynomial. We also discuss time evolution for the system under investigation and supply the propagator."

Franz Schwabl