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In this paper we develop a new approach to semiclassical dynamics which exploits the fact that extended wavefunctions for heavy particles (or particles in harmonic potentials) may be decomposed into time-dependent wave packets, which spread minimally and which execute classical or nearly classical trajectories. A Gaussian form for the wave packets is assumed and equations of motion are derived for the parameters characterizing the Gaussians. If the potential (which may be nonseparable in many coordinates) is expanded in a Taylor series about the instantaneous center of the (many-particle) wave packet, and up to quadratic terms are kept, we find the classical parameters of the wave packet (positions, momenta) obey Hamilton's equation of motion. Quantum parameters (wave packet spread, phase factor, correlation terms, etc.) obey similar first order quantum equations. The center of the wave packet is shown to acquire a phase equal to the action integral along the classical path. State-specific quantum information is obtained from the wave packet trajectories by use of the superposition principle and projection techniques. Successful numerical application is made to the collinear He + H, system widely used as a test case. Classically forbidden transitions are accounted for and obtained in the same manner as the classically allowed transitions; turning points present no difficulties and flux is very nearly conserved.

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Images of electron flow from the quantum point contact (QPC) are obtained by raster scanning a negatively charged SPM tip above the surface of the device and simultaneously measuring the position-dependent conductance of the device. The negatively charged tip capacitively couples to the 2DEG, creating a depletion region that backscatters electron waves. When the tip is positioned over areas with high electron flow from the QPC the conductance is decreased, whereas when the tip is over areas of relatively low electron flow the conductance is unmodified. By raster scanning the tip over the sample and simultaneously recording the effect the tip has on device conductance, a two dimensional image of electron flow can be obtained.The quantum point contact sample is mounted in an atomic force microscope and cooled to liquid He temperatures. The QPC is formed in the 2DEG inside a GaAs/AlGaAs heterostructure by negatively biasing two gates on the surface -a negative potential on these gates creates two depletion regions that define a variable width channel between them as shown in Fig. 1a. The conductance of the QPC is measured using an ac lock-in amplifier at 11kHz. The heterostructure for the devices used in this experiment was grown by molecular beam epitaxy on an n-type GaAs substrate.The 2DEG resides 57 nm below the surface with mobility µ = 1.0x10 6 cm 2 /Vs and density n = 4.5x10 11 /cm 2 . These values of mobility and density correspond to a mean free path l = 11 µm, Fermi wavelength λ F = 37 nm, and Fermi energy E F = 16 meV. The root mean square voltage across the QPC was chosen so as to not heat electrons -0.2 mV for 1.7K scans. The conductance of the quantum point contact, shown in Fig. 1b, increases as the width of the channel is increased (by changing the gate voltage V g ) and shows well defined conductance plateaus at integer multiples of the conductance quantum 2e 2 /h 1,2 . When probing the electron flow, the SPM tip was held at a negative potential relative to the 2DEG and was scanned at 10nm above the surface of the heterostructure. Figures 2a and 2b show images of electron flow from two different quantum point contacts at the temperature 1.7K; both QPCs are biased on the G = 2e 2 /h conductance plateau. Figure 2b shows the flow patterns on each side of a quantum point contact (the gated region at the center was not scanned), and Figure 2a shows a higher-resolution image of flow from one side of a different QPC.In both these images, the current exits the point contact in a central lobe, as would be expected from an exact quantum-mechanical calculation of electron flow through an ideal QPC without impurities or non-uniform distributions of dopant atoms. Rather than continuing out as a smoothly widening fan, it quickly forks into several different paths and continues to branch off into ever smaller rivulets for the full width of the scan. This branching behavior was observed in all of the 13 QPC exit patterns observed so far. Previously, there have been suggestions of an unexpected narrowness in observe...

A time-dependent picture of vibrational Raman scattering in the weak field limit is presented. From this viewpoint we can separate the static effects, due to the coordinate dependence of the electronic transition dipole, from the dynamic effects that arise from wave packet propagation on the Born–Oppenheimer surfaces. Away from resonance, the energy uncertainty relation gives the propagation time necessary to obtain the cross section as being inversely proportional to the mismatch of the excitation frequency with the excited surface. The wave packet, given by the initial vibrational wave function times the transition dipole, hardly moves on the excited surface when the excitation frequency is far from resonance. As the excitation frequency is tuned closer to resonance, the propagated wave packet samples a larger portion of the surface. Using the short time approximation to the propagator, we obtain formulas for the cross section that are applicable for Raman scattering by polyatomics. The short time approximation is expected to be good away from resonance independent of the nature of the surface, and also on resonance with a repulsive surface. For an attractive surface, the approximation gives the average resonant cross section useful in the case when the vibrational structures cannot be observed.