The theory of explicitly time-dependent invariants is developed for quantum systems whose Hamiltonians are explicitly time dependent. The central feature of the discussion is the derivation of a simple relation between eigenstates of such an invariant and solutions of the Schrödinger equation. As a specific well-posed application of the general theory, the case of a general Hamiltonian which settles into constant operators in the sufficiently remote past and future is treated and, in particular, the transition amplitude connecting any initial state in the remote past to any final state in the remote future is calculated in terms of eigenstates of the invariant. Two special physical systems are treated in detail: an arbitrarily time-dependent harmonic oscillator and a charged particle moving in the classical, axially symmetric electromagnetic field consisting of an arbitrarily time-dependent, uniform magnetic field, the associated induced electric field, and the electric field due to an arbitrarily time-dependent uniform charge distribution. A class of explicitly time-dependent invariants is derived for both of these systems, and the eigenvalues and eigenstates of the invariants are calculated explicitly by operator methods. The explicit connection between these eigenstates and solutions of the Schrödinger equation is also calculated. The results for the oscillator are used to obtain explicit formulas for the transition amplitude. The usual sudden and adiabatic approximations are deduced as limiting cases of the exact formulas.
A formulation of the energy eigenvalue problem for a many-particle system is presented in several alternative forms and, following Brueckner and his collaborators, the e»ergy of the system is expressed in terms of solutions to the two-body problem. This work bears considerable resemblance to Brueckner's "linked cluster" expansion. The present expansion is derived with the aid of an ordering operator and involves a sequence of "nearest neighbor" interactions. The particles tend to form "chains," the sequence of approximations involving "two-particle chains," "three-particle chains," etc. The rapidity of convergence of the method depends on the particle density and "temperature" and on the nature of the force between particles. The method lends itself readily to use in statistical mechanics. Some applications to the calculation of the equation of state of a gas are included.
A diagnostic experiment has been carried out on the d-d reactions produced by fast magnetic compression of a deuterium plasma. A determination of the velocity spectra of protons and of tritons from the d-d reaction was made by magnetic analysis and nuclear plate detection of the particles. The observed distributions are Gaussian, with widths which correspond to a deuteron temperature of 1.3 kev. Comparison of the mean proton and triton momenta indicates that no plasma drift in the (axial) direction of observation is present, nor any potential difference between the source plasma and detector greater than a few volts. These results, coupled with previous ones on the neutron yield, duration, source extent, and lack of circumferential drift argue against any of the simple, physically plausible non-Maxwellian acceleration mechanisms for the d-d reactions so far proposed.
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