An experiment has been completed which demonstrated quantum mechanical tunneling of electrons between two gold electrodes separated in vacuum. The tunneling current between the gold electrodes has been measured, for fixed voltages of 0.1 and 0.01 volts, as the electrode spacing was varied from a distance of approximately 2.0 nm down to a point where the electrodes touched. Current-voltage characteristics for fixed electrode spacing in the direct tunneling region have also been measured. Numerical calculations of the tunneling current based on the free-electron model of the electrodes and the barrier, an image-potential reduced barrier, and a WKB approximation for the tunneling probability have been performed and compared with Simmons' theory and with the experimental results. Within experimental error the results indicate that an image potential reduced barrier with the modifications suggested by Lang and Kohn gives a close approximation to the true barrier for metal-vacuum-metal tunneling. For the first time, the work function of the electrodes in a tunneling experiment has been deduced from experimental parameters independent of the tunneling device.

The Bogoliubov theory of excitations in superfluid helium is used to study collective modes at zero temperature. A repulsive delta function shell potential is used in the quasiparticle excitation energy spectrum to fit the observed elementary excitation spectrum, except in the plateau region. The linearized equation of motion method is used to obtain the secular equation for a collective mode consisting of a linear combination of one and two free quasiparticles of zero total momentum. It is shown that in this case for high-lying collective modes, vertices involving three quasiparticles cancel, and only vertices involving four quasiparticles are important. A decomposition into various angular momentum states is then made. Bound roton pairs in the angular momentum D-state observed in light-scattering experiments exist only for an attractive coupling between helium atoms in this oversimplified model. Thus, the interaction between particles can be reinterpreted as a phenomenological attractive coupling between quasiparticles, in order to explain the Raman scattering from bound roton pairs in superfluid helium.

The gauge-invariant formulation of quantum mechanics is compared to the conventional approach for the case of a one-dimensional charged harmonic oscillator in an electromagnetic field in the electric dipole approximation. The probability of finding the oscillator in the ground state or excited states as a function of time is calculated, and the two approaches give different results. On the basis of gauge invariance, the gauge-invariant formulation of quantum mechanics gives the correct probability, while the conventional approach is incorrect for this problem. Therefore, expansion coefficients or a wave function cannot always be interpreted as probability amplitudes. For a physical interpretation as probability amplitudes the expansion coefficients must be gauge invariant.