The effect of a localized time-dependent potential on a Fermi gas is discussed. A simple bosonlike model for the particle-hole excitations is developed. Its use appears justified for the dissipative part of the response of the Fermi gas to localized time-dependent potentials. General expressions are obtained for the probability that the Fermi gas remains in the ground state and for the energy spectrum of excited states. Several illustrative examples are discussed. We consider the effect of an exponentially switched-on localized potential. It is shown that the state reached when the potential attains its full strength $V$ is orthogonal to the ground state of the Fermi gas with the potential $V$, no matter how slow the switching. It consists of a spectrum of excited states of the full Hamiltonian with a width proportional to the switching rate. Other aspects of this failure of adiabaticity such as nonreversibility and path dependence are investigated. Some possible applications are pointed out. The potential source may have a finite mass. The effect of its recoil is calculated for the special case of a suddenly switched-on potential. The probability of finding the Fermi gas in its final ground state after the perturbation is switched on is found to be $P_G=\exp \left[\frac{{\tilde{V}}^2\ln \gamma }{(1-{\gamma }^2)}\right]$, where $\tilde{V}=V{\rho }_{\epsilon F}$ and is a measure of the coupling of the localized perturbation to the electron gas. $\gamma$ is the mass ratio $\frac{m}{M}$. The potential source is initially at rest. The above expression is critically discussed and corrections to it are calculated. They are found to be small. The spectrum of excitations produced is also calculated. In conclusion, connections are pointed out with other particle field problems exhibiting a similar infrared singularity.

• Received 7 July 1970

DOI:https://doi.org/10.1103/PhysRevB.3.1102