A microscopic model of wave-function decoherence in the double-slit experiment

The act of measurement on a quantum state is supposed to “decohere” and “collapse” the state into one of several eigenstates of the operator corresponding to the observable being measured. This measurement process is sometimes described as outside standard quantum-mechanical evolution and not calculable from Schr¨odinger’s equation [2]. Progress has, however, been made in studying this problem with two main calculation tools - one uses a time-independent Hamiltonian [18], while a rather more general approach proving that decoherence occurs under some generic conditions [21]. The two general approaches to the study of wave-function collapse are as follows. The first approach, called the “consistent” or “decoherent”’ histories approach [11], studies microscopic histories that diverge probabilistically and explains collapse as an event in our particular history. The other, referred to as the “environmental decoherence” approach[8, 21] studies the effect of the environment upon the quantum system, to explain wave-function decoherence. Then collapse is produced by irreversible effects of various sorts. In the “environmental decoherence” approach, one writes down a Markovian-approximated Master equation to study the time-evolution of the reduced density matrix and obtains the long-time dependence of the off-diagonal elements of this matrix. The calculation in this paper studies the evolution of a quantum system under the “environmental” approach, with a rather important analytic difference. We start from the Schr¨odinger equation for the state of the system, with a time-dependent Hamiltonian that reflects the actual microscopic interactions that are occurring. Then we systematically solve for the time-evolved state, without invoking a Markovian approximation when writing out the effective time-evolution equation, i.e., keeping the evolution unitary until the end. This approach is useful and it allows the system wave-function to explicitly “un-collapse” if the measurement apparatus is sufficiently small. However, in the limit of a macroscopic system, collapse is a temporary state that will simply take extremely long (of the order of multiple universe lifetimes) to reverse. While this has been attempted previously [12], we study a particularly simple and calculable example. We make some connections to the work by Linden et al [21] while doing so. The calculation in this paper has interesting implications for the interpretation of the Wigner’s friend experiment, as well as the Mott experiment, which is explored in Sections V and VI (especially the enumerated points in Section VI). The upshot is that as long as Wigner’s friend is macroscopically large (or uses a macroscopically large measuring instrument), no one needs to worry that Wigner would see something different from his friend. Indeed, Wigner’s friend does not even need to be conscious during the measurement that she conducts. In particular, as a result of the mathematical analysis, the short-time behavior of a collapsing system, at least the one considered in this paper, is not exponential. Instead, it is the usual Fermigolden rule result. The long-term behavior is, of course, still exponential. This is a second novel feature of the paper - we connect the short-term Fermi-golden rule (quadratic-in-time behavior) transition probability to the exponential long-time behavior of a collapsing wave-function in one continuous mathematical formulation.