Understanding Quantum Physics: An Advanced Guide for the Perplexed

not during a
measurement. However, these problems are generally connected to each other. In
particular, in order to know what physical state the wave function of a quantum
system describes, we need to measure the system in the first place, while the
measuring process and the measurement result are necessarily determined by the
evolution law for the wave function. Fortunately, it has been realized that the
conventional measurement that leads to the collapse of the wave function is
only one kind of quantum measurement, and there also exists another kind of measurement
that avoids the collapse of the wave function, namely the protective
measurement proposed by Aharonov, Vaidman and Anandan in 1993. Protective
measurement is a method to measure the expectation values of observables on a
single quantum system, and its mechanism is independent of the controversial
process of wavefunction collapse and only depends on the established parts of
quantum mechanics. As a result, protective measurement can not only measure the
physical state of a quantum system and help to unveil the meaning of the wave
function, but also be used to examine the solutions to the measurement problem
before experiments give the last verdict. A full exposition of these ideas will
be given in the subsequent chapters.
    In Chapter 2, we
first investigate the physical meaning of the wave function. According to
protective measurement, the mass and charge density of a quantum system as one
part of its physical state can be measured as expectation values of certain
observables, and it turns out that they are proportional to the modulus square
of the wave function of the system. The key to unveil the meaning of the wave
function is to find the origin of the mass and charge density. It is shown that
the density is not real but effective; it is formed by the time average of the
ergodic motion of a localized particle with the total mass and charge of the
system. Moreover, it is argued that the ergodic motion is not continuous but
discontinuous and random. Based on this result, we suggest that the wave
function represents the state of random discontinuous motion of particles, and
in particular, the modulus square of the wave function gives the probability
density of the particles appearing in certain positions in real space.
    In Chapter 3, we
further analyze the linear evolution law for the wave function. It is shown
that the linear non-relativistic evolution of the wave function of an isolated
system obeys the free Schrödinger equation due to the requirements of spacetime
translation invariance and relativistic invariance. Though these requirements
are already well known, an explicit and complete derivation of the free
Schrödinger equation using them is still missing in the literature. The new
integrated analysis, which is consistent with the suggested interpretation of
the wave function, may help to understand the physical origin of the
Schrödinger equation, as well as the meanings of momentum and energy for the
random discontinuous motion of particles. In addition, we also analyze the
physical basis and meaning of the principle of conservation of energy and
momentum in quantum mechanics.
    In Chapter 4, we
investigate the implications of protective measurement and the suggested
interpretation of the wave function based on it for the solutions to the
measurement problem. To begin with, we argue that the two no-collapse quantum
theories, namely the de Broglie-Bohm theory and the many-worlds interpretation,
are inconsistent with protective measurement and the picture of random
discontinuous motion of particles. This result strongly suggests that
wavefunction collapse is a real physical process. Secondly, we argue that the
random discontinuous motion of particles may provide an appropriate random
source to collapse the wave function. The key point is to realize that the instantaneous
state of a particle not only includes its wave function but also includes its
random position, momentum and energy that undergo the discontinuous motion, and
these random variables can have a stochastic influence on the evolution of the
wave function and further lead to the collapse of the wave function. Moreover,
it is argued that the principle of conservation of energy (for an ensemble of
identical systems) requires that the random variable that influences the
evolution of the wave function is not position but energy, and due to the
discontinuity of motion the influence can