Understanding Quantum Physics: An Advanced Guide for the Perplexed

i |a i >, then an (impulse)
measurement of the observable A will instantaneously, discontinuously, and
randomly collapse the state into one of the eigenstates |a i > with
probability |c i | 2 . [3] This is usually called the collapse
postulate, and the nonlinear stochastic process is called the reduction of the
state vector or the collapse of the wave function.
    The link between
the mathematical formalism and experiment is provided by the Born rule. It says
that the probability of the above measurement of the observable A yielding the
result a i is |c i | 2 . Note that the Born rule
can be derived from the collapse postulate by resorting to the
eigenvalue-eigenstate link, but it does not necessarily depend on the postulate.
Different from the controversial collapse postulate, the Born rule has been
confirmed by precise experiments and is an established part of quantum
mechanics.
    The conventional
impulse measurements can be further formulated as follows. According to the standard
von Neumann procedure, measuring an observable A in a quantum state |ψ>
involves an interaction Hamiltonian
    H I =
g(t)PA (2.1)
    coupling the
measured system to an appropriate measuring device, where P is the conjugate
momentum of the pointer variable. The time-dependent coupling strength g(t) is
a smooth function normalized to ∫dtg(t) = 1 during the interaction interval τ,
and g(0) = g(τ) = 0. The initial state of the pointer is supposed to be a
Gaussian wave packet of width w 0 centered at initial position 0,
denoted by |φ(0)> .
    For an impulse
measurement, the interaction H I is of very short duration and so
strong that it dominates the rest of the Hamiltonian (i.e. the effect of the
free Hamiltonians of the measuring device and the measured system can be
neglected). Then the state of the combined system at the end of the interaction
can be written as

    By expanding |ψ in
the eigenstates of A, |a i , we obtain

    where c i are the expansion coefficients. The exponential term shifts the center of the
pointer by a i :

    This is an
entangled state, where the eigenstates of A with eigenvalues a i get
correlated to measuring device states in which the pointer is shifted by these
values a i (but the width of the pointer wavepacket is not changed).
Then by the collapse postulate, the state will instantaneously and randomly
collapse into one of its branches |a i >|φ(a i )> with
probability |c i | 2 . This means that the measurement result
can only be one of the eigenvalues of measured observable A, say a i ,
with a certain probability |c i | 2 . The expectation value
of A is then obtained as the statistical average of eigenvalues for an ensemble
of identical systems, namely =Σ i |c i | 2 a i .
    2.2 Weak measurements
    The conventional
impulse measurements are only one kind of quantum measurements, for which the
coupling between measured system and measuring device is very strong, and thus
the results are the eigenvalues of measured observable. In fact, we can also
obtain other kinds of measurements by adjusting the coupling strength. An
interesting example is weak measurements (Aharonov, Albert and Vaidman 1988),
for which the measurement result is the expectation value of the measured
observable. In this section, we will introduce the basic principle of weak
measurements.
    A weak measurement
is a standard measuring procedure with weakened coupling. As in the
conventional impulse measurement, the Hamiltonian of the interaction with the
measuring device is also given by Eq. (2.1) in a weak measurement. The weakness
of the interaction is achieved by preparing the initial state of the measuring
device in such a way that the conjugate momentum of the pointer variable is
localized around zero with small uncertainty, and thus the interaction
Hamiltonian (2.1) is small [4] .
The explicit form of the initial state of the pointer in position space is:

    The corresponding
initial probability distribution is

    Expanding the
initial state of the system |ψ> in the eigenstates |a i > of the
measured observable A, |ψ>=Σ i c i |a i >,
then after the interaction (2.1) the state of the system and the measuring
device is:
    The probability distribution of the pointer variable corresponding to the final
state (2.7) is:
    In case of a conventional
impulse measurement, this is a weighted sum of the initial probability
distribution localized around various eigenvalues a i . Therefore, the
reading of the pointer variable in the end of the measurement always