Understanding Quantum Physics: An Advanced Guide for the Perplexed

1993).
    2.4.3
A specific example
    Consider the
spatial wave function of a single quantum system with negative charge Q (e.g. Q
= -e)
    ψ(x, t) = aψ 1 (x,
t) + bψ 2 (x, t), (2.27)
    where ψ 1 (x,
t) and ψ 2 (x, t) are two normalized wave functions respectively
localized in their ground states in two small identical boxes 1 and 2, and |a| 2 + |b| 2 = 1. An electron, which initial state is a Gaussian wave
packet narrow in both position and momentum, is shot along a straight line near
box 1 and perpendicular to the line of separation between the boxes. The
electron is detected on a screen after passing by box 1. Suppose the separation
between the boxes is large enough so that a charge Q in box 2 has no observable
influence on the electron. Then if the system were in box 2, namely |a| 2 = 0, the trajectory of the electron wave packet would be a straight line as
indicated by position “0” in Fig.1. By contrast, if the system were in box 1,
namely |a| 2 = 1, the trajectory of the electron wave packet would be
deviated by the electric field of the system by a maximum amount as indicated
by position “1” in Fig.1.
    We first suppose
that ψ(x, t) is unprotected, then the wave function of the combined system
after interaction will be
    ψ(x, x , t) = a ϕ 1 (x , t)ψ 1 (x, t) + b ϕ 2 (x , t)ψ 2 (x, t), (2.28)
    where ϕ 1 (x , t) and ϕ 2 (x
, t) are the wave functions of the electron influenced by the electric fields
of the system in box 1 and box 2, respectively, the trajectory of ϕ 1 (x , t) is deviated by a maximum amount, and the
trajectory of ϕ 2 (x , t) is not deviated and still a straight line.
When the electron is detected on the screen, the above wave function will
collapse to ϕ 1 (x , t)ψ 1 (x, t) or ϕ 2 (x , t)ψ 2 (x, t). As a result, the detected
position of the electron will be either “1” or “0” in Fig.1, indicating that
the system is in box 1 or 2 after the detection. This is a conventional impulse
measurement of the projection operator on the spatial region of box 1, denoted
by A 1 . A 1 has two eigenstates corresponding to the system
being in box 1 and 2, respectively, and the corresponding eigenvalues are 1 and
0, respectively. Since the measurement is accomplished through the
electrostatic interaction between two charges, the measured observable A 1 ,
when multiplied by the charge Q, is actually the observable for the charge of
the system in box 1, and its eigenvalues are Q and 0, corresponding to the
charge Q being in boxes 1 and 2, respectively. Such a measurement cannot tell
us the charge distribution of the system in each box before the measurement.
    Fig.1 Scheme of a protective measurement of the charge density of a quantum
system
    Now let’s make a
protective measurement of A 1 . Since ψ(x, t) is degenerate with its
orthogonal state ψ (x, t) = b ∗ ψ 1 (x, t)−a ∗ ψ 2 (x, t), we need an artificial
protection procedure to remove the degeneracy, e.g. joining the two boxes with
a long tube whose diameter is small compared to the size of the box [13] . By this protection ψ(x, t) will be a
nondegenerate energy eigenstate. The adiabaticity condition and the weakly
interacting condition, which are required for a protective measurement, can be
further satisfied when assuming that (1) the measuring time of the electron is
long compared to /∆E, where ∆E is the smallest of the energy differences
between ψ(x, t) and the other energy eigenstates, and (2) at all times the
potential energy of interaction between the electron and the system is small
compared to ∆E. Then the measurement of A 1 by means of the electron
trajectory is a protective measurement, and the trajectory of the electron is
only influenced by the expectation value of the charge of the system in box 1.
In particular, when the size of box 1 can be ignored compared with the
separation between it and the electron wave packet, the wave function of the
electron will obey the following Schrödinger equation:

    where m e is the mass of electron, k is the Coulomb constant, r 1 is the
position of the center of box 1, and |a| 2 Q is the expectation value
of the charge Q in box 1. Correspondingly, the trajectory of the center of the
electron wave packet, r c (t), will satisfy the following equation by
Ehrenfest’s theorem:

    Then the electron
wave packet will reach the position “ |a| 2 ” between “0” and “1” on
the screen as denoted in Fig.1. This shows that the result of the protective
measurement is the