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where H is the Hamiltonian operator that gives the total energy of a particle
(potential plus kinetic energy) H D
p
2
2m
C V.x/. Equation (
7.43
) denotes that the
particle with momentum p and energy E is diffused in the wave with a probability
density proportional to
2
denotes the probability for the particle to be at position x at time t. The external
potential V is defined as V D
2
. Equivalently, it can be stated that
j .x;t/j
j .x;t/j
p
2
2m
C E where E is the eigenvalue (discrete energy
level) of the Hamiltonian H.ForV D 0 or constant, the solution of Eq. (
7.43
)isa
superposition of plane waves of the form
i.px
Et/
j .x;t/ >D e
(7.44)
„
where i is the imaginary unit, x is the position of the particle, and „ is Planck's
constant. These results can be applied to the case of an harmonic potential if only
the basic mode was taken into account.
The probability to find the particle between x and x C
dx
at the time instant t
is given by P.x/
dx
2
dx
. The total probability should equal unity, i.e.
Dj .x;t/j
R
1
1
j .x;t/j
2
dx
D 1. The average position x of the particle is given by
Z
1
Z
1
.
x /
dx
<x>D
P.x/
xdx
D
(7.45)
1
1
where
is the conjugate value of . Now, the wave function .x;t/ can be
analyzed in a set of orthonormal eigenfunctions in a Hilbert space
X
1
.x;t/ D
c
n
n
(7.46)
nD1
The coefficients c
n
are an indication of the probability to describe the particle's
position x at time t by the eigenfunction
n
and thanks to the orthonormality of
the
n
's, the c
n
's are given by c
n
D
R
1
n
dx
. Moreover, the eigenvalues and
eigenvectors of the quantum operator of position x can be defined as x
n
D a
n
n
,
where
n
is the eigenvector and a
n
is the associated eigenvalue. Using Eqs. (
7.45
)
and (
7.46
) the average position of the particle is found to be
1
X
nD1
jjc
n
jj
1
2
a
n
<x>D
(7.47)
2
denoting the probability that the particle's position be described by the
eigenfunction
n
. When the position x is described by
n
the only measurement
of x that can be taken is the associated eigenvalue a
n
. This is the so-called filtering
problem, i.e. when trying to measure a system that is initially in the state D
with jjc
n
jj
P
nD1
c
n
n
the measurement effect on state is to change it to an eigenfunction
n
with measurable value only the associated eigenvalue a
n
. The eigenvalue a
n
is
chosen with probability P /jjc
n
jj
2
.