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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 .
 
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