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and potential energy, which is given by H D p
2
=2m C V , with p being the
momentum of the particle, m the mass, and V an external potential. The solution
of Eq. (
13.1
) is given by .x;t/ D e
iHt
.x;0/ [
34
].
However, cartesian coordinates are not sufficient to describe the particle's
behavior in a magnetic field and thus the spin variable taking values in SU(2)
has been introduced. In that case the solution of Schrödinger's equation can
be represented in the basis jr; > where r is the position vector and is the
spin's value which belongs in f
2
;
2
g (fermion). Thus vector which appears in
Schrödinger's equation can be decomposed in the vector space jr; > according to
1
j >D
P
R
d
3
rjr; >;< r;j >. The projection of j >in the coordinates
system r; is denoted as <r;j >D
.r/. Equivalently one has
C
.r/ D<
r; Cj >and
.r/ D<r;j >. Thus one can write .r/ D Œ
C
.r/;
.r/
T
.
13.2.2
Measurement Operators in the Spin State-Space
1
2
It has been proven that the eigenvalues of the particle's magnetic moment are ˙
1
or ˙„
2
. The corresponding eigenvectors are denoted as jC > and j >. Then the
relation between eigenvectors and eigenvalues is given by
z
jC >DC.„=2/jC >
z
j >DC.„=2/j >
(13.2)
which shows the two possible eigenvalues of the magnetic moment [
34
]. In general
the particle's state, with reference to the spin eigenvectors, is described by
(13.3)
j >D ˛jC > Cˇj >
with j˛j
D 1 while matrix
z
has the eigenvectors jC >D Œ1;0 and j >D
Œ0;1 and is given by
2
Cjˇj
2
10
0 1
2
z
D
(13.4)
Similarly, if one assumes components of magnetic moment along axes x and
z
, one
obtains the other two measurement (Pauli) operators
01
10
;
y
D
0 i
i0
2
2
x
D
(13.5)
