Physics 6210/Spring 2007/Lecture 3
Lecture 3
Relevant sections in text:
§
1.2, 1.3
Spin states
We now model the states of the spin 1/2 particle. As before, we denote a state of the
particle in which the component of the spin vector
S
along the unit vector
n
is
±
¯
h/
2 by

S
·
n
,
±
. We define our Hilbert space of states as follows. We postulate that
H
is spanned
by

S
·
n
,
±
for any choice of
n
, with different choices of
n
just giving different bases.
Thus, every vector

ψ
∈ H
can be expanded via

ψ
=
a
+

S
·
n
,
+ +
a


S
·
n
,

.
(1)
We define the scalar product on
H
by postulating that each set

S
·
n
,
±
forms an
or
thonormal basis
:
S
·
n
,
±
S
·
n
,
±
= 1
,
S
·
n
,
∓
S
·
n
,
±
= 0
.
Since every vector can be expanded in terms of this basis, this defines the scalar product
of any two vectors (exercise). Note that the expansion coefficients in (1) can be computed
by
a
±
=
S
·
n
,
±
ψ .
This is just an instance of the general result for the expansion of a vector

ψ
in an
orthonormal (ON) basis

i
,
i
= 1
,
2
, . . . , n
, where the ON property takes the form
i

j
=
δ
ij
.
We have (exercise)

ψ
=
i
c
i

i ,
c
i
=
i

ψ .
We can therefore write

ψ
=
i

i
i

ψ .
If we choose one of the spin bases, say,

S
·
n
,
±
, and we represent components of
vectors as columns, then the basis has components
S
·
n
,
±
S
·
n
,
+
=
1
0
S
·
n
,
±
S
·
n
,

=
0
1
.
More generally, a vector with expansion

ψ
=
a
+

S
·
n
,
+ +
a


S
·
n
,

1
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Physics 6210/Spring 2007/Lecture 3
is represented by the column vector
S
·
n
,
±
ψ
=
a
+
a

.
The bra
ψ

corresponding to

ψ
has components forming a row vector:
ψ

S
·
n
,
±
= (
a
*
+
a
*

)
.
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 Spring '07
 M
 Physics, Linear Algebra, mechanics, linear operator, Aij Bjk

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