dimensions are described by 3
×
3 orthogonal matrices with positive determinant, which is
what SO(3) means. Conversely, our spin1/2 rotation operators
ˆ
R
φ,k
are 2
×
2 unitary ma
trices with positive determinant, which is what SU(2) means. It is interesting that SU(2)
being a twofold cover of SO(3) (requiring two complete turns to get back to where you
started) has nothing to do with quantum mechanics, but rather is built into the structure
of rotations in three dimensions. See for
a nice discussion of this. Mathematicians figured out all this stuff before Planck came up
with his constant in 1900, setting in motion the discovery of quantum mechanics.
3
(b) Show that [
ˆ
A,
ˆ
B
] = 0 and [
ˆ
B,
ˆ
C
] = 0 but [
ˆ
A,
ˆ
C
]
6
= 0. This means
that
A
and
B
are compatible observables and also
B
and
C
are
compatible observables, but
A
and
C
are incompatible observ
ables.
Find a basis (two normalized and mutually orthogonal
spinors) which is composed of eigenspinors of both
ˆ
B
and
ˆ
C
, and
show that these basis vectors are
not
eigenspinors of
ˆ
A
.
(c) The basis vectors found in part 5b are states of definite
B
and
C
,
but not states of definite
A
.
Show this explicitly by computing
the uncertainties Δ
A
, Δ
B
, Δ
C
in each of the two basis states.
Use
h
A
2
i
=
h
e
k

ˆ
A
2

e
k
i
, etc.
4
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 Fall '15
 mechanics, Power Series, Work, Photon, Polarization, Townsend Sec., spin1/2 quantum state