α
)
β
invariant:
braceleftBig
R
T
(
α
)
R
(
β
)
R
(
α
)
bracerightBig
β
′
=
R
T
(
α
)
R
(
β
)
β
=
R
T
(
α
)
β
=
β
′
.
(4
.
66)
Hence
β
′
is the axis of this rotation. Therefore
R
T
(
α
)
R
(
β
)
R
(
α
) =
R
(
β
′
) =
R
(
R
(
−
α
)
β
)
.
(4
.
67)
In Box 4.3 we showed that when

α

is infinitesimal,
R
(
−
α
)
β
≃
β
−
α
×
β
,
so when
β
is also infinitesimal, equation (4.67) can be written in terms of
the classical generators (4.63) as
(1 + i
α
· J
) (1
−
i
β
· J
) (1
−
i
α
· J
)
≃
1
−
i(
β
−
α
×
β
)
· J
.
(4
.
68)
The zeroth order terms (‘1’) and those involving only
α
or
β
cancel, but the
terms involving both
α
and
β
cancel only if
α
i
β
j
[
J
i
,
J
j
] = i
α
i
β
j
summationdisplay
k
ǫ
ijk
J
k
.
(4
.
69)
This equation can hold for all directions
α
and
β
only if the
J
i
satisfy
[
J
i
,
J
j
] = i
summationdisplay
k
ǫ
ijk
J
k
,
(4
.
70)
which is identical to the ‘quantum’ commutation relation (4.33). Our red
erivation of these commutation relations from entirely classical considerations
is possible because the relations reflect the fact that the order in which you
rotate an object around two different axes matters (Problem 4.6).
This is
a statement about the geometry of space that has to be recognised by both
quantum and classical mechanics.
In Appendix D it is shown that in classical statistical mechanics, each
component of position,
x
i
, and momentum,
p
i
, is associated with a Hermi
tian operator ˆ
x
i
or ˆ
p
i
that acts on functions on phase space. The operator
ˆ
p
i
generates translations along
x
i
, while ˆ
x
i
generates translations along
p
i
(boosts). The operators
ˆ
L
i
associated with angular momentum satisfy the
commutation relation [
ˆ
L
x
,
ˆ
L
y
] = i
H
ˆ
L
z
, where
H
is a number with the same
dimensions as ¯
h
and a magnitude that depends on how ˆ
x
i
and ˆ
p
i
are nor
malised.
If the form of the commutation relations is not special to quantum me
chanics, what is? In quantum mechanics, complete information about any
system is contained in its ket

ψ
)
. There is nothing else. From

ψ
)
we can
evaluate amplitudes such as
(
x
,μ

ψ
)
for the system to be found at
x
with
orientation
μ
. If we do not care about
μ
, the total probability for

ψ
)
to be
found at
x
is
Prob(at
x

ψ
) =
summationdisplay
μ
vextendsingle
vextendsingle
(
x
,μ

ψ
)
vextendsingle
vextendsingle
2
.
(4
.
71)
Eigenstates of the
x
operator with eigenvalue
x
0
are states in which the
system is definitely at
x
0
, while eigenstates of the
p
operator with eigenvalue
¯
h
k
are states in which the system definitely has momentum ¯
h
k
.
By contrast, in classical statistical mechanics we declare at the outset
that a well defined state is one that has definite values for all measurable
quantities, so it has a definite position, momentum, orientation etc.
The
eigenfunctions of ˆ
p
or
ˆ
L
do not represent states of definite momentum or
angular momentum, because we have already defined what such states are.
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 Spring '15
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 Physics, mechanics, The Land, David Skinner, probability amplitudes, James Binney, Physics of Quantum Mechanics