Assume that the basic laws of physics are the same everywhere. Then the mechanical
properties of a closed system are unchanged by any parallel displacement of the entire
system in space. This is the assumption of the **homogeneity of
space**.

Let T_{a }describe an **operation** called a **translation**.
Every part of the closed system is displaced by
the same amount **a** by this operation. To every state vector |ψ> before the translation corresponds a state vector |ψ’> after the translation.
Let U(T_{a})
be the **operator** that maps the state vector before the translation onto the state
vector after the translation, |ψ’> = U(T_{a})|ψ>.
**Note the difference between the operation and the
operator**. If the laws of physics are the same everywhere, then the general formalism
describing the properties of the system is invariant under the translation. In particular
we have

|ψ> = c_{1}|ψ_{1}> + c_{2}|ψ_{2}> -->
|ψ'> = c_{1}|ψ_{1}'> + c_{2}|ψ_{2}'>,

i.e. U(T_{a}) is a linear operator, and <Φ|ψ> = <Φ’|ψ’>,
U^{†}(T_{a})U(T_{a})
= 1, U(T_{a}) is a unitary operator. We say that translation is
a **unitaristic operation**.

With our assumption of the homogeneity of space, the time evolution is also invariant
under translation.

If |ψ(t)> satisfies the
Schroedinger equation, so does |ψ’(t)> = U(T_{a})|ψ(t)>,

iħ (∂/∂t) U(T_{a})|ψ(t)>
= HU(T_{a})|ψ(t)>. But

iħ (∂/∂t) U(T_{a})|ψ(t)>
= iħ U(T_{a})(∂/∂t)|ψ(t)> = U(T_{a})H|ψ(t)>.

Therefore HU(T_{a})|ψ(t)> = U(T_{a})H|ψ(t)> for any |ψ(t)>,
HU(T_{a}) = U(T_{a})H, [H,U(T_{a})]
= 0.

When **a** = 0, then U(T_{0}) = I.

Assume that U(T_{a}) is a differentiable function of **a**.
Then for **a **= d**a
**U(T

I + ∑

Since UU^{†} = I, we have
(∂U/∂a_{i})U^{†}
+ U(∂U^{†}/∂a_{i})
= 0. If we evaluate this expression at **a** = 0 we obtain

(∂U/∂a_{i})|_{a=0}
+ (∂U^{†}/∂a_{i})|_{a=0}
= 0, D_{i} + D_{i}^{†}
= 0. D_{i }is anti-Hermitian.

Let P_{i} = iħD_{i},
then P_{i }is
Hermitian, it is qualified to describe an observable.

**Let us identify P _{i}.
**U(T

ψ’(

We write

U(T

The effect of U(T

ψ(

We therefore have D_{i}ψ(**r**)
= -∂ψ(**r**))/∂x_{i},
P_{i} = iħD_{i}
= -iħ∂/∂x_{i}, i.e. the observables P_{i} are the components of the momentum of the
closed system.

Since [H,U(T_{a})] = 0, we have [H,P_{i}]
= 0, **P** is a constant of motion. **Homogeneity of space implies momentum
conservation**.

Let us now generalize this result. If U(λ) is
a family of unitary operators depending differentiably on the real parameter λ, with U(0) = I, then
X = iħ(∂U/∂λ)|_{λ=0} is a Hermitian operator.
If U(λ)
describes the invariance Q_{λ}, then X is
a constant of motion.

**Rules:**

U(λ + μ)|ψ> = U(λ)U(μ)|ψ>,

U(λ + dλ)|ψ> = U(λ)U(dλ)|ψ> = U(λ)(I + (∂U/∂λ)|_{λ=0}
dλ)|ψ> = U(λ)(I - (i/ħ)X dλ)|ψ>.

U(λ + dλ) - U(λ) = - (i/ħ)X dλ U(λ), U(λ) = exp(-iXλ/ħ)

X is called the Hermitian generator of the unitary family U(λ).

**P** is the Hermitian generator of the unitary family of translation operators
U(T_{a}) = exp(-i**p∙a**/ħ). The evolution operator is U(t) =
exp(-iHt/ħ). It is of the form U(λ) = = exp(-iXλ/ħ), with X = H and λ = t.
The Hamiltonian is the** Hermitian
generator of time translation operators**.

Assume that the basic laws of physics are the same in all directions. Then the
mechanical properties of a closed system are unchanged by any rotation of the entire
system in space. This is the assumption of the **isotropy of space**. Like translations, rotations are therefore unitaristic operations on the state space of
any physical system. There are unitary operators U(R(**n**,θ)) representing the effect of a rotation about the
**n** axis through an angle θ. Let us denote the Hermitian generator of rotation by **J**. Then

U(R(**n**,θ)) = exp(-i**J****∙n**θ/ħ).

Rotations about different axes do not commute. As a consequence, the different
components of **J** do not commute.

We find [J_{i},J_{j}] =
iħε_{ijk}J_{k}.
**J ** satisfies the
commutation relations for angular momentum.

A rotation is an operation upon 3-dimensional vectors. The matrix for rotating a system counterclockwise (ccw) through an angle θ about the z-axis is

We are describing an active rotation. The corresponding passive rotation rotates the
coordinate system clockwise (cw) an angle θ about the z-axis.
Consider a system without any internal degrees of freedom, for example a spinless
particle. U(R) acting on the state vector |**r**> yields |(R**r**)>,
U^{†}(R) acting on the state vector |**r**> yields |(R^{-1}**r**)>.
Therefore

ψ’(**r**) = <**r**|U(R)|ψ> =<U^{†}(R)**r**|ψ> = <(R^{-1}**r**)|ψ>
= ψ(xcosθ + ysinθ, -xsinsθ + ycosθ, z).

We write ψ’(**r**) = U(R)ψ(**r**) = ψ(R^{-1}**r**)

The effect of U(R) on a wave function ψ is to
produce a new wave function ψ’, whose value at R**r**
is the same as the value of ψ at **r**.
For an
infinitesimal rotation we have

ψ’(**r**) = <**r**|I - i**J∙n**dθ/ħ|ψ> = ψ(**r**)
- (i**J∙n**dθ/ħ)ψ(**r**).

But we also have
ψ'(**r**) = ψ(**r**) - (∂ψ(**r**)/∂θ)dθ

= ψ(**r**) - [(∂ψ(**r**)/∂x)(∂x/∂θ) + (∂ψ(**r**)/∂y)(∂y/∂θ)]dθ

= ψ(**r**) - [y(∂ψ(**r**)/∂x) - x(∂ψ(**r**)/∂y)]dθ.

Therefore we find J_{z} = -iħ[x(∂/∂y) - y(∂/∂x)].

Similarly we find J_{x} and J_{y}. We have **J **=** r **
x** p**, the classical definition of angular momentum. We therefore have expressed the rotation operator for a spinless particle in terms of
its orbital angular momentum **L**.

U(R(**n**,θ)) = exp(-i**J∙n**θ/ħ) = exp(-i**L****∙n**θ/ħ),

where **L** = **J** = total angular momentum. In deriving this
expression we assumed that ψ(**r**) completely
describes the state of the particle. This is not true for a particle with spin, a system
which has no classical analog. For such a system we write the rotation operator as
U(R(**n**,θ)) = exp(-i**J∙n**θ/ħ), and consider this relation to be the definition of the total angular momentum **J**
of the system.

Besides rotating the system itself, we can also rotate the instruments with which we
observe it. We now define the law for the transformation of the observables representing
the various measurements that can be made on the system. Let A be an observable and
let A’ be its transform in the rotation, i.e. A describes what is
measured with a certain apparatus, and A’ describes what is measured after the
apparatus has been rotated. The average value of A in the state |ψ> must equal the average value of A’ in |ψ’>.
(Isotropy of space implies that if the system and
the observer are rotated together, the physical predictions remain unchanged.)

<ψ|A|ψ> = <ψ'|A'|ψ'> = <Uψ|A'|Uψ> = <ψ|U^{†}A'U|ψ> for all |ψ>.

Therefore
A = U^{†}(R)A'U(R), A' = U(R)A'U^{†}(R).

An observable A is a scalar observable if

A’ = A, U(R)AU

[I - i

i.e. if A commutes with all components of

A vector observable

For an infinitesimal ccw rotation R about the z-axis we have V’

We have

V’

We therefore have V

Similarly we show [V

This leads to an alternate definition of a vector observable

[V

An example of a vector observable is

Solution:

Properties of an isolated physical system do not vary when it is rotated as a whole in any
manner in space. This is the assumption of isotropy of space. Assume a system is rotated
about the **u** axis by an infinitesimal
amount dα.

(i) Classically, the equations of motion must be invariant under the rotation. The
equations of motion can be derived from the Lagrangian **L**(**r**,**v**,t).
Therefore the Lagrangian must remain unchanged by the rotation.

Under the rotation **r** --> **r** + dα **u **x **r**, **
v** --> **v** + dα **u **x** v**.

Therefore δ**r** = dα **u **x **r**, ** **δ**v** = dα
**u **x** v**.

Isotropy of space implies that

Using the definition ∂L/∂**v**_{n} = **p**_{n} we obtain
from

.

We therefore have, (using **a **× (**b **x** c**) =** b **×(**c **x** a**) =** c **× (**a **x** b**)),

δα** u ∙ **(d/dt)∑_{n}(**r**_{n}** **x** p**_{n})
= 0 for all
**u**.

This implies (d/dt)∑_{n}(**r**_{n}** **x** p**_{n})
= 0, angular momentum is conserved.

(ii) Quantum mechanically, we consider an isolated system in the state |ψ(t)>. After an arbitrary rotation the state of the
system becomes |ψ’(t)> = U(R)|ψ(t)>.
Rotational invariance implies that the time
evolution is not affected by the rotation, i.e. [H,U(R)] = 0.

Therefore [H,(I - i**J∙u**dα/ħ)] = 0, [H,**J∙u**] =
0, [H,**J**] = 0.

H commutes with all components of **J**, therefore **J** is
a constant of motion.

Rotational invariance implies that the total angular momentum of an isolated physical
system is a constant of motion. We therefore always can find common eigenstates of H, J^{2},
and J_{z}. We denote the common eigenbasis by {|k,j,m>}.
Using this standard basis we find that **the matrix elements of any scalar operator
are non zero only between states with the same values of j and m**.

Let A be a scalar observable, [A,U(R)] = 0, [A,J^{2}]
= 0, [A,J_{z}] = 0. Then

Similarly

Moreover, **the matrix elements
of a scalar observable are independent of m**.

.