It is not hard to show that [tex] \sqrt 2[/tex] cannot be represented by a rational number. Suppose [tex] \sqrt 2[/tex] is rational, then we must have
[tex] \sqrt 2 = \frac{p}{q}[/tex]
where [tex] p [/tex] and [tex] q [/tex] are both integers.

We further suppose that [tex] p[/tex] and [tex] q[/tex] have no common factors since we could cancel these factors out and replace [tex] p[/tex] and [tex] q[/tex] with [tex] p’[/tex] and [tex] q’[/tex] which are both integers and have no common factors.

We now square our equation to get
[tex] 2 = \frac{p^2}{ q^2}[/tex]
which implies [tex] p^2 = 2 q^2[/tex].

This means that [tex] p^2 [/tex]is divisible by [tex] 2[/tex]. The square of an odd number is always odd, so that [tex] p [/tex] can’t be odd. Hence [tex] p[/tex] must be even. Since [tex] p[/tex] is even, [tex] p = 2 k[/tex] for some integer [tex] k[/tex].

This gives
[tex] p^2 = (2 k)^2 = 4 k^2 = 2 q^2 [/tex]
and hence
[tex] q^2 = 2 k^2[/tex].
This means [tex] q^2[/tex] is divisible by [tex] 2[/tex], and by the same argument as before, [tex] q[/tex] must be even.

At the start we assumed [tex] \sqrt 2[/tex] was rational, so that [tex] \sqrt 2 = \frac{p}{q}[/tex] where [tex] p[/tex] and [tex] q[/tex] are integers with no common factors, but this leads us to conclude that both [tex] p[/tex] and [tex] q[/tex] are even, so they have the common factor [tex] 2[/tex].

We have a contradiction. The asumption that led us to the contradiciton is that [tex] \sqrt 2 [/tex] is rational. Either we accept the contradiction or we reject the assumption. As we can’t accept the contradiction, we must reject the assumption that [tex] \sqrt 2[/tex] is rational.

Electrons are elementary particles. Experiment indicates that they have no sub-structure.

They have a mass of [tex]9.11 \times 10^{-31}[/tex] kg and an electric charge of [tex]-1[/tex]. They have a spin of [tex]\frac{1}{2}[/tex].

The most interesting thing is the (intrinsic) spin [tex]\frac{1}{2}[/tex]. This spin is not like the spin we usually associate to a spinning object.

For a classical object, the spin can take any (reasonable) value we wish. For an electron, the spin is quantised. The experiment that demonstrates this is the Stern-Gerlach experiment, where a beam of silver atoms is passed through an inhomogeneous magnetic field. The experiment reveals that rather than fanning out as would be expected classically, the beam is split into two beams, one going up in the direction of the z-component of the magnetic field and the other down.

These two spins are called up [tex]\frac{1}{2}[/tex] and down [tex]-\frac{1}{2}[/tex]. The funny number [tex]\frac{1}{2}[/tex] comes from the Lie algebra [tex]su(2)[/tex].

In an effort to understand the nature of electron spin we turn to the study of the Lie algebra [tex]su(2)[/tex].

We begin with the defining relations of [tex]su(2)[/tex]. The algebra has three generators, [tex]S_x, S_y[/tex] and [tex]S_z[/tex] which obey the following commutation relations

[tex] [S_x, S_y] = i S_z [/tex] (and cyclic permutations).

We are working in natural units here, so that [tex]\hbar[/tex] is set to unity.

It is convenient to introduce the ladder operators [tex]S_+[/tex] and [tex]S_-[/tex] defined by
[tex] S_+ = S_x + i S_y [/tex]
[tex] S_- = S_x - i S_y . [/tex]

We also set

[tex]S_0 = S_z [/tex].

In terms of these new operators, the [tex]su(2)[/tex] commutation relations become

[tex] [S_0 ,S_\pm]= \pm S_\pm [/tex]
and
[tex] [S_+ ,S_-]= 2 S_0 . [/tex]

Let [tex]V[/tex] be a vector space over a field [tex]F[/tex]. Since we are doing quantum mechanics, we take the field to be [tex]C[/tex], the field of complex numbers.

Let [tex]L[/tex] be a Lie algebra, in our case [tex]L = su(2)[/tex]. What we want to do is make the elements of our Lie algebra act on the elements of the vector space [tex]V[/tex] in a way that is consistent with the commutation relations of the algebra. In other words we want a homomorphism from [tex]L[/tex] into [tex]V[/tex].

Now since we are working over the complex field [tex]C[/tex], we can be sure that there is at least one solution to the eigenvector equation

[tex] S_0 v_k = k v_k. [/tex]

Now, [tex] [ S_0 , S_+ ] = S_+ [/tex], so

[tex] S_0 S_+ = S_+ S_0 + S_+ [/tex]

and the action of [tex]S_0[/tex] on the vector [tex]S_+ v_k[/tex] is

[tex] S_0 (S_+ v_k ) = k S_+ v_k + S_+ v_k = (k+1) S_+ v_k .[/tex]

So, [tex]S_+ v_k[/tex] is an eigenvector of [tex]S_0[/tex], with eigenvalue [tex]k+1[/tex].

We define

[tex] S_+ v_k = v_{k+1} , [/tex]

giving

[tex] S_0 v_{k+1} = (k+1) v_{k+1} . [/tex]

Now we see why [tex]S_+[/tex] is called a ladder operator (raising operator), since it raises the eigenvalue of [tex]v_k[/tex].

We can do a similar thing with [tex]S_-[/tex], going throught the calculations, we find

[tex] S_0 (S_- v_k ) = (k-1) S_- v_k [/tex]

and we set

[tex] S_- v_k = v_{k-1} . [/tex]

The ladder operator [tex]S_-[/tex] is called a lowering operator, since it acts on [tex]v_k[/tex] , to reduce its eigenvalue by 1.

To summarise, we have three equations describing how our Lie algebra operators, [tex]S[/tex] act on vectors from [tex]v \in V[/tex],

[tex] S_0 v_k = k v_k [/tex]

[tex] S_+ v_k = v_{k+1} [/tex]

[tex] S_- v_k = v_{k-1} .[/tex]

We introduce the following operator, called the Casimir operator for [tex]su(2)[/tex]

[tex] S^2=S_x ^2+S_y ^2+S_z ^2. [/tex]

This operator isn’t really part of [tex]su(2)[/tex], however. The algebra [tex]su(2)[/tex] doesn’t contain elements like [tex]S_x ^2[/tex]. We need a “bigger” algebra, the Universal Eneveloping algebra [tex]U(su(2))[/tex] of [tex]su(2)[/tex], which consists of all products of the form [tex]S_x ^p S_y ^q S_z ^r[/tex], where [tex]p,q,r[/tex] are integers. For now, we’ll just ignore this and pretend [tex]S^2[/tex] is an element of [tex]su(2)[/tex].

Consider now the commutator,

[tex] [S^2, S_x ] = [S_x^2+S_y^2+S_z^2, S_x ] [/tex]
[tex] = [S_x^2, S_x ] + [S_y^2, S_x ] + [S_z ^2, S_x ][/tex]
[tex] = [S_x S_x ,S_x ]+[S_y S_y ,S_x ]+[S_z S_z , S_x ]. [/tex]

To expand the last expression we note that

[tex] [ab,c] = abc - cab = abc - acb + acb -cab [/tex]

[tex] = a(bc-cb) + (ac -ca)b [/tex]

[tex] = a[b,c] + [a,c]b. [/tex]

Hence,

[tex] [S^2, S_x ] = S_x [S_x ,S_x ] + [S_x ,S_x ]S_x + S_y [S_y ,S_x ] + [S_y ,S_x ]S_y [/tex][tex] + S_z [S_z , S_x] + [S_z ,S_x ]S_z[/tex]

[tex]= -i S_y S_z - i S_z S_y + i S_z S_y + i S_y S_z [/tex]

[tex] = -i S_y S_z + i S_y S_z + i S_z S_y - i S_z S_y = 0. [/tex]

We have shown that [tex]S^2[/tex] commutes with [tex]S_x[/tex] . A similar calculation shows that [tex]S^2[/tex] commutes with [tex]S_y[/tex] and [tex]S_z[/tex] .

We’ve seen that the [tex]su(2)[/tex] Casimir, namely [tex]S^2[/tex] commutes with the generators of [tex]su(2)[/tex]. In order to proceed we now express [tex]S^2[/tex] in a form which will make our calculations easier.

Recall that,

[tex] S^2 = S_x^2 + S_y^2 + S_z^2 [/tex]

and that our raising and lowering operators were defined as

[tex] S_+ = S_x + i S_y [/tex]

[tex] S_- = S_x - i S_y . [/tex]

(We also set [tex]S_0 = S_z[/tex] .)

Consider the following,

[tex] S_+ S_- = (S_x + i S_y )(S_x - i S_y ) [/tex]
[tex] = S_x ^2 -i S_x S_y + iS_y S_x + S_y ^2 [/tex]
[tex] = S_x ^2 -i [S_x ,S_y ] + S_y ^2 [/tex]
[tex] = S_x ^2 -i^2 S_z + S_y ^2 [/tex]
[tex] = S_x ^2 + S_y ^2 + S_z [/tex]

similarly

[tex] S_- S_+ = S_x ^2 + S_y ^2 - S_z . [/tex]

This gives us two expressions for [tex]S^2[/tex],

[tex] S^2 = S_+ S_- + S_0 ^2 - S_0 = S_+ S_- + S_0 (S_0 - 1) [/tex]

and

[tex] S^2 = S_- S_+ + S_0 ^2 + S_0 = S_- S_+ + S_0 (S_0 + 1). [/tex]