Spin Eigenvalues and Eigenfunctions

We've already found for kinetic moment:

L ² Yl,m = h² l(l +1) Yl,m and Lz Yl,m = h m Yl,m

where L ² is the squared kinetic moment operator and h²l(l +1) is the eigenvalue of eigenfunction Yl,m.

The z-component has its own eigenvalue which is equal to hm.

Now let's have a look at spin operators and eigenvalues. Here we can write down the following:

s² χ = h² s(s + 1) χ and szχ = h msχ

where h²s(s + 1) is the eigenvalue of the squared spin and hms is the eigenvalue of its z-component. We obtain for s = ½ the following eigenvalues ms = ± ½ with corresponding eigenfunctions χ+ (for ms = +½) and c- (for ms = −½):

s² χ± = ¾ h²χ± and szχ± = ± ½ h χ±

The Dirac style for spin description is utterly visual:

χ+ = ( | 1 | ) = |+> |  | c- = ( | 0 | ) = |->
0 | 1

with

sx = h/2{ |+><-| + |-><+| } = h/2( | 0 1 | )
1 0
sy = ih/2{− |+><-| + |-><+| } = h/2( | 0 −i | )
i 0
sz = h/2{ |+><+| − |-><-| } = h/2( | 1 0 | )
0 −1

One can easily write down for |+> and |-> <+| = (1 0) <-| = (0 1) for instance

|+> <-| = ( | 1 0 | ) .( | 0 1 | ) = ( | 0 1 | )
0 0 | 0 0 | 0 0

Certainly we must know the matrix multiplication !

s² = sx² + sy² + sz² = ¾ h² ( | 1 0 | )
0 1

s² |+> = ¾ h² ( | 1 0 | ) ( | 1 | ) = ¾ h² ( | 1 | ) = ¾ h² |+>
0 1 | 0 | 0
s² |-> = ¾ h² ( | 1 0 | ) ( | 0 | ) = ¾ h² ( | 0 | ) = ¾ h² |->
0 1 | 1 | 1
sz |+> = h/2 ( | 1 0 | ) ( | 1 | ) = h/2 ( | 1 | ) = h/2 |+>
0 −1 | 0 | 0
sz |-> = h/2 ( | 1 0 | ) ( | 0 | ) = h/2 ( | 0 | ) = −h/2 |->
0 −1 | 1 | −1

Sometimes the designation χms is used for ms = +½ and ms = −½.

The total wavefunction Y in the radial symmetrical field is then as follows:

Y nlm l ms= Rnl(r) Ylml(J,j) cms

and an electron can be described by four quantum numbers n, l, ml und ms.

The total kinetic moment is the sum of orbital moment and intrinsic kinetic moment (Spin) of an electron:

= +