PH 401
Dr. Cecilia Vogel
Review
Prove the radial H-atom solution
Spin
evidence
spin angular momentum
Outline
Spin
spin angular momentum
not really spinning
simultaneous eigenstates and
measurement
Schrödinger's cat
Spin Angular Momentum
 Spin is like other forms of angular momentum,
in the sense that
it acts like a magnet, affected by B-fields
it contributes to the angular momentum,
when determining conservation thereof.
The eigenvalues of the magnitude of the

vector are | S | s(s 1)

for electron, s=1/2, so | S | 23 
And the eigenvalues of the z-component are
ms where ms ranges from –s to s in integer steps
for electron, s=1/2, so ms =+½
“Spinning” is an imperfect model
 Spin is UNlike other forms of angular
momentum, in the sense that nothing is
physically spinning!
For one thing, the electron is a point particle;
how can a point spin?
For another thing, assuming that there is a spin
angle, fs leads to contradiction.
Let’s begin by assuming that there is a
physical angle of rotation, fs, corresponding to
spin rotation
in the same way that f corresponds to orbital
angular momentum.
Pf by contradiction
If fs corresponds to spin rotation
in the same way that f corresponds to orbital
angular momentum, then g (fs )  eim f
would hold true (like g (f )  eimf for orbital)
s s
OK, then what is the value of the function at f=0?
e0=1
And what is the value of the function at f=2p?
 eim f  ei ( )(2p )  eip  1
So, which is it? It’s the same point in space, but is
the function 1 or -1?
Wavefunction should be single-valued
CONTRADICTION! Cross it out!
s s
1
2
Spin Commutators
 Spin is like other forms of angular momentum,
in one more way…
it obeys the same type of commutation
relations.
 [S x , S y ]  iS z
and similarly for cyclic permutations of x, y, z
and [S 2 , Si ]  0
where i =x or y or z
Spin Simultaneous Eigenstates
 Because [S 2 , Sz ]  0
there exists a complete set of simultaneous
eigenstates of S2 and Sz,
 with quantum numbers s and ms.
Because [S x , S y ]  iS z
(and similarly for cyclic permutations of x, y, z)
there are NO simultaneous eigenstates of two
different components of spin of electron
If electron is in an eigenstate of Sz (ms=+1/2, for ex)
then Sz is certain, but
Sx and Sy are uncertain!
Simultaneous Eigenstates
Revisited
 Recall
there exists a complete set of simultaneous
eigenstates of two operators, only if they
commute
 [Si , S j ]  0 and[Li , Lj ]  0 if i  j
so there is not a complete set of simultaneous
eigenstates of different components of spin OR orbital
angular momentum
But, just because there is not a complete set, does not
mean there are none.
Simultaneous Eigenstates
Revisited
 Recall
there exists a simultaneous eigenstate, |ab>
of two operators, A and B, if [ A, B] | ab  0
 Is that possible for two components of spin?
suppose [Sx , S y ] | ab  0
using the commutation relation,
this means iS z | ab  0
which means |ab> is an eigenstate of Sz, with
eigenvalue zero
For electron, Sz has eigenvalues +½ only.
CONTRADICTION again
Simultaneous Eigenstates
Revisited
 Recall
there exists a simultaneous eigenstate, |ab>
of two operators, A and B, if [ A, B] | ab  0
 Is that possible for two components of orbital
angular momentum?
suppose [Lx , Ly ] | ab  0
using the commutation relation,
this means iLz | ab  0
which means |ab> is an eigenstate of Lz, with
eigenvalue zero
That’s cool.
Just means that the state is one with mℓ=0
Simultaneous Eigenstate of Ang
Mom components
 In the previous slide, we showed that a
simultaneous eigenstate of Lx and Ly could exist
so long as it was also an eigenstate of Lz
with Lz=0
 That means it’s a simultaneous eigenstate of Lz
and Lx (and Ly)
thus [Lx (and Ly ), Lz ] | ab  0
which means  iLy (and iLx ) | ab  0
which means |ab> is an eigenstate of Lx, Ly, and Lz,
ALL with eigenvalue zero
That’s cool. Then L2=0
Just means that the state is one with ℓ=mℓ=0
Simultaneous Eigenstates
The punchline is
there are NO simultaneous eigenstates of two
different components of spin of electron
but there are simultaneous eigenstates of two
different components of orbital angular
momentum of electron,
and those are the states with ℓ=mℓ=0
Simultaneous Eigenstates &
Measurement
Suppose an electron is in a superposition state of spinup and spin-down
it has an uncertain Sz
Then we measure Sz and find Sz= - ½ 
now Sz is no longer uncertain
the measurement collapsed the wavefunction
into an eigenstate of what we were measuring.
Since Sz is certain, Sx and Sy are uncertain
but there is nothing to stop us from measuring Sx.
What happens if we measure Sx and find +½ ?
….
Simultaneous Eigenstates &
Measurement
We measured Sz and found Sz= - ½ 
Then we measured Sx and found Sx=+½ ? ….
So our electron has Sz= - ½  and Sx =+½ ?
NO – that would be a simultaneous eigenstate of Sx
and Sz, which is impossible!
When we measured Sx, we collapsed the
wavefunction again
it is not in the same state it was in
it no longer has Sz = - ½ 
instead it has collapsed into an eigenstate of Sx
If we measure Sz now, we have no idea what we’ll
find!
Review Schrödinger's Cat
http://en.wikipedia.org/wiki/Schroedinger's_cat#The
_thought_experiment