hello in this video we're going to take
a look
at spin one half it's a particularly
simple example
of a finite dimensional hilbert space
which we've encountered
in the previous two videos so spin
is a quantum property which is also
known as
intrinsic angular momentum so if we
think kind of classically for a minute
and think of the electron
in the atom as like the earth orbiting
the sun so kind of orbiting like this
well that has angular momentum but
you can also have the earth spinning on
its own axis
and that's what intrinsic angular
momentum is like okay slightly spinning
on its own axis
but things are obviously going to have
to get a little bit weird because while
the earth has a finite radius
the radius of the electron as far as we
can tell is effectively zero
so we can identify properties of the
spin of an electron using what's called
a Stern-Gerlach 0:00:56.840,0:01:00.320 apparatus so there's an experiment that
was carried out
in 1922 originally i believe and it
looks something like this
that is a very a large magnet
we've got the north pole up here in the
south pole up here this is somewhat
schematic
and in particular the apparatus
generates a large field gradient in this
direction
so we have a gradient of the magnetic
field B
directed in this direction now the
electron
if it has this intrinsic angular
momentum if it's spinning well
it's got an electric charge so we
might expect it to have a magnetic field
and we can shoot a beam of electrons
down through here
i think this was done by just taking a
heated element and it will emit
electrons naturally [actually Silver atoms, sorry!]
and we can shoot our beam of electrons [Ag atoms]
down through here and they'll deflect
in the direction of the gradient
so classically if we have a screen over
here
some kind of measurement device for
measuring where the electrons land
classically we'd expect some kind of
spread that would look like this
and this would all be filled in
so it spreads out left to right just
because there's going to be some natural
spread of the beam
and in the top bottom direction
the electrons are being separated
according to the projection
of their angular momentum along the
field gradient direction
so if the if the electron happens to
be spinning
at 90 degrees to that field gradient
direction it has no magnetic field
in the direction of the gradient and so
it won't accelerate and we'll get it
into the middle
and on the other hand if it's completely
lined up with it either along or against
the gradient
it'll go to the top or bottom and we'd
expect it to take any value in between
but what they found when they did the
experiment is that it didn't look like
this at all
in fact the spread looks like something
like this that is
every electron either goes up or down
and there's nothing in between it's
quantized this is a very clear example
of the quantization of quantum mechanics
recalling that
quantum means discrete so it seems that
whenever we measure the spin of the
electron along any direction
it always takes one of two values and
those values are either
plus h bar over two or
minus h bar over two
so plus or minus a half in units of the
reduced planck's constant h bar
and so the electron is what we call
spin-half
so we can deduce various things from
applying these Stern-Gerlach
apparatuses to beams of electrons here
are the experimental observations
taking a Stern-Gerlach apparatus we can
place a block in front of one of the two
beams so for example we could block off
this beam down here
and then we'll be guaranteed to have
spin
plus a half or what's called spin up
in this direction and let's define this
direction to be z
in this case the Stern-Gerlach apparatus has a
direction in which it's going to split
the beam
and by blocking one of the paths we can
guarantee that our electrons are spin
polarized
in this case it would give us spin up we
could also choose the spin down
direction
so we make the following observation
measurement of spin
yields the values plus or minus h bar
over two only
consecutive measurements of the spin in
the same direction yield consistent
results
so if we pass this electron beam
selected out as plus h bar over two in
this direction
through a second Stern-Gerlach filter
in the same direction
we'll all the electrons that go
through the first will get through the
second
similarly if we put a second one in
the same direction and we block off all
the ones that have plus h bar over two
only allowing the ones with minus h bar
over two none will get through
because we'll get consistent results
however subsequent measurement in a
perpendicular direction
yields either the value plus or minus h
bar over two with equal probability
so that's the real clincher here
if we measure it in the spin in z and
then
we pass our spin polarized beam plus h
bar over two through a measurement in
x we'll have equal probability for it
to be plus a half or minus a half means
h bar but that's really weird because
that means if we perform a measurement
on in the z direction and then we
perform a measurement
in the x direction and then we form a
second measurement in the z direction
say we get plus h phi over 2 in the
first measurement of z
whatever we measure an x we select one
or the other of the two spin
polarizations in x and then we pass it
back to another z filter
well it said plus h bar over 2 the first
time but this time it has a 50:50
chance that any electrons coming
through this
to this have a 50:50 chance of going h bar
over 2 and 50:50
chance of giving the opposite so by
making an intermediate measurement in a
different direction
we can actually change the answer it
gives in the z direction and that's
one of the fundamental weird things
about quantum mechanics
so we're going to take a look at a
demonstration of that in a separate
video
for now let's just notice that there is
some precedent for this in classical
mechanics
if you think of let's take a spinning
classical object again
if we say this has some angular momentum
in this direction spinning like this
and we also want to say so this is a
some well defined value and i can work
out what that is
but if i also wanted to have a
well-defined angular momentum in this
direction
but it's got to spin like this and like
this so now what is its
spin in the first direction because when
it gets down to this it's now spinning
like this and so now it's got zero
angular momentum in this direction
but as it spins back round to here
then it's now got
the opposite of what it had before so
it's become time dependent
so there is some classical precedent
for this weirdness
but it is ultimately a fundamentally
quantum thing
spin and it's very weird
thank you