Quantum spin of the Electron, Why is it 1/2?
Name: Matthew Johnson
Hi, I am Matthew Johnson from West Chicago High School. Why is the
electron given the quantum spin of +1/2 or -1/2? Is this number arbitrary
or is the 1/2 a value of "something?"
An excellent question.
Indeed it does...and its interpretation is not trivial at all.
A short answer is the following; atomic orbitals have
integer amounts of angular momentum, right? An s orbital
has l = 0, a p orbital has l = 1, etc...what's more,
if you put a 2p electron into a magnetic field, you see
a separation of the states in energy; these correspond to
the "magnetic" quantum number, m_l, having values 1, 0, -1...
these numbers represent how much of the orbital momentum
can be projected along the z axis; if m_l = 1, then the
projection = +h; if m_l = -1, the projection = -h; if zero,
the projection = 0. With me so far?
As it happens, if you turn up the strength of the magnetic field,
you start to see each of the m_l states split into two states;
each of which corresponds to an electron having m_s = 1/2 or
m_s = -1/2...thus, if m_s = 1/2 there is an additional component
of angular momentum along the z axis = +h/2; if m_s = -1/2
there's an additional component = -h/2.
So, the electron behaves as if it has "built-in" angular momentum
which can add to, or subtract from, the orbital angular
momentum...and it adds/subtracts half as much angular momentum
projection (in units of h) as orbital angular momentum does.
Whew!! I hope that helped. Some short answer!!!
There is a very basic reason why the electron must be defined to have
spin angular momentum L_spin = 1/2. It is that there are two and only
two possible values for the component m_L of this angular momentum along
any given direction. The number of values of m_L for any value of L in
quantum mechanics is always equal to 2 times L plus 1, so L must be 1/2
when exactly two values of m_L are seen. The 2 L + 1 rule is familiar in
the "shell" picture of the atom: recall each subshell (s,p,d,f,...) with an
angular momentum L has 2 times L plus 1 subsubshells, into each of which
exactly two electrons can fit. Thus an s subshell, for which L = 0, has
only 2 times 0 + 1 = 1 subsubshell, which can accomodate two electrons. A
p subshell, for which L = 1, has 2 times 1 + 1 = 3 subsubshells (p_x, p_y,
p_z orbitals) into which fit a total of 2 times 3 = 6 electrons, and so on.
How does one know there are exactly 2 allowed m_L values? The Stern-
Gerlach experiment, which you may read about in nearly any quantum
mechanics textbook, measures the number of values of m_L directly.
Alternatively, each energy level of an atom with one valence electron (e.g.
hydrogen, sodium, potassium etc.) is split into two very closely spaced
energy levels ("a doublet") due to the different energies the two states of
different m_L values have in the magnetic field created by the orbiting
motion of the electron. Explaining the "fine structure" this produces in
the spectra of these atoms is exactly why Uhlenbeck and Goudsmit in 1925
suggested that an electron has spin angular momentum of value 1/2.
Ummm..isn't that the same thing I said, chris?
Or did I miss something?
After all, the Stern-Gerlach experiment sends the atom through
an inhomogenous magnetic field, eh? And withut that field,
the spin states are pretty much degenerate.
Oops, I guess I wasn't clear. I thought Mr. Johnson posed implicitly
the following question: can't you redefine the elementary unit of angular
momentum such that the spin of an electron is 1? It would still have half
the angular momentum of a p-electron, which in the new units would be 2
(d-electrons would have angular momentum 4, etc.) What I wanted to
emphasize is that this isn't possible, that the fraction cannot be scaled
away, and highlight the reason (the relationship between number of m_L
values and value of L), because it illustrates the unusual and powerful
counting rules that quantization gives you.
The spin energy levels are not exactly degenerate in the absence of a
magnetic field even for light atoms, because of the spin-orbit interaction.
You get emission from two P states to the ground state in sodium, for
example, giving the famous yellow doublet "sodium D line" at 5896 and 5890
angstroms that gives streetlights their color. I mentioned these doublets
mostly because they appear to have been a primary motivation for the
original proposal that the electron had spin 1/2.
It might be worth adding that the Zeeman splitting (of, e.g. the three p
states in hydrogen) that occurs when you turn on an external magnetic field
is usually *smaller* than the spin-orbit splitting. This is because of the
larger Lande g factor for spin magnetic moment versus orbital magnetic
moment, and because laboratory magnetic fields are weak by atomic
standards. A crude estimate of the magnetic field generated by an orbiting
1s electron in an H atom, which is what gives the small spin-orbit
splitting, is 20,000 gauss. (Earth's magnetic field is about 0.5 gauss.)
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Update: June 2012