Simple Nature - Light and Matter

Beta decay example 27

When a free neutron undergoes beta decay, we have

n→p + e−+ν ̄.

All four of these particles have spin 1/2, so the angular momenta

go like

half-integer→half-integer + half-integer + half-integer,

which is possible, e.g., 1/2 = 3/ 2 − 5 /2 + 3/2. Because the

neutrino has almost no interaction with normal matter, it normally

flies off undetected, and the reaction was originally thought to be

n→p + e.

With hindsight, this is impossible, because we can never have

half-integer→half-integer + half-integer.

The reasoning holds not just for the beta decay of a free neu-

tron, but for any beta decay: a neutrino or antineutrino must be

emitted in order to conserve angular momentum. But historically,

this was not understood at first, and when Enrico Fermi proposed

the existence of the neutrino in 1934, the journal to which he first

submitted his paper rejected it as “too remote from reality.”

States in hydrogen, with spin

Taking electron spin into account, we need a total of four quan-

tum numbers to label a state of an electron in the hydrogen atom:n,

`,`z, andsz. (We omitsbecause it always has the same value.) The

symbols`and`zinclude only the angular momentum the electron

has because it is moving through space, not its spin angular mo-

mentum. The availability of two possible spin states of the electron

leads to a doubling of the numbers of states:

n= 1, `= 0, `z= 0, sz= +1/2 or− 1 / 2 two states

n= 2, `= 0, `z= 0, sz= +1/2 or− 1 / 2 two states

n= 2, `= 1, `z=−1, 0, or 1, sz= +1/2 or− 1 / 2 six states

...

A note about notation

There are unfortunately two inconsistent systems of notation

for the quantum numbers we’ve been discussing. The notation I’ve

been using is the one that is used in nuclear physics, but there is a

different one that is used in atomic physics.

936 Chapter 13 Quantum Physics