At this stage, it becomes necessary to look somewhat closer at the
various particles involved in quantum mechanics themselves. The
analysis so far already used the fact that particles have a property
called mass, a quantity that special relativity has identified as
being an internal amount of energy. It turns out that in addition
particles have a fixed amount of
spin. Spin reflects itself, for
example, in how a charged particle such as an electron interacts with
a magnetic field.
To keep it apart from spin, from now the angular momentum of a
particle due to its motion will on be referred to as
orbital angular momentum. As was discussed in chapter
4.2, the square orbital angular momentum of a particle is
The square spin angular momentum of a particle is given by a similar
Particles with half integer spin are called “fermions.” For example, electrons, protons, and neutrons all three have spin and are fermions.
Particles with integer spin are called “bosons.” For example, photons have spin 1. The -mesons have spin 0 and gravitons, unobserved at the time of writing, should have spin 2.
The spin angular momentum in an arbitrarily chosen -direction is
The common particles, (electrons, protons, neutrons), can only have spin angular momentum or in any given direction. The positive sign state is called “spin up”, the negative one “spin down”.
It may be noted that the proton and neutron are not elementary particles, but are baryons, consisting of three quarks. Similarly, mesons consist of a quark and an anti-quark. Quarks have spin , which allows baryons to have spin or . (It is not self-evident, but spin values can be additive or subtractive within the confines of their discrete allowable values; see chapter 12.) The same way, mesons can have spin 1 or 0.
Spin states are commonly shown in “ket notation” as . For example, the spin-up state for an electron is indicated by and the spin-down state as . More informally, and are often used.
- Most particles have internal angular momentum called spin.
- The square spin angular momentum and its quantum number are always the same for a given particle.
- Electrons, protons and neutrons all have spin . Their spin angular momentum in a given direction is either or .
- Photons have spin one. Possible values for their angular momentum in a given direction are , zero, or , though zero does not occur in the direction of propagation.
- Particles with integer spin, like photons, are called bosons. Particles with half-integer spin, like electrons, protons, and neutrons, are called fermions.
- The spin-up state of a spin one-half particle like an electron is usually indicated by or . Similarly, the spin-down state is indicated by or .
Delta particles have spin . What values can their spin angular momentum in a given direction have?
Delta particles have spin . What is their square spin angular momentum?