Quantum operator methods
p. 1
SETUP:
Consider a quantum particle with some
new properties
that we can measure.
We won’t talk much about the physics of these measurements yet, but the formalism of quantum
mechanics will teach us a great deal, just from operator methods!
Consider a hermitian operator S
x
which yields only 2 possible measurement outcomes,
+1 or 1.
(It will represent the measurement of “the component of spin in the x direction”. For now think of it
as a measurement of some property of a particle which can be “rightwards” or “leftwards”…)
With only two possible measurement outcomes, S
x
has
only
two (orthonormal) eigenvectors.
People get tired of writing out complicated “ket names”, so a simple, common notation is
S
x
+
= +
,
S
x

=  1

I suppose you might refer to the +> state as a “spin right” state, and > as “spin left”, does that seem
reasonable to you?
Stare at these two eigenequations and make sure you, and your group, understand the notation:
which symbols are the eigenvectors here, what are the eigenvalues, in those equations?
What can you say about, e.g.
 +
? (Explain)
Now consider a second observable, with corresponding Hermitian operator, S
z.
This operator is NOT the same as S
x,
although it does have the same spectrum.
Since the eigenvectors of S
z
are different from those of S
x,
we need to give them a different name.
Here, people conventionally name the ket with a
symbol
:
S
z
=
,
S
z
=  1
(Since S
z
measures the vertical, or z, component of spin, I suppose you might refer to the 
↑
> state as a
“spin up”, and 
↓
> as “spin down” particle, does that seem reasonable to you?)