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Quantum operator methods
p. 1
Phys 3220 Tutorials
©
S. Pollock and the Physics Education Group
2008
University of Colorado, Boulder
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
ˆ
S
x
+
= +
,
ˆ
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.
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 Fall '08
 STEVEPOLLOCK
 mechanics

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