Lecture I:
Dirac Notation
•
To describe a physical system, QM assigns a
complex number (`amplitude’) to each distinct
available physical state.
–
(Or alternately: two real numbers)
–
What is a `distinct physical state’?
•
Consider a system with M distinct available
states
–
The 2M real numbers can be viewed as a vector in
an 2Mdimensional realvalued vector space
–
Or alternatively as a vector in an Mdimensional
complexvalued vector space
–
We will refer to this abstract vector space as
`Hilbert Space
’ or
`state space’
–
Any vector in this space corresponds to a possible
quantummechanical state. The number of such
quantum states is uncountable infinity
•
Just as
calculus
provides the mathematical basis
for Classical Mechanics, the mathematical basis
for QM is
linear algebra
–
Vectors, matrices, eigenvalues, rotations, etc… are
key concepts
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Various common vector notations:
1.
Vector notation:
–
Just a name, an abstraction that refers to
something physical
2.
Unit vectors:
–
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 Fall '09
 M.Moore
 Linear Algebra, mechanics, Vector Space, Complex number, Hilbert space, unit vectors, rr rr vectors

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