Lecture I:
Dirac Notation
•
To describe a physical system, QM assigns a
complex number (`amplitude’) to each distinct
possible physical state.
–
i.e. 2 real numbers per state
–
In Classical mechanics, a system must have a single
state, while in QM arbitrary superpositions are
allowed
•
Consider a system with M distinct possible states
–
The 2xM real numbers can be viewed as a vector in
an 2xMdimensional 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
•
Just as
calculus
is necessary to do Classical
Mechanics, the mathematical basis for QM is
linear algebra
–
Vectors, matrices, eigenvalues, rotations, etc… are
key concepts
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 '08
 MichaelMoore
 Linear Algebra, mechanics, Vector Space, Complex number, Hilbert space, unit vectors

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