Linear Vector Spaces in Quantum Mechanics
We have observed that most operators in quantum mechanics are linear operators.
This is fortunate because it allows us to represent quantum mechanical operators as
matrices and wavefunctions as vectors in some linear vector space. Since computers
are particularly good at performing operations common in linear algebra
(multiplication of a matrix times a vector, etc.), this is quite advantageous from a
practical standpoint.
In an
dimensional space we may expand any vector
as a linear combination of
basis vectors
(80)
For a general vector space, the coefficients
may be complex; thus one should not
be too quick to draw parallels to the expansion of vectors in threedimensional
Euclidean space. The coefficients
are referred to as the ``components'' of the state
vector
, and for a given basis, the components of a vector specify it completely. The
components of the sum of two vectors are the sums of the components.
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 Fall '10
 LauraChoudry
 Chemistry, Linear Algebra, Vector Space, Hilbert space, basis vectors

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