What is a Vector Space?
The prototypical vector space is of course the set of real vectors in ordinary three-
dimensional space, these vectors can be represented by trios of real numbers
measuring the
Vector Space Dimensionality
The vectors
are linearly independent if
implies
A vector space is n-dimensional if the maximum number of linearly independent
vectors in the space is n.
Such a space is oft
It is straightforward to verify the following properties from the definition as a limit of
a Gaussian wavepacket:
Principal Value Integral
There is no unique way to define the delta function, and othe
Constructing an Orthonormal Basis: the Gram-Schmidt Process
To have something better resembling the standard dot product of ordinary three
vectors, we need
that is, we need to construct an orthonormal
Linear Operators
A linear operator A takes any vector in a linear vector space to a vector in that
space,
and satisfies
with c1, c2 arbitrary complex constants.
The identity operator I is (obviously!)
The generalization from
operators is that
becomes:
where ijkp is summed over all permutations of 132n, and the symbol is zero if
any two of its suffixes are equal, +1 for an even permutation and 1 for
Inner Product Spaces
The vector spaces of relevance in quantum mechanics also have an operation
associating a number with a pair of vectors, a generalization of the dot product of two
ordinary three-d
Eigenkets and Eigenvalues
If an operator A operating on a ket
gives a multiple of the same ket,
then
is said to be an eigenket (or, just as often, eigenvector, or eigenstate!)
of A with eigenvalue .
E
Determinants
We review in this section the determinant of a matrix, a function closely related to the
operator properties of the matrix.
Lets start with
matrices:
The determinant of this matrix is def
Diracs Delta Function
Now we have taken both N and L to infinity, what has happened to our function
? Remember that our procedure for finding
gave the equation
and from this we found
Following the sam
The Adjoint Operator and Hermitian Matrices
As weve discussed, if a ket
in the n-dimensional space is written as a column
vector with n (complex) components, the corresponding bra is a row vector havi