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 components in the x, y and zdirections respectively.
T
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 often called V n(C), or V n(R) if only real numbers are us
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 other cutoff procedures can
give useful insights. For examp
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 basis in the
space. There is a straightforward procedu
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!) defined by:
For an n-dimensional vector space with an
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 an odd
permutation. (Note: any permutation can be writ
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-dimensional vectors,
Following Dirac, we write the inner
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 .
Eigenkets and eigenvalues are of central importance in q
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 defined by:
Writing the two rows of the matrix as vectors:
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 same formal procedure with the (
) Fourier transforms, we
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 having as
elements the complex conjugates of the ket elemen