wavefuntion that
provides the amplitude to find the particle in state 
. Thus 
is the probability of
finding the particle in state .
wavefuntion that
provides the amplitude to find the particle in state 
. Thus 
is the probability of
finding the particle in state i.e. at a given
location x.
set of complex numbers
that provide the amplitude to find the particle in state 
. Thus 
is a real number that
represent the probability of finding the particle in state 
.
is the probability of
measuring the particle to be in state 1.
Normalization
For the general state
Lets imagine that there are a set of discrete locations
where the particle can be found so that the general state 
is a sum over the
complete set of possible locations. The
amplitude for each position state 
is labeled with and i.
This would probably be considered as poor notation but it says that if we have a
general state labeled by 
then we will label the
amounts of this state for each of the possible basis states 
as . This choice is based on the notation used when the states
are viewed as continuous.
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This is the 2-d analog. To illustrate the fact that we have two “associated” vectors I show the dot product in matrix form. This requires that you introduce a column and row vector representation of the vector. This is so trivial that it not mentioned when first introduced but we are extending the difference by requiring for the Hilbert space.
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This is the normalization condition. The key point is that it is equivalent to the
much simpler result obtained above for a simpler system
.
We are going to extend this way of building states to the
situation where we describe quantum states in terms of the amplitude for the
particle to be at each location in space. Thus rather that providing two
coefficients 
and 
we provide the function 
. This allows us to
say that the particle is spread over all space with contributions from each
location possibly different. To maintain
a reasonable normalization we require
This should be recognized is the same approach taken above
when assigning values
to the coefficients. The integral is required because of the
continuous nature of the locations x.
can be interpreted as a probability density.
let us examine the
What is the wave function [Position space representation].
Insert a complete set of states
For vector spaces one can always project a vector onto a compete basis.
For example:
