Groups, rings, modules, vector spaces, Hilbert spaces

Consider a set of elements X= {a,b,c,...}and an operator *.

Group Theory requires four axioms. They are:
i) closure law: for any a and b in X, then a*b is also in X,
ii) associative law: a*(b*c) = (a*b)*c,
iii) Identity law: there is an element e such that a*e = e*a = a,
iv) Inverse law: for any a there is an element a-1 such that a * a-1= a-1*a = e. (For ordinary multiplication, the element a cannot be equal to zero).

If a group obeys the following rule: a*b = b*a then it is said to be commutative or abelian.

A ring has two operators such as * and +.
In addition to the above four axioms for each of the two operators, it has a distributive law:
a*(b+c) = a*b+a*c.

This is the essence of high school algebra.

A module A is like a set of primitive vectors. They are multiplied by a suitable ring R of scalars.

Consider a module A= {a,b,c...} and a ring R= {k,l,m...}
Definition: An R-module A is an additive commutative group together with a function that maps (k,a) into ka, subject to the following axioms:
i) k(a+b) =ka+kb
ii) (k+m)a= ka+ma
iii) (km)a=k(ma)
iv) ea=a, where e is the identity element

More explicitly, such a module is a left module because in forming ka the scalar k is written on the left of the module a. One can re-define a right module using the same procedure to define ak, with the scalar on the right.

Now if the ring is the whole field of real numbers or the complex numbers, then the module is a vector space.

Now a Hilbert space is defined as previously mentioned, with a slightly different language.(See Hilbert Space and Quantum Mechanics). Every inner product (.,.) between two vectors on a real or complex vector space H gives rise to a norm or the length of the vector, as follows:

x = sqrt (x,x)

We say that H is a Hilbert space if it is complete with respect to this norm. Completeness in this context means that any sequence of elements of the space converges to an element in the space, in the sense that the norm of differences approaches zero.

Hilbert Space And Quantum Mechanics

Consider the class of infinite real sequences (a1, a2...),
such that the sum of all (an)squared< infinite.

Example:
i) p=(1,1,...) does not converge, that is, 1+1+1+1+...does not converge to a point.
ii) q=(1,1/2,1/4,1/8,) converges.

Consider two points for which its infinite real sequence converges.
p=(an) and q =(bn)

Now if one defines the function d, also called a metric, as
d(p,q)=square root of the sum of all ¦ an - bn ¦squared

This space is called a Hilbert space.

Eigenvalues

We define eigenvectors and eigenvalues as follows. Let A be an n-by-n matrix of real number or complex numbers. We say that k is an eigenvalue of A with eigenvector v if v is not zero and

Av = kv, where k is a number (real or complex).

note: we have a matrix A -- often called an operator -- multiplying a vector v equals a number k times the same vector v.


The set of all k's is called the spectrum.

In the mathematically rigorous formulation of quantum mechanics, developed by Paul Dirac and John von Neumann, the possible states of a quantum mechanical system are represented by unit vectors (called "state vectors") residing in a Hilbert space. The exact nature of this Hilbert space is dependent on the system; for example, the state space for position and momentum states is the space of square-integrable functions, while the state space for the spin of a single electron is just the product of two complex planes. Each observable ( position, momentum, energy...)is represented by a densely-defined Hermitian linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate.

For example if H, called the hamiltonian, represents the linear operator of the observable quantity energy and lv> is its eigenvector. Then

iii)Hlv> = Elv>, where E, the energy of that system, is a number.

In one instance, as in the case of the hydrogen atom,

iv)H = p2 /2m + V, where p is momentum, m is the mass and V is the Coulomb potential energy.

When solving that equation by substituting iv) in iii) it yields a set of constants (Ei), which will be the energy spectrum.

It turns out that in this case, Ei has discrete values. This is one feature, discreteness, that makes quantum mechanics different from classical physics. Such observables as angular momentum, energy and spin of a particle in a bound state will have discrete values.

In every day life, a body -- take a car -- can have any energy value from zero to infinity. But an electron bounded in the hydrogen atom can sustain only discrete values. Certain values are forbidden to it.