Wednesday, October 26, 2005

Hilbert Space And Quantum Mechanics

Consider the class of infinite real sequences (a1, a2...),such that the sum of all (an)2 < 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)= (Σ |an - bn|2)½

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 quantum mechanics, 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 | v > is its eigenvector. Then

iii) H | v > = E | v >, 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. However an electron bounded in the hydrogen atom can sustain only discrete values. Certain values are forbidden to it.
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