Hilbert space

The most part in the study of Quamtum Physics is purely 
algebraic manipulations.

1. Definitions

Hilbert space is a vector space, that 
is a space in wich we operate with vectors of one, or 
two or three or more dimentions. It is different from other 
vector spaces by its properties. 
In Physics, Hilbert space is used mainly in quantum Mechanics 
by the means of operators acting on wave functions.

Properties of Hilbert space:

Hilbert space H works over the C set (complex numbers). It is 
then a complex vector space.
On this space, we define an inner product < , >  as follows:
- < f, g > = < g, f>* (* denotes the complex conjugate)
- < α f + βg , h > = α < f , h > + β < g , h >
- If f ≠ 0 then < f, f > > 0 
This space is complete with the definition of the norm:
- ||f|| = < f, f > 1/2

f, g, and h in H and α and β in C.

2. The 1-Operator

We will use the Dirac notation by calling <ψ| a bra, 
and |ψ> a ket, so: <ψ|ψ> is a braket.

Let's assume  {|φn >} a set of linear independent vectors in 
the Hibert space H; and this set is an orthogonal vectors set, 
that is < φm|φn >= δnm. This set constitutes the basis 
in the Hilebert space. Therefore, we can expand every arbitrary 
vector in H according this basis as:

|ψ> = Σ an |φn>  n from 1 to ∞

Thus:
<φm | ψ > =  Σ an <φm |φn>  n from 1 to ∞ 
= an δnm = am 

Hence, each vector can be represented as: 


|ψ> = Σ <φn | ψ > |φn > 
n from 1 to ∞


Or:
|ψ> = Σ |φn ><φn | ψ >   n from 1 to ∞ 

From this relation we get:

Σ |φn><φn|  = 1 
n from 1 to ∞
Called the 1-Operator


||ψ||2 = <ψ|ψ> = <ψ|Σ |φn ><φn | ψ > = 
 Σ <ψ|φn ><φn | ψ >  =  Σ <φn|ψ>* <φn | ψ > = 
 Σ |<φn|ψ>|2 
n from 1 to ∞ 

3. Linear Oprators in Hilbert space

3.1. Action of an operator:

An operator A in Hilbert space acts on a 
vector |ψ> as follows:
A|ψ> = |A ψ>  = |ψ'> 

A|ψ> = |A ψ>  = |ψ'> 

3.2. Adjoint operator:

A+ is the adjoint of A if:

<φ|A|ψ> = <A+φ|ψ>

<φ|A|ψ> = <A+φ|ψ>
A+ is the adjoint of A

3.3 Hermitian operator:

The operator A is hermitian if:

<φ|A|ψ> = <Aφ|ψ>

<φ|A|ψ> = <Aφ|ψ>
A is hermitian

4. Cauchy-Schwarz inequality)
|<x,y>| <= ||x||.||y||

In a real vector space, then for any real λ, we have:
0 <=  <x + λy, x+ λy> = <x,x +λy> +λ <y,x + λy> = 
<x,x> + 2λ<x,y> + λ2<y,y> = ||x||2 + 2λ<<x,y> + λ2||y||2 .

Minimizing ( deriving with respect to λ), we get: 

2<x,y> + 2λ||y||2 = 0, that is:

λ = - <x,y>/||y||2. 

Substituting this value in the inequality, yields: 

0 <=  ||x||2 - 2(<x,y>/||y||2)<x,y> + (<x,y>2/||y||4)||y||2

0 <=  ||x||2 - 2(<x,y>2/||y||2 + <x,y>2/||y||2

0 <=  ||x||2 - <x,y>2/||y||2 

<x,y>2 <=  ||x||2||y||2  

<x,y> <=  ||x||||y|| 

<x,y> <=  ||x||.||y||