Operators

In quantum mechanics, an operator has the same fundamental meaning as a wavefunction because an operator is attached to each experimentally observable value (such as impulse or energy).


Mathematically, an operator is a mathematical function that can be applied to another function. Here are a few examples of common operators:
x· , sin , exp , ∂/∂x , ∫ , Ö¾

We have a relatively easy time in quantum mechanics because we only need to use linear operators (look here for comparison). Operators are referred to as linear operators when the superposition principle holds true, i.e. one can write the following
  A (φ1+φ2)  =  Aφ1 + Aφ2      A (|1> + |2>)  =  A |1> + A |2> 

and 

A (c · φ)  =  c · Aφ      A (c |f>)  =  c A |f>

these rules should hold true for any function.
When applying the operator A on a  function f(x) we create a new function g(x):

A f(x) = g(x)

For instance, if we apply the operator of differential ∂/∂x on function sin x we will obtain new function cos x:

∂/∂x(sin x) = cos x

It is an important special case when the resultant function g(x) is proportional to f(x) i.e.: g(x) = A f(x). Here it will be the following:

A f(x) = A f(x)

For instance, when applying the differential operator ∂/∂x on the function eax one will obtain

 ∂/∂x(eax ) = a eax

This holds true for any constant value a.
If we establish special conditions for our function, or boundary conditions, then it is only possible to obtain definite functions Yn which satisfy our equation A f(x) = A f, i.e. we can write:
 

    operator · eigenfunction  =  eigenvalue · eigenfunction 

    A·Yn =  An·Yn

The functions Yn are also referred to as eigenfunctions and values An are calledeigenvalues of this operator with the given boundary conditions. This equation is also called the the eigenvalue equation and has great meaning in quantum mechanics since the basic principle of quantum mechanics states:
 

   The eigenvalues An are identical to experimentally measured values
 

Comparison table of classical quantities and quantum mechanical operators:
 
Classical Operators
p  =  (px, py, pz) p  =  h/i(∂/∂x,∂/∂y,∂/∂z)
E  =  p²/2m + V H  =  −h²/2mΔ + V
Lx  =  ypz − zpy Lx  =  h/i(y∂/∂z− z∂/∂y)
Ly  =  zpx − xpz Ly  =  h/i(z∂/∂x− x∂/∂z)
Lz  =  xpy − ypx Lz  =  h/i(x∂/∂y− y∂/∂x)
 

Operators features

Operators are referred to as linear operators when the superposition principle holds true, i.e. one can write the following
 

  A (φ1+φ2)  =  Aφ1 + Aφ2      A (|1> + |2>)  =  A |1> + A |2> 

and 

A (c · φ)  =  c · Aφ      A (c |f>)  =  c A |f>

for any function.

Since eigenvalues A are experimentally observed values, then A should be real, i.e. A = A*. It follows for quantum mechanical operators:

ò y*AydV  =  ò y A*y*dV     or     A  = A†

And in general:

ò y*AφdV  =  ò fA*y*dV

Moreover the operation hermitian conjugation, i.e. the transition from left integral to right integral, is assigned as "†".

Examples of linear and non-linear operators:

Operator (∂/∂x + a) and a is a constant:

 (∂/∂x + a) (f(x) + g(x))  =  ∂/∂xf + af + ∂/∂xg + ag   ®  linear operator

Operator  ∫dx   is also linera: ∫(f +g)dx  =  ∫fdx + ò gdx

Quadrature operator  (   )2     isn't linear operator, since  (f(x) +g(x))2 ¹  f2 + g2
 

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