Operators

An operator have the same fundamental meaning as a wavefunction in quantum mechanics, since an operator is attached to each experimentally observable value (such as impulse or energy).

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

It's a bit easier in quantum mechanics because we need only linear operators (look here for comparison) here. Operators are referred to as linear operators when the superposition principle holds true for such kind of operators, i.e. one can write down 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) there should be in general 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 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

It holds true for any constant number a.
If we establish special conditions on 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 down:

operator · eigenfunction = eigenvalue · eigenfunction A·Yn = An·Yn

The functions Yn are also referred to as eigenfunctions and values An are eigenvalues of this operator with the given boundary conditions. This equation is also called as 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 for such kind of operators, i.e. one can write down 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*A ydV = ò y A*y*dV or A = A†

And in general:

ò y*AφdV = ò f A*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