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).
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 |
Classical | Operators |
p = (px, py, pz) | p = |
E = p²/2m + V | H = − |
Lx = ypz − zpy | Lx = |
Ly = zpx − xpz | Ly = |
Lz = xpy − ypx | Lz = |
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*AydV = ò y A*y*dV or A = A†
And in general:
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:
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