Function, Operator, Functional

FUNCTION
Functions is a prescription which maps one or more numbers to another number. For example: take a number and multiply it by itself: , or take two numbers and add them together: z = g(x, y) = x + y. Sometimes function does not have a value for some numbers, and only certain numbers can be used as an argument for a function. E.g., square root is only defined for nonnegative numbers (if you want to have a real number as a result).

OPERATOR
Operator (usually written with a hat, e.g., or in calligraphic style ) is a prescription which maps one function to another function. For example: take a function and square its value: , e.g., . Or calculate second derivative of a function versus x: , and . Nabla is a popular differenial operator in 3 dimensions. In cartesians it is:

It is used to calculate forces (which are vectors), i.e., gradients of potential energy: . Its square is called laplacian, and represents the sum of second derivatives:

The or appears in the kinetic energy operator. The prescription of forming the quantum mechanical operators, are called Jordan rules. For cartesian coordinate representation (coordinate space), they are obtained as follows:

1.

write a classical expression for the physical quantity and rearrange it in such a way, that everything depends either on coordinates, or momenta (e.g., if something depends on the component of velocity, , change it to ).

2.

replace coordinates with the operators of multuplying by the coordinate:

3.

replace components of momenta with their operators:

The Schrödinger equation (1) is an example of an eigenproblem, i.e., the equation in which an operator acts on a function and as a result it returns the same function multiplied by a constant. For some operators, there are no nontrivial solutions (trivial means: ), but for operators which correspond to some physical quantity of some physical system, these equations have solutions in principle, i.e., one can find a set of functions, and corresponding constants, which satisfy them. While these eigenproblems have solutions in principle, these equations may not be easy to solve. They frequently have the form of partial second order differential equations, or integro-differential equations, and they may not be analytically solved in general, some special cases may have an analytic solution (e.g., one particle, or two particles which interact in a special way).

FUNCTIONAL
Functional takes a function and provides a number. It is usually written with the function in square brackets as F[f] = a. For example: take a function and integrate it from to : . Note that the formula for the expectation value (3) is the total energy functional , since it takes some function and returns the value of energy for this .

Functionals can also have derivatives, which behave similarly to traditional derivatives for functions. The differential of the functional is defined as:

The functional derivatives have properties similar to traditional function derivatives, e.g.:

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