PlanetPhysics/Quantum Operator Concept

Consider the function $$\frac{\partial \Psi}{\partial t}$$, the derivative of $$\Psi$$ with respect to time; one can say that the operator $$\frac{\partial}{\partial t}$$ acting on the function $$\Psi$$ yields the function $$\frac{\partial \Psi}{\partial t}$$. More generally, if a certain operation allows us to bring into correspondence with each function $$\Psi$$ of a certain function space, one and only one well-defined function $$\Psi^{\prime}$$ of that same space, one says the $$\Psi^{\prime}$$ is obtained through the action of a given operator $$A$$ on the function $$\Psi$$, and one writes

$$ \Psi^{\prime} = A \Psi. $$

By definition $$A$$ is a linear operator if its action on the function $$\lambda_1 \Psi_1 + \lambda_2 \Psi_2$$, a linear combination with constant (complex) coefficients, of two functions of this function space, is given by

$$ A\left( \lambda_1 \Psi_1 + \lambda_2 \Psi_2 \right) = \lambda_1 \left( A \Psi_1 \right ) + \lambda_2 \left ( A \Psi \right ). $$

Among the linear operators acting on the wave functions

$$ \Psi := \Psi(\mathbf{r},t) := \Psi(x,y,z,t) $$

associated with a particle, let us mention:


 * 1) the differential operators $${\partial} / {\partial} x$$,$${\partial} / {\partial} y$$,$${\partial} / {\partial} z$$,$${\partial} / {\partial} t$$, such as the one which was considered above;


 * 1) the operators of the form $$f(\mathbf{r},t)$$ whose action consists in multiplying the function $$\Psi$$ by the function $$f(\mathbf{r},t)$$

Starting from certain linear operators, one can form new linear operators by the following algebraic operations:


 * 1) multiplication of an operator $$A$$ by a constant $$c$$:

$$ (cA)\Psi := c(A\Psi) $$


 * 1) the sum $$S = A + B$$ of two operators $$A$$ and $$B$$:

$$ S\Psi := A \Psi + B \Psi $$


 * 1) the product $$P=AB$$ of an operator $$B$$ by the operator $$A$$:

Note that in contrast to the sum, the product of two operators is not commutative. Therein lies a very important difference between the algebra of linear operators and ordinary algebra.

The product $$AB$$ is not necessarily identical to the product $$BA$$; in the first case, $$B$$ first acts on the function $$\Psi$$, then $$A$$ acts upon the function $$(B\Psi)$$ to give the final result; in the second case, the roles of $$A$$ and $$B$$ are inverted. The difference $$AB-BA$$ of these two quantities is called the commutator of $$A$$ and $$B$$; it is represented by the symbol $$[A,B]$$:

$$ [A,B] := AB - BA $$

If this difference vanishes, one says that the two operators commute:

$$AB = BA$$

As an example of operators which do not commute, we mention the operator $$f(x)$$, multiplication by function $$f(x)$$, and the differential operator $${\partial} / {\partial x}$$. Indeed we have, for any $$\Psi$$,

$$ \frac{\partial}{\partial x} f(x) \Psi = \frac{\partial}{\partial x} (f \Psi) = \frac{ \partial f}{\partial x} \Psi + f \frac{\partial \Psi}{\partial x} = \left ( \frac{\partial f}{\partial x} + f \frac{\partial}{\partial x} \right ) \Psi $$

In other words

$$ \left [ \frac{\partial}{\partial x},f(x) \right ] = \frac{\partial f}{\partial x} $$

and, in particular

$$ \left [ \frac{\partial}{\partial x},x \right ] = 1 $$

However, any pair of derivative operators such as $${\partial} / {\partial} x$$,$${\partial} / {\partial} y$$,$${\partial} / {\partial} z$$,$${\partial} / {\partial} t$$, commute.

A typical example of a linear operator formed by sum and product of linear operators is the Laplacian operator

$$ \nabla^2 := \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} $$

which one may consider as the scalar product of the vector operator gradient $$\nabla := \left( \frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right )$$, by itself.