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

$\displaystyle \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

$\displaystyle 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

$\displaystyle \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;

  2. 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$ :

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

  2. the sum $ S = A + B$ of two operators $ A$ and $ B$ :

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

  3. 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]$ :

$\displaystyle [A,B] := AB - BA$ (1)

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

$\displaystyle AB = BA$

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$ ,

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

In other words

$\displaystyle \left [ \frac{\partial}{\partial x},f(x) \right ] = \frac{\partial f}{\partial x}$ (2)

and, in particular

$\displaystyle \left [ \frac{\partial}{\partial x},x \right ] = 1$ (3)

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

$\displaystyle \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.

References

[1] Messiah, Albert. "Quantum mechanics: volume I." Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62.

This entry is a derivative of the Public domain work [1].



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