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[parent] quantum operator concept (Topic)

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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See Also: Observables and States, Lie algebras, wave function space, constants of the motion time dependence of the statistical distribution, commutator algebra

Also defines:  commutator, commute, linear operator

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Cross-references: work, domain, volume, quantum mechanics, gradient, vector, scalar product, Laplacian, algebraic, wave, operation, operator, function
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This is version 5 of quantum operator concept, born on 2009-03-11, modified 2010-02-14.
Object id is 588, canonical name is QuantumOperatorConcept.
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Physics Classification: 03.65.Ca (Formalism)
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