PlanetPhysics/Quantum Operator Concept

Consider the [[../Bijective/|function]] $\frac{\partial \mathrm{\Psi }}{\partial t}$, the derivative of $\mathrm{\Psi }$ with respect to time; one can say that the [[../QuantumOperatorAlgebra4/|operator]] $\frac{\partial }{\partial t}$ acting on the function $\mathrm{\Psi }$ yields the function $\frac{\partial \mathrm{\Psi }}{\partial t}$. More generally, if a certain [[../Cod/|operation]] allows us to bring into correspondence with each function $\mathrm{\Psi }$ of a certain function space, one and only one well-defined function ${\mathrm{\Psi }}^{\prime }$ of that same space, one says the ${\mathrm{\Psi }}^{\prime }$ is obtained through the action of a given operator $A$ on the function $\mathrm{\Psi }$, and one writes

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

By definition $A$ is a linear operator if its action on the function ${\lambda }_1{\mathrm{\Psi }}_1+{\lambda }_2{\mathrm{\Psi }}_2$, a linear combination with constant (complex) coefficients, of two functions of this function space, is given by

$A({\lambda }_1{\mathrm{\Psi }}_1+{\lambda }_2{\mathrm{\Psi }}_2)={\lambda }_1\left(A{\mathrm{\Psi }}_1\right)+{\lambda }_2\left(A\mathrm{\Psi }\right).$

Among the linear operators acting on the [[../CosmologicalConstant2/|wave]] functions

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

associated with a [[../Particle/|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 $f(𝐫,t)$ whose action consists in multiplying the function $\mathrm{\Psi }$ by the function $f(𝐫,t)$

Starting from certain linear operators, one can form new linear operators by the following [[../CoIntersections/|algebraic]] operations:

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

$(cA)\mathrm{\Psi }:=c(A\mathrm{\Psi })$

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

$S\mathrm{\Psi }:=A\mathrm{\Psi }+B\mathrm{\Psi }$

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 $\mathrm{\Psi }$, then $A$ acts upon the function $(B\mathrm{\Psi })$ 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 $\mathrm{\Psi }$,

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

In other words

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

and, in particular

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

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 [[../LaplaceOperator/|Laplacian]] operator

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

which one may consider as the [[../DotProduct/|scalar product]] of the [[../Vectors/|vector]] operator [[../Gradient/|gradient]] $\mathrm{\nabla }:=(\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z})$, by itself.

References

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

This entry is a derivative of the Public [[../Bijective/|domain]] [[../Work/|work]] [1].