Conjugate variables

In quantum mechanics, the complementarity of two measurable quantities (observables) is understood as the property that the operators belonging to the associated observables have a commutator that takes the value $\pm \mathrm {i} \hbar$ . Here denotes $\hbar$ the reduced Planck quantum of action. Therefore, for two complementary operators and holds:

$[A,B]=AB-BA=\pm \mathrm {i} \hbar$

Due to the generalized Heisenberg uncertainty principle, it follows that both observables cannot be measured simultaneously with arbitrary accuracy, but that for the variance of their measurement always

$\sigma _{A}\sigma _{B}\geq {\frac {\hbar }{2}}$

is valid. In particular, if the first quantity is completely known, nothing at all can be said about the result of a quantum mechanical measurement of the second quantity (all possible measurement results are equally probable).

A well-known pair of mutually complementary observables are the location and momentum of an object. Since the classical trajectory is described by location and momentum, the complementarity of these two quantities means that the concept of a classical orbital motion in quantum mechanics must be abandoned.

The different components of angular momentum are not complementary observables in this sense: they also cannot be measured simultaneously, but the commutator of the components of the angular momentum operator is not a number, but an operator itself. Quantities which cannot be measured simultaneously with arbitrary precision, but which are not complementary, are called incommensurable.

Questions and Answers

Q: What are conjugate variables?

A: Conjugate variables are special pairs of variables (like x, y, z) that don't give the same result when you do a certain mathematical operation with them. This means that x*y is not equal to y*x.

Q: Who discovered conjugate variables?

A: Physicist Werner Heisenberg and his co-workers used equations studied in classical physics to describe and predict events from quantum physics. He discovered that the momentum (mass times velocity, represented by P) and position (represented by Q) are conjugate variables.

Q: What equation can be used to calculate the product of momentum and position?

A: The first equation could be used to find out the product of momentum and position: Y(n,n-b)=∑a p(n,n-a)q(n-a,n-b).

Q: What equation can be used to calculate the product of position and momentum?

A: The second equation could be used to calculate the product of position and momentum: Z(n,n-b)=∑a q(n,n-a)p(n-a, n-b).

Q: What did Max Born discover about conjugate variables?

A: Max Born found out that because P*Q is not equal to Q*P, the result of Q*P minus P*Q is not zero. He also found out that Q•P - P•Q = ih/2π.

Q: How does Planck's constant show up in quantum mechanics?

A: Planck's constant shows up in quantum mechanics a lot as it appears in Max Born's equation for calculating conjugate variable products; specifically as h/2π on one side of the equals sign.

Q: In what areas do conjugate variables have applications?

A: Conjugate variables have applications all over Physics, Chemistry and other areas of science.