Conjugate variables

Conjugate variables are special pairs of variables that do not commute, If two variables, let us call them m and n, commute, then mn = nm. If two variables, let us call them p and q do not commute, then pq ≠ qp.

One of the most well known pairs of conjugate variables was discovered in 1925 by Werner Heisenberg and his colleagues. Heisenberg adapted equations from classical physics to model quantum events. In so doing he produced an equation that could be used to calculate the product momentum and position:

$Y(n,n-b)=\sum _{a}^{}\,p(n,n-a)q(n-a,n-b)$ (n, n-a, n-b, etc. refer to energy levels of an electron in a hydrogen atom)

The same kind of equation could be used to calculate the product of position and momentum:

$Z(n,n-b)=\sum _{a}^{}\,q(n,n-a)p(n-a,n-b)$

Heisenberg realized right away that these two results were going to be different, and it bothered him. He was very tired after making his breakthrough work, which he wrote up as a paper for publication. So after he gave the paper to Max Born for editing and then to send off to the publisher, Heisenberg went on vacation. Born realized that these strange equations were blueprints for making a matrix for momentum and a matrix for position and then multiplying the two. He also saw that not only would multiplying the matrix p by the matrix q give a different answer matrix than multiplying the matrix q by the matrix p, but that the difference between the two could be computed from the other equations that were involved in Heisenberg's paper, and "Immediately there stood before me the strange formula:

${QP-PQ={\frac {ih}{2\pi }}}$."

[The symbol Q is the matrix for displacement, P is the matrix for momentum, i stands for the square root of negative one, and h is Planck's constant.]

References

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