Heisenberg's entryway to matrix mechanics

Werner Heisenberg used the idea that since classical physics is correct when it applies to phenomena in the world of things larger than atoms and molecules, it must stand as a special case of a more inclusive quantum theoretical model. So he hoped that he could modify quantum physics in such a way that when the parameters were on the scale of everyday objects it would look just like classical physics, but when the parameters were pulled down to the atomic scale the discontinuities seen in things like the widely spaced frequencies of the visible hydrogen bright line spectrum would come back into sight.

The one thing that people at that time most wanted to understand about hydrogen radiation was how to predict or account for the intensities of the lines in its spectrum. Although Heisenberg did not know it at the time, the general format he worked out to express his new way of working with quantum theoretical calculations can serve as a recipe for two matrices and how to multiply them.[6]

Heisenberg's groundbreaking paper of 1925 neither uses nor even mentions matrices. Heisenberg's great advance was the "scheme which was capable in principle of determining uniquely the relevant physical qualities (transition frequencies and amplitudes)"[7] of hydrogen radiation.

After Heisenberg wrote his ground breaking paper, he turned it over to one of his senior colleagues for any needed corrections and went on a well-deserved vacation. Max Born puzzled over the equations and the non-commuting equations that Heisenberg had found troublesome and disturbing. After several days he realized that these equations amounted to directions for writing out matrices. Matrices were a bit off the beaten track, even for mathematicians of that time, but how to do math with them was already clearly established. He and a few colleagues took up the task of working everything out in matrix form before Heisenberg returned from his time off, and within a few months the new quantum mechanics in matrix form formed the basis for another paper.

When quantities such as position and momentum are mentioned in the context of Heisenberg's matrix mechanics, it is essential to keep in mind that a statement such as pq ≠ qp does not refer to a single value of p and a single value q but to a matrix (grid of values arranged in a defined way) of values of position and a matrix of values of momentum. So multiplying p times q or q times p is really talking about the matrix multiplication of the two matrices. When two matrices are multiplied, the answer is a third matrix.

Max Born saw that when the matrices that represent pq and qp were calculated they would not be equal. Heisenberg had already seen the same thing in terms of his original way of formulating things, and Heisenberg may have guessed what was almost immediately obvious to Born — that the difference between the answer matrices for pq and for qp would always involve two factors that came out of Heisenberg's original math: Planck's constant h and i, which is the square root of negative one. So the very idea of what Heisenberg preferred to call the "indeterminacy principle" (usually known as the uncertainty principle) was lurking in Heisenberg's original equations.

Paul Dirac decided that the essence of Heisenberg's work lay in the very feature that Heisenberg had originally found problematical — the fact of non-commutativity such as that between multiplication of a momentum matrix by a displacement matrix and multiplication of a displacement matrix by a momentum matrix. That insight led Dirac in new and productive directions.[8]

Intensities of the visible spectrum of a Hydrogen plasma obtained with Ocean Optics USB2000 low resolution spectrometer. Alpha, Beta, Gamma Balmer lines are visible, other lines are indistinguishable from the noise.

By means of an intense series of mathematical analogies that some physicists have termed "magical," Werner Heisenberg wrote out an equation that is the quantum mechanical analog for the classical computation of intensities. The equation below appears in his paper of 1925.[11][12] Its general form is as follows:

${\displaystyle C(n,n-b)=\sum _{a}^{}\,A(n,n-a)B(n-a,n-b)}$

This general format indicates that some term C is to be computed by summing up all of the products of some group of terms A by some related group of terms B. There will potentially be an infinite series of A terms and their matching B terms. Each of these multiplications has as its factors two measurements that pertain to sequential downward transitions between energy states of an electron. This type of rule differentiates matrix mechanics from the kind of physics familiar in everyday life because the important values are where (in what energy state or "orbital") the electron begins and in what energy state it ends, not what the electron is doing while in one or another state.

The formula looks rather intimidating, but if A and B both refer to lists of frequencies, for instance, all it says to do is perform the following multiplications and then sum them up:

Multiply the frequency for a change of energy from state n to state n-a by the frequency for a change of energy from state n-a to state n-b. and to that add the product found by multiplying the frequency for a change of energy from state n-a to state n-b by the frequency for a change of energy from state n-b to state n-c,
and so forth:
Symbolically that is:
f(n, n-a) * f(n-a,n-b)) +
f(n-a,n-b) * f(n-b,n-c) +
etc.

(According to the convention used, n-a represents a higher energy state than n, so a transition from n to n-a would indicate that an electron has accepted energy from an incoming photon and has risen to a higher orbital, while a transition from n-a to n would represent an electron falling to a lower orbital and emitting a photon.)

It would be very easy to do each individual step of this process for some measured quantity. For instance, the boxed formula at the head of this section gives each needed wavelength in sequence. The values calculated could very easily be filled into a grid as described below. However, since the series is infinite, nobody could do the entire set of calculations.

Heisenberg originally devised this equation to enable himself to multiply two measurements of the same kind (amplitudes), so it happened not to matter in which order they were multiplied. Heisenberg noticed, however that if he tried to use the same schema to multiply two variables, such as momentum, p, and displacement, q, then "a significant difficulty arises."[13] It turns out that multiplying a matrix of p by a matrix of q gives a different result from multiplying a matrix of q by a matrix of p. It only made a tiny bit of difference, but that difference could never be reduced below a certain limit, and that limit involved Planck's constant, h. More on that later. Below is a very short sample of what the calculations would be, placed into grids that are called matrices. Heisenberg's teacher saw almost immediately that his work should be expressed in a matrix format because mathematicians already were familiar with how to do computations involving matrices in an efficient way. (Since Heisenberg was interested in photon radiation, the illustrations will be given in terms of electrons going from a higher energy level to a lower level, e.g., n←n-1, instead of going from a lower level to a higher level, e.g., n→n-1)

${\displaystyle Y(n,n-b)=\sum _{a}^{}\,p(n,n-a)q(n-a,n-b)}$ (Equation for the conjugate variables momentum and position)

Matrix of p

Matrix of q

The matrix for the product of the above two matrices as specified by the relevant equation in Heisenberg's 1925 paper is:

Where:

A=p(n︎←n-a)*q(n-a︎←n-b)+p(n︎←n-b)*q(n-b︎←n-b)+p(n︎←n-c)*q(n-c︎←n-b)+.....

B=p(n-a︎←n-a)*q(n-a︎←n-c)+p(n-a︎←n-b)*q(n-b︎←n-c)+p(n-a︎←n-c)*q(n-c︎←n-c)+.....

C=p(n-b︎←n-a)*q(n-a︎←n-d)+p(n-b︎←n-b)*q(n-b︎←n-d)+p(n-b︎←n-c)*q(n-d︎←n-d)+.....

and so forth.

If the matrices were reversed, the following values would result:

A=q(n︎←n-a)*p(n-a︎←n-b)+q(n︎←n-b)*p(n-b︎←n-b)+q(n︎←n-c)*p(n-c︎←n-b)+.....
B=q(n-a︎←n-a)*p(n-a︎←n-c)+q(n-a︎←n-b)*p(n-b︎←n-c)+q(n-a︎←n-c)*p(n-c︎←n-c)+.....
C=q(n-b︎←n-a)*p(n-a︎←n-d)+q(n-b︎←n-b)*p(n-b︎←n-d)+q(n-b︎←n-c)*p(n-d︎←n-d)+.....

and so forth.

Note how changing the order of multiplication changes the numbers, step by step, that are actually multiplied.

• Aitchison, Ian J. R.; MacManus, David A.; Snyder, Thomas M. (2004). "Understanding Heisenberg's "magical" paper of July 1925: A new look at the calculational details". American Journal of Physics. 72 (11): 1370–1379. arXiv:quant-ph/0404009. Bibcode:2004AmJPh..72.1370A. doi:10.1119/1.1775243. S2CID 53118117. Direct download for Aitchison et al. on this subject.