Born’s discovery of the quantum-mechanical matrix calculus

J. Tran, Leanne Doughty, James K. Freericks

In 1925, Heisenberg, Born, and Jordan developed matrix mechanics as a strategy to solve quantum-mechanical problems. While finite-sized matrix formulations are commonly taught in quantum instruction, following the logic and detailed steps of the original matrix mechanics has become a lost art. In preparation for the 100th anniversary of the discovery of quantum mechanics, we present a modernized discussion of how matrix mechanics is formulated, how it is used to solve quantum-mechanical problems, and how it can be employed as the starting point for a postulate-based formulation of quantum-mechanics instruction. We focus on the harmonic oscillator to describe how quantum mechanics advanced from the Bohr–Sommerfeld quantization condition, to matrix mechanics, to the current abstract ladder-operator approach. We also describe a number of different activities that can be included in the quantum mechanics classroom to celebrate this centennial.

Robert J. Kolenkow

Carl Eckart

A formal calculus is developed which includes the Born and Jordan matrix dynamics, and also the remarkable quantum condition of Schroedinger. A method for the calculation of the matrices which is in close analogy to the classical Hamilton-Jacobi method of solving dynamical problems is explained. These results have been obtained independently by E. Schroedinger [Ann. d. Physik. 79, 734 (1926)].

Aurel Wintner

Altayev Namaz Karabalievich

From the very beginning of his development, Einstein was very critical of the main results of quantum mechanics. That is why in 1925, when the main papers on matrix mechanics were published, he wrote to his friend Besso [1] as follows: "The most interesting theoretical achievement of recent times is the Heisenberg-Born-Jordan theory of quantum states. A real witchcraft calculus in which infinite determinants (matrices) appear instead of Cartesian coordinates. It is highly witty and, due to its complexity, is insured against proving erroneous." In this connection I wish to say the following. A lot of time has passed since this thought was made. Therefore it makes sense to try to prove that the results of matrix mechanics actually contain defects. In the paper an attempt is made to prove that it is really so.