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In July 1925 Heisenberg published a paper [Z. Phys. 33, 879-893 (1925)] whichended the period of `the Old Quantum Theory' and ushered in the new era ofQuantum Mechanics. This epoch-making paper is generally regarded as beingdifficult to follow, perhaps partly because Heisenberg provided few clues as tohow he arrived at the results which he reported. Here we give details ofcalculations of the type which, we suggest, Heisenberg may have performed. Wetake as a specific example one of the anharmonic oscillator problems consideredby Heisenberg, and use our reconstruction of his approach to solve it up tosecond order in perturbation theory. We emphasize that the results areprecisely those obtained in standard quantum mechanics, and suggest that somediscussion of the approach - based on the direct computation of transitionamplitudes - could usefully be included in undergraduate courses in quantummechanics.Comment: 24 pages, no figures, Latex, submitted to Am. J. Phy

Understanding Heisenberg’s “magical” paper of July 1925: A new look at the calculational details