  In July 1925 Heisenberg published a paper [Z. Phys. 33, 879-893 (1925)] which
ended the period of `the Old Quantum Theory' and ushered in the new era of
Quantum Mechanics. This epoch-making paper is generally regarded as being
difficult to follow, perhaps partly because Heisenberg provided few clues as to
how he arrived at the results which he reported. Here we give details of
calculations of the type which, we suggest, Heisenberg may have performed. We
take as a specific example one of the anharmonic oscillator problems considered
by Heisenberg, and use our reconstruction of his approach to solve it up to
second order in perturbation theory. We emphasize that the results are
precisely those obtained in standard quantum mechanics, and suggest that some
discussion of the approach - based on the direct computation of transition
amplitudes - could usefully be included in undergraduate courses in quantum
mechanics.
