Memories of electrons in crystals olved an old puzzle by demonstrating the great reduction from the classical value in the specific heat of a degenerate Fermi gas and, further, had developed the new onsequences for the ratio of the electric and thermal conductivity of metals Except for the replacement of classical statistics and the inclusion of the spin however, Pauli and Sommerfeld both accepted the old ideas of Drude and Lorentz who treated the conduction electrons as an ideal gas of free particles. The high conductivity and reflectivity of metals of course strongly supported the assumption of very mobile electrons but I had never understood how anything like free motion could be even approximately true. After all, a metal wire with all its densely packed ions is far from being a hollow tube and as I started to think about it, I felt that the rst thing to be done in my thesis was to face this striking parado From the beginning I was convinced that the answer, if at all, could be found only in the wave nature of the electron, particularly since Heitler London as well as Hund had shown before that the valency electrons in a molecule were not confined to stay on a single atom. The fact that the periodicity of a crystal would be essential was somehow suggested to me by remembering a demonstration in elementary physics where many equal and equally coupled pendula were hanging at constant spacing from a rod and the motion of one of them was seen to 'migrate along the rod from pendulum to pendulum. Returning to my rented room one evening in early January, it was with such vague ideas in mind that I began to use pencil and paper and to treat the easiest case of a single electron in a one- dimensional periodic potential. By straight Fourier analysis I found to my delight that the solutions of the Schrodinger equation differed from the de broglie wave of a free particle only by a modulation with the period of the potential. The generali zation to three dimensions was obvious and while it certainly made me happy, the whole thing was so simple that I did not think it amounted to much of a discovery indeed, I saw later that wittmer and Rosenfeld had come before to the same conclusion and eventually I was even told by real scholars that something called Floquet's theorem had been known for a long time. But my findings were news to heisenberg, too when i told him about them the next day and in his usual optimism he thought that the problem was now essentially in the bag. Actually, it took me a further half a year before my thesis was finished and submitted for oublication in the Zeitschrift fair Physik. The following remarks about my thoughts during that time might be of some historical interest In the first place, I considered my original proof for the modulated wavefunctions to be not sufficiently elegant and since the application of group theory to quantum mechanics had just become fashionable, I presented it in that form rather than by the use of the more primitive Fourier method Next and more important, I did not think of conduction bands in the sense in which they are now commonly understood, although they clearly appeared in my result for a periodic potential with deep minima. This must have been the cause of my misconception of the essential difference between insulators and conductors later pointed out by A. H. wilson. In retrospect it seems rather obvious that closed shells have their analogue in filled bands. Instead, I thought that the differencelYI emorie8 of electron8 in crY8tal8 25 solved an old puzzle by demonstrating the great reduction from the classical value in the specific heat of a degenerate Fermi gas and, further, had developed the new consequences for the ratio of the electric and thermal conductivity of metals. Except for the replacement of classical statistics and the inclusion of the spin, however, Pauli and Sommerfeld both accepted the old ideas of Drude and Lorentz, who treated the conduction electrons as an ideal gas of free particles. The high conductivity and reflectivity of metals of course strongly supported the assumption of very mobile electrons but I had never understood how anything like free motion could be even approximately true. Mter all, a metal wire with all its densely packed ions is far from being a hollow tube and as I started to think about it, I felt that the first thing to be done in my thesis was to face this striking paradox. From the beginning I was convinced that the answer, if at all, could be found only in the wave nature of the electron, particularly since Heitler & London as well as Hund had shown before that the valency electrons in a molecule were not confined to stay on a single atom. The fact that the periodicity of a crystal would be essential was somehow suggested to me by remembering a demonstration in elementary physics where many equal and equally coupled pendula were hanging at constant spacing from a rod and the motion of one of them was seen to 'migrate' along the rod from pendulum to pendulum. Returning to my rented room one evening in early January, it was with such vague ideas in mind that I began to use pencil and paper and to treat the easiest case of a single electron in a one￾dimensional periodic potential. By straight Fourier analysis I found to my delight that the solutions of the Schrodinger equation differed from the de Broglie wave of a free particle only by a modulation with the period of the potential. The generali￾zation to three dimensions was obvious and while it certainly made me happy, the whole thing was so simple that I did not think it amounted to much of a discovery; indeed, I saw later that Wittmer and Rosenfeld had come before to the same conclusion and eventually I was even told by real scholars that something called 'Floquet's theorem' had been known for a long time. But my findings were news to Heisenberg, too, when I told him about them the next day and in his usual optimism he thought that,the problem was now essentially in the bag. Actually, it took me a further half a year before my thesis was finished and submitted for pUblication in the Zeitschrift fur Physik. The following remarks about my thoughts during that time might be of some historical interest. In the first place, I considered my original prooffor the modulated wavefunctions to be not sufficiently elegant and since the application of group theory to quantum mechanics had just become fashionable, I presented it in that form rather than by the use of the more primitive Fourier method. Next and more important, I did not think of conduction bands in the sense in which they are now commonly understood, although they clearly appeared in my result for a periodic potential with deep minima. This must have been the cause of my misconception of the essential difference between insulators and conductors, later pointed out by A. H. Wilson. In retrospect it seems rather obvious that closed shells have their analogue in filled bands. Instead, I thought that the difference