Quantum_ Einstein, Bohr and the Great Debate About the Nature of Reality - Manjit Kumar [125]
Heisenberg chose not to send Bohr either a copy of the paper or the details of what he had done. It was a sign of how strained their relationship had become. 'I wanted to get Pauli's reactions before Bohr was back because I felt again that when Bohr comes back he will be angry about my interpretation', he explained later.54 'So I first wanted to have some support, and see whether somebody else liked it.' Five days after Heisenberg posted his letter, Bohr was back in Copenhagen.
Refreshed after his month-long vacation, Bohr dealt with pressing institute business before carefully reading the uncertainty paper. When they met to discuss it, he told a stunned Heisenberg that it was 'not quite right'.55 Bohr not only disagreed with Heisenberg's interpretation, but he had also spotted an error in the analysis of the gamma-ray microscope thought experiment. The workings of the microscope had nearly proved to be Heisenberg's undoing as a student in Munich. Only the intervention of Sommerfeld had secured his doctorate. Afterwards, a contrite Heisenberg had read up on microscopes, but he was about to discover that he still had some more to learn.
Bohr told Heisenberg it was wrong to place the origin of the uncertainty in the momentum of the electron in the discontinuous recoil it suffers due to the collision with the gamma-ray photon. What prohibits the precise measurement of the momentum of the electron is not the discontinuous and uncontrollable nature of the momentum change, Bohr argued, but the impossibility of measuring that change exactly. The Compton effect, he explained, allows the change in momentum to be calculated with pinpoint accuracy as long as the angle by which the photon is scattered after the collision through the aperture of the microscope is known. However, it is impossible to fix the point where the photon enters the microscope. Bohr identified this as the source of the uncertainty in the momentum of the electron. The electron's position when it collides with the photon is uncertain, since the finite aperture of any microscope limits its resolving power and therefore its ability to locate any microphysical object exactly. Heisenberg had failed to take all this into account, and there was worse to come.
Bohr maintained that a wave interpretation of the scattered light-quantum was indispensable for the correct analysis of the thought experiment. It was the wave-particle duality of radiation and matter that was at the heart of quantum uncertainty for Bohr as he linked Schrödinger's wave packets with Heisenberg's new principle. If the electron is viewed as a wave packet, then for it to have a precise, well-defined position requires it to be localised and not spread out. Such a wave packet is formed from the superposition of a group of waves. The more tightly localised or confined the wave packet is, the greater the variety of waves needed, the greater the range of frequencies and wavelengths involved. A single wave has a precise momentum, but it was an established fact that a group of superimposed waves of differing wavelengths cannot have a well-defined momentum. Equally, the more precisely defined the momentum of a wave packet, the fewer component waves it has and the more spread out it is, thereby increasing the uncertainty in its position. The simultaneously precise measurement of position and momentum is impossible, as Bohr showed that the uncertainty relations could be derived from the wave model of the electron.
Figure 12: (a) Position of the wave can be precisely determined but not the wavelength (and hence momentum); (b) wavelength can be measured accurately but not the position, since the wave is spread out
What troubled Bohr was that Heisenberg had adopted an approach based exclusively