5 0 8 D O C U M E N T 5 1 3 A P R I L 1 9 2 7 habits, lead us into temptation here, because it had hitherto always been possible for us to stay afloat between the realities, as long as we are prepared to sacrifice any customary wish. Precisely this circumstance, that the limitation of our concepts so exactly coincides with the limitation of our capacity to observe, permits, as Heisenberg stresses, the avoidance of contradictions. In connection with the question of light quanta, then, it is essential to set the well-known paradoxes in relation to the physical limitation of the concept of a monochromatic, plane train of waves. Purely geometrically, to this description of a wave train belongs a certain indefiniteness about the wavelength, which allows a description of a finite extension in the direction of propagation, just as an impreci- sion in the parallelism of the rays defines a limitation of the cross section of a wave train all this according to the well-known laws of optical determination of time and of imaging by optical instruments. If the uncertainty of the oscillation frequency is denoted by Δν, then the time needed by the waves to pass by a particular location is, of course, at least of order of magnitude . If, furthermore, Δλ denotes the uncertainty of the wavelength, then the order of magnitude of the minimal length of the train , and the minimal width , where ε is an angle that indicates the divergence of the light rays. This uncertainty in the geometric de- scription of waves, and consequently in the possibility to observe light quanta, thus stands in a peculiarly inverse relation to the precision with which energy and momentum of quanta are defined. Thus, we have and and , all in agreement with the general relation of the simultane- ous uncertainty of conjugate variables that, according to Heisenberg, is a direct consequence of the mathematical laws of quantum mechanics. The new formulation offers the possibility to harmonize the requirement of the conservation of energy with the consequences of the wave theory of light, in that, in keeping with the character of the account, the various aspects of the problem never emerge simultaneously. In this connection, I believe that even the paradoxes in the spectral decomposition of light emitted from a moving atom and observed through a slit vertical to the direction of motion, discussed by you in the Berlin Academy, can be circumvented.[4] If we first consider the problem from the point of view of wave theory, we find that the indefiniteness of the frequency, which Δt 1 Δν ------ -= Δx λ2 Δλ ------ -= Δy λ ε -- -= E hν = I h λ -- -= ΔEΔt hΔν 1- Δν ------ ⋅ h = = ΔIxΔx hΔλ λ2 ---------- λ2- Δλ ------ ⋅ h = = ΔIyΔy hε λ ----- λ ε -- - ⋅ h = =