3 4 8 D O C U M E N T 1 9 8 F E B R U A R Y 1 9 2 6 Colloquium in Berlin vorzutragen vielleicht dürfte ich Sie in jenen Tagen dann auch einmal besuchen. Ihr aufrichtig ergebener Werner Heisenberg ALS. [12 172]. There are perforations for a loose-leaf binder at the right margin of the document. [1]According to Jordan, he had been the sole author of the section on radiation fluctuations of Born, Heisenberg, and Jordan 1926 (see Doc. 98, note 3). [2]In a letter to Paul Ehrenfest of a week earlier (Doc. 194), Einstein also conflated commutation relations and components of angular momentum. [3]In the letter mentioned in the preceding note, Einstein questioned whether in matrix mechanics the energy matrix is invariant under canonical transformations and claimed that he had found a simple example suggesting that it is not. [4]The transformation given here can be written as and , with , and neglecting terms of the order , as can be verified with the help of the formula and the commutation relation . Since p and q are Hermitian operators, U is unitary and Q and P are Hermitian. [5]The terms of order λ in are all products of three matrices, each of which is either P or Q. Since the only nonvanishing elements of both P and Q change the energy level by ±1, the product of three matrices cannot have diagonal elements. Hence, to first order in λ, which is as far as Heisen- berg’s calculation goes, the canonical transformation does not change the energy spectrum of the har- monic oscillator. [6]The proofs of Born and Wiener 1926. [7]As in Heisenberg 1925, Heisenberg analyzes the rigid rotator in two dimensions (see also Doc. 132, note 10). The angular momentum is the momentum variable conjugate to the angle ϕ. [8]The radiation emitted by the system will depend on the time variation of the Cartesian coordi- nates and , or, more compactly, on the combination . Heisenberg calculates the time variation of the associated matrix for the more general case of , where τ is an integer. [9]Born, Heisenberg, and Jordan 1926, p. 563, eq. (6). [10]The derivation of the expression for given here can also be found in a letter from Heisenberg to Wolfgang Pauli of 27 January 1926 (see Pauli 1979, [117]). [11]In Heisenberg 1925, p. 886, the condition that the coordinate matrix elements connecting a state to lower states is introduced as a criterion to identify that state as the ground state (see also Doc. 112, note 4). [12]To find the ground state for the rigid rotator in two dimensions Heisenberg sets and . It then follows from the equation for ν derived earlier in this letter that , so that . On the (incorrect) assumption that only states of angular momentum differing by one unit of ƫ are connected by radiative transitions, Heisenberg concludes that only states with are allowed. Note that this implies that the energy is proportional to , whereas in Heisenberg 1925 a proportionality to is found (see also Doc. 112, note 2). See also the corrected treatment of radiative transitions in Dirac 1927b. [13]A reference to the quantum theory of radiation of Niels Bohr, Hendrik Kramers, and John Slater (see Doc. 49, note 5, for more details). [14]Max von Laue. Q U†qU = P U†pU = U iλqpq ƫ) exp( = λ2 eABe–A B A B] , [ ½[ A A B]] , [ , + + + = q p] , [ = H P Q) , ( p mr2ϕ · = x t) ( y t) ( x iy) + ( r exp[iϕ(t)] = iτϕ(t)] exp[ iτϕ(t)] exp[ ν 0= τ 1= H p) ( H p ƫ) ( = p ½ƫ = p τ ½+ ( = τ ½+ ( )2 τ ½+ ( )2 ¼+
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