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Unformatted text preview: We can re-express the coefficient with only fundamental constants of nature: R := 1 2 ke 2 hc a = 1 2 ke 2 hc × 4 π 2 mc 2 ke 2 ( hc ) 2 = 2 π 2 mc 2 ( ke 2 ) 2 ( hc ) 3 which we can evaluate numerically: R = 2 π 2 × 5 . 11 × 10 5 eV × (1 . 44 eV · nm) 2 (1240 eV · nm) 3 = 1 . 097 × 10- 2 1 nm = 1 . 097 × 10 7 1 m which agrees precisely with the empirical value found by Balmer, R empirical . The coefficient R , the Rydberg constant, is given in Bohr’s model in term of only the fundamental constants of nature m, k, e, c , and has no degrees of freedom to fit with the data. We now understand the Balmer series as the transitions to the n = 2 energy level from higher ones. We can now use the prediction of the Bohr model even if n low = 2, and predict hydrogen spectral lines that haven’t been observed yet. This is the hallmark of new models and theories: explain what’s already known, and in addition make predictions of phenomena not yet observed. The Balmer lines are obtained as follows: Take the first transition to n = 2, i.e., from n high = 3 to n low = 2: 1 / 2 2- 1 / 3 2 = 1 / 4- 1 / 9 = 5 / 36, so that λ = 36 / (5 R ) = 656 . 3 nm which is the first spectral line in the Balmer series. Despite its great successes, the Bohr model suffers from serious problems. Despite many attempts, it was impossible to extend the Bohr model even to the next atom, namely helium. (It is possible to use it for hydrogen–like atoms, such as once ionized helium, say.) Moreover, the Bohr model gives an incorrect conceptual image of a solar–system like model, with electrons orbiting the nucleus. In Quantum Mechanics there is no such thing as a classical trajectory, because of the Heinsenberg Uncertainty Principle. The electron does not really move around the nucleus. The Schr¨odinger equation The Schr¨odinger equation is a first principle . Specifically, it means we cannot prove it. Just like Newton’s second law, that governs classical dynamics, the Schr¨odinger equation governs quantum dynamics. We can justify it, thereby make it plausible. But this is not a proof. At the end of the day, the sole arbiter is the confrontation of the predictions with experiment and observation. We do the same thing also with Newtons second law: there is no proof, but we can make plausibility arguments. The important thing is the agreement with experiment....
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This note was uploaded on 07/19/2011 for the course PH 113 taught by Professor Liorburko during the Spring '09 term at University of Alabama - Huntsville.
- Spring '09