Stationary States
المؤلف:
Franklin Potter and Christopher Jargodzki
المصدر:
Mad about Modern Physics
الجزء والصفحة:
p 67
25-10-2016
618
Stationary States
In the Bohr model of the hydrogen atom, the angular momentum for the orbital motion of the electron of mass m at distance r is quantized in integral units of Planck’s constant h that is, assuming the proton position to be fixed, mvr = nh/2π, where n is an integer and v the electron velocity. Using mv = h/λ, de Broglie was able to derive Bohr’s quantization rule and nλ = 2πr. If f1 and f2 are the frequencies of the Bohr orbital motion of the electron in energy states E1 and E2, then if an electron jumps down from state 2 to state 1, why isn’t the energy of the emitted photon the difference energy hf1 – hf2?
Answer
In the Bohr model of the hydrogen atom, one would calculate the frequency f = 2πr/v of the electron’s orbital motion. The virial theorem states that twice the kinetic energy plus the potential energy add to zero, so mv2 = ke2 r, from which the electron’s frequency of orbit is f = n3h3/(4π2me4). The actual Bohr energy E = –2π2me4/(n2h2) is clearly a different quantity, and for an electron jump between two energy states, E2 – E1 ≠ hf2 – hf1.
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