- #1
v_pino
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Homework Statement
In Bohr’s hydrogen atom, the frequency of radiation for transitions between adjacent orbits
is νBohr = (En+1−En)/h. Classically, a charged particle moving in a circular orbit radiates
at the frequency of the motion, νorbit = v/2πr. Find the ratio νBohr/νorbit for the Bohr
orbits as a function of n. Evaluate the ratio for n = 1, 2, 5, 10, 100, and 1000, and thus show that νBohr/νorbit → 1 as n → ∞. This is an example of Bohr’s correspondence principle—that in the limit of large quantum numbers quantum-mechanical results agree with classical results.
Homework Equations
[tex] \nu_{Bohr}=(E_{n+1}-E_n)/h [/tex]
[tex] \nu_{orbit}=\frac{v}{2 \pi r} [/tex]
[tex] E_n= \frac{Z^2}{n^2}E_1 [/tex]
The Attempt at a Solution
From equation 3, can I say [itex] E_{n+1}= \frac{Z^2}{(n+1)^2}E_1 [/itex] ?
Then doing the algebra gives me
[tex] \frac{\nu_{Bohr}}{\nu_{Orbit}}=(\frac{1}{(n+1)^2}-\frac{1}{n^2} )\frac{Z^2E_1}{h} \frac{2 \pi r}{v} [/tex]
Does that seem correct?
However, the values from the brackets of [itex] \frac{1}{(n+1)^2}-\frac{1}{n^2} [/itex] gives me -3/4, -5/36, ... which don't tend to 1.