I think there is a deeper question to the answer Martha seeks.
I don't see why Bohr's achievement is any more impressive than Balmer's
(1885), which also fit the phenomena astonishingly well. Balmer's
observation made it evident that hydrogen was the correct place to look for
a simple theory of atomic spectra. He might not have made his discovery had
it not been for Hartley's observation (1883) that one should look at the
wave number (the reciprocal of the wavelength), rather than the wavelength,
to find some order in spectra. The groundwork for Bohr was also greatly
helped by the discovery of Ritz (1908) that related regularities and
simplifications could be seen in the spectra of other elements. I could
continue to cite contributions without which Bohr probably couldn't have
made his, but I won't. I'll simply point out that some old books are worth
looking at. In this case I mean the Introduction to "The Theory of Atomic
Spectra" by Condon & Shortley, a book I haven't seen cited before in this
group, though that would have been unthinkable even twenty years ago.*
It seems to me what Martha is asking is "Why does Bohr's theory fit so well
when we now see that it is incorrect; Schr=F6dinger's is better?" I've tried
to indicate above that Bohr's theory was just one of a number of steps
along the path through theories with increasing ability to fit the
appearances. It should always be kept in the front of one's mind that none
of these theories is real in the sense that it is revealed truth. Every
theory, including the spectacularly accurate quantum electrodynamics, is
merely a contemporary mathematical best fit to the phenomena. We observe
that Nature appears to have underlying mathematical symmetries, and it
should now be completely unsurprising (though still worthy of wonder) that
solutions to mathematical relations can be isomorphic to numbers we obtain
directly from Nature.
I think that I have largely unasked Martha's question, but I, too, think it
is a good one because it makes me think. I believe students and teachers
should be encouraged to take part in thinking more than they do at present.
Leigh
*I note with satisfaction that this book, published in 1935, the year I was
born, is still in print and listed by Amazon at $57.95 in paperback. My
copy, printed in 1964, says 27s. 6d. net; US$3.95.