QUANTUM PHYSICS FUNDAMENTALS - Chapter 3, Exercise 2 Solution
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Hydrogen Spectral Line: n=5 to n=2

PROBLEM
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Using the Rydberg formula, calculate the wavelength of light emitted
when a hydrogen electron falls from n = 5 to n = 2, and confirm it
matches the real, observed Balmer line at 434 nm.

SOLUTION
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Using the real Rydberg formula:

    1/lambda = R_H (1/n1^2 - 1/n2^2)

Substitute the given values (n1 = 2, n2 = 5):

    R_H = 1.097 x 10^7 m^-1

    1/lambda = (1.097 x 10^7) x (1/2^2 - 1/5^2)
    1/lambda = (1.097 x 10^7) x (0.25 - 0.04)
    1/lambda = (1.097 x 10^7) x 0.21
    1/lambda = 2.3037 x 10^6 m^-1

    lambda = 1 / (2.3037 x 10^6)
    lambda = 4.34 x 10^-7 m
    lambda = 434 nm

ANSWER: The wavelength is approximately 434 nm - confirming a match
with the real, observed Balmer line.

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WHY THIS WORKS AS AN ANSWER

The calculated value, 434 nm, lands exactly on one of the four real,
independently observed hydrogen spectral lines listed in the chapter.
This is a genuine, meaningful confirmation rather than a coincidence:
the Rydberg formula was itself originally derived by fitting real,
measured spectral data, so any transition it predicts between the
n=2 level and a higher level should, and does, correspond to an
actually observed line - exactly the same kind of agreement that let
Bohr's later theoretical model be tested and confirmed against
decades-old empirical measurements.