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The Hydrogen Spectrum
PHYS 1301 F99
Prof. T.E. Coan
version: 24 Nov ‘98
Introduction
In a previous laboratory experiment on diffraction, you should have noticed that
the light from the mercury discharge tube was composed of only three colors, or three
distinct wavelengths of light. Indeed, each element of the periodic table emits its own
characteristic wavelengths of light. The collection of the different distinct wavelengths
emitted by an atom is called the emission spectrum of the atom. Spectra composed from
white light but with distinct wavelengths absorbed or removed are called absorption
spectra. Each element’s unique emission spectra can be thought of as a kind of
“fingerprint” for the element. The element helium was first discovered in this manner
through the spectroscopic analysis of light from the Sun and was only later discovered in
natural gas deposits on Earth.
But why are distinct wavelengths observed? And why are they different for
particular elements? There is nothing distinct about the light from an incandescent
source. In an empirical study of the spectrum of hydrogen, it was discovered that the
precise frequencies and wavelengths of the light produced could be described by an
equation involving a constant and an integer. This equation was then expanded to
describe the entire spectrum of hydrogen, including the ultra-violet and the infrared
spectral lines. This equation is called the Rydberg equation:
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l
= R (- ),
where R is the Rydberg constant, and n1 and n2 are integers. That such a simple formula
describes the emission spectrum of hydrogen is nothing short of amazing. You don’t need
to know the speed of the electron or the proton, and you do not need to know how rapidly
the atom is moving. All you need to know are the constant R and two integers. The
presence of integers in this equation created a real problem for physicists until the
development of the quantum theory of the atom by Neils Bohr. Bohr's theory suggested
that the electron orbiting the nucleus could only have certain quantized angular momenta.
Ignoring the precise technical meaning of this quantity, the implication of it is that the
electron can orbit only at discrete radii and speeds around the nucleus and subsequently
can only possess certain discrete energies. The discrete radii can be labeled by integers
and the integers are the ones found in the Rydberg equation. The integer n1 is the
quantum number of the initial state or energy level, and n2 is the quantum number of
the final state.
Transitions of an electron from one orbit to another of smaller size produce a
burst of light. It is this light which you eventually see with your own eye. Since the
