A hydrogen atom is energized so that its electron is excited to the n = 3 energy state. How
many different frequencies of electromagnetic radiation could it emit in returning to its
ground state?Electromagnetic radiation is emitted whenever an electron drops to a lower energy state,
and the frequency of that radiation depends on the amount of energy the electron emits
while dropping to this lower energy state. An electron in the n = 3 energy state can either
drop to n = 2 or drop immediately to n = 1. If it drops to n = 2 , it can then drop once more
to n = 1. There is a different amount of energy associated with the drop from n = 3 to n =
2 , the drop from n = 3 to n = 1 , and the drop from n = 2 to n = 1 , so there is a different
frequency of radiation emitted with each drop. Therefore, there are three different
possible frequencies at which this hydrogen atom can emit electromagnetic radiation.
Wave-Particle Duality
The photoelectric effect shows that electromagnetic waves exhibit particle properties
when they are absorbed or emitted as photons. In 1923, a French graduate student
named Louis de Broglie (pronounced “duh BRO-lee”) suggested that the converse is also
true: particles can exhibit wave properties. The formula for the so-called de Broglie
wavelength applies to all matter, whether an electron or a planet:
De Broglie’s hypothesis is an odd one, to say the least. What on earth is a wavelength
when associated with matter? How can we possibly talk about planets or humans having
a wavelength? The second question, at least, can be easily answered. Imagine a person of
mass 60 kg, running at a speed of 5 m/s. That person’s de Broglie wavelength would be:
We cannot detect any “wavelength” associated with human beings because this
wavelength has such an infinitesimally small value. Because h is so small, only objects
with a very small mass will have a de Broglie wavelength that is at all noticeable.
De Broglie Wavelength and Electrons
The de Broglie wavelength is more evident on the atomic level. If we recall, the angular
momentum of an electron is. According to de Broglie’s formula, mv = h/
. Therefore,
The de Broglie wavelength of an electron is an integer multiple of , which is the length
of a single orbit. In other words, an electron can only orbit the nucleus at a radius where
it will complete a whole number of wavelengths. The electron in the figure below