Clas Blomberg - Physics of life-Elsevier Science (2007)

Chapter 6. Quantum mechanics 47
The formula Planck suggested for the dipole vibration energies is the famous one, which
we also considered in the chapter on radiation, Section 5E:
Energy Integer Frequency Basic constant
The basic constant is the famous h 6.626 10 34 J/sec. What is noteworthy with Planck’s
formula is that it contains two constants that provide direct relation to microscopic features,
this constant h and the constant k B , which normally is called Boltzmann’s constant but for
the first time was explicitly introduced at this stage. It can be mentioned here that Planck’s
formula provides a relation between atomic quantities and what could be directly measured.
As Boltzmann’s constant is directly given from the formula, this provides also a measure
of Avogadro’s constant, the number of molecules in what is called one mole.
I will point out here that one speaks about two kinds of frequencies, which can lead to
confusion. Frequencies stand for a periodic process, and the normal definition is to give the
number of periods per second, what also corresponds to the sort “hertz”. The expression above
shall be regarded in that way. In the formal developments of quantum mechanics or any
dynamics, one preferably speaks about an angular frequency. If one considers a rotating wheel,
the period frequency means number of full turns per second, while the angular frequency
refers to the change of angles per second. The latter should mean angles in radians, which
means that a full turn corresponds to an angle 2p. Thus, the angular frequency is equal to the
period frequency multiplied by 2p. If the energy formula is written with an angular frequency,
the constant h should be replaced by what is written as ħ h/2p 1.05 10 34 J/sec.
ħ may be the most relevant constant in the formal development.
It shall also be emphasised that Planck had no other motivation for his assumption of discrete
vibration energies other than that he could derive his important formula.
The next progress was due to Einstein. He is mainly famous for the relativity theories,
but his accomplishments with this kind of progress (for which he got the Nobel Prize) are
as important and pioneering. For the physics of phenomena that is close to our lives, as
what this book is about, they are more important. We may see here an attitude that the more
spectacular parts of physics (as relativity theory) arouse more interest and glamour than
those that appeal for the every day phenomena.
Certainly Einstein was worth two prizes.
One of his famous works in 1905 was about what is called photoelectric effect. To explain
that, he took the audacious step to assume that electromagnetic radiation is quantised.
Radiation with a creation frequency has a smallest energy, proportional to the frequency.
He thus gave a formula for radiation similar to that Planck has suggested for the vibrating
dipoles, indeed what we have in Eq. (5.22):
Energy of radiation Integer Frequency Fundamental constant h
This formula is often ascribed to Planck, but as said here, he had another approach. This is
Einstein’s formula and this formula is extremely important. Einstein may be famous for his