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

Chapter 6. Quantum mechanics 53
“van der Waals forces” and are important for the condensation of gases. In principle, that
effect is always present, but it is most important in cases where there are no direct electrostatic
interactions.
It is also possible to have atoms with an even number of electrons in coherent states, in
what is called “Bose–Einstein condensation”. In low-energy situations with little interaction
between the atoms, these can for a very special state where again, all atoms are the same,
thus behave as a unit, coherently. This coherence gives rise to special features, in particular
as they all move as a unity, we get special possibilities of motion and this relates to superconductivity
of some metals where there are currents with no resistance or special superfluid
properties of liquid helium at very low temperatures. The possibility to get a state of the
Bose–Einstein condensation character is suggested for photons in solids under special conditions,
in principle a state similar to a laser state.
We consider ideas about the relevance of the latter effects to phenomena of life in a
later part.
6C
The hydrogen atom
In order to illustrate the quantum mechanical concepts, I will show here some features of
the formalism of the treatment of the hydrogen atom. The intention is not to go in any depth
of quantum mechanics but rather to illustrate its kind of concepts and basic results. Parts
may here look quite advanced, but the importance lies in the main results.
Basically, quantum mechanics as classical mechanics works with energy and momentum
concepts. One regards the hydrogen atom as a heavy nucleus (proton) at the centre and an
electron in a suggested circular orbit around the proton. Classically, the problem is much
the same as the Newton (Kepler) description of planetary orbits around the sun. Thus consider
a circular motion of the electron around and at the distance r from a centre (the nucleus).
The electron velocity is v, at least classically along a circular orbit.
Kinetic energy: mv 2 /2 p 2 /2 m; in the last expression expressed by momentum p mv.
Electrostatic potential energy: q 2 /4p 0 r, where q is electron and proton charge.
Angular momentum: mvr.
As we shall show, it is meaningful to describe the quantum mechanical motion in these
terms and we can get numerical values of various quantities that are much the same as the
corresponding classical quantities. However, if we begin to investigate the features in more
detail, we find that it has very little to do with planetary motion.
The primary principle of quantum mechanics is that the main quantities, energies and
momentums are not, as in the classical theory, well-defined concepts with well-defined values,
but represented by what is called ‘operators’. This means quantities that work on the central
concept in this formalism, the wave function, with no correspondence in classical physics.
The wave function or rather the square of the wave function provides a probability distribution
function of the position or of the momentum. The wave function is generally symbolised
by the Greek letter c, and is a function of the space variables x, y, z. In the formalism,
a component of the momentum in the x-direction is represented by a derivative of the wave
function with respect to x. More precisely, the effect of the momentum component operator
on the wave function is given by dc/dx multiplied by the Planck’s constant ħ (1.05 1 34 ),