2. Basic Ideas of Quantum Mechanics

Abstract

In this chapter the basic ideas of quantum mechanics are described and then two examples are worked out.

Before embarking on this chapter, do take note of the very sage advice given by Richard Feynman, Nobel-prize winning pioneer of relativistic quantum mechanics:

Do not keep saying to yourself, if you can possibly avoid it, “But how can it be like that?” because you will get “down the drain,” into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that. [Richard P. Feynman (1965) The Character of Physical Law]

And it may be uncertain whether Neils Bohr, Nobel-prize winning pioneer of early quantum mechanics actually said it to Albert Einstein, and if so, exactly what he said, but it may be the sanest statement about quantum mechanics of all:

Stop telling God what to do.

First of all, in this chapter the classical picture of particles with positions and velocities will be thrown out. Completely.

Quantum mechanics substitutes instead a function called the “wave function” that associates a numerical value with every possible state of the universe. If the “universe” that you are considering is just a single particle, the wave function of interest associates a numerical value with every possible position of that particle, at every time.

The physical meaning of the value of the wave function, or “quantum amplitude,” itself is somewhat hazy. It is just a complex number. However, the square magnitude of the number has a clear meaning, first stated by Born: The square magnitude of the wave function at a point is a measure of the probability of finding the particle at that point, if you conduct such a search.

But if you do, watch out. Heisenberg has shown that if you turn the position of a particle into certainty, its linear momentum explodes. If the position is certain, the linear momentum has infinite uncertainty and vice-versa. In reality neither position nor linear momentum can have a definite value for a particle. And usually other quantities like energy do not either.

Which brings up the question what meaning to attach to such physical quantities. Quantum mechanics answers that by associating a separate Hermitian operator with every physical quantity. The most important ones will be described. These operators act on the wave function. If, and only if, the wave function is an eigenfunction of such a Hermitian operator, only then does the corresponding physical quantity have a definite value: the eigenvalue. In all other cases the physical quantity is uncertain.

The most important Hermitian operator is called the “Hamiltonian,” and is associated with the total energy of the particle. The chapter will conclude by finding the eigenvalues of the Hermitian for two basic cases. These eigenvalues describe the only possible values that the total energy of the particle can have for those systems.

Because the two systems are so basic, much quantum mechanical analysis starts with one or the other. The first system is a particle stuck in a pipe of square cross section. While relatively simple, this case describes some of the quantum effects encountered in nano technology. In later chapters, it will be found that this case also provides a basic model for such systems as valence electrons in metals and ideal gases.

The second system is the quantum version of the simple spring mass system, the harmonic oscillator. It provides a model not just for vibrations of atoms in crystals, but also for the creation of the photons of electromagnetic radiation.