3. Basic Ideas of Quantum Mechanics


Abstract

In this chapter the basic ideas of quantum mechanics are described and then a basic but very important example is 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 129. BBC/Penguin]

“So do not take the lecture too seriously, ..., but just relax and enjoy it.” [ibid.]

And it may be uncertain whether Niels 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. [Neils Bohr, reputed].

First of all, this chapter will throw out the classical picture of particles with positions and velocities. 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, also called the 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 eliminate the uncertainty in the position of a particle, its linear momentum explodes. If the position is precise, the linear momentum has infinite uncertainty. The same thing also applies in reverse. Neither position nor linear momentum can have an precise 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. It is associated with the total energy of the particle. The eigenvalues of the Hamiltonian describe the only possible values that the total energy of the particle can have.

The chapter will conclude by analyzing a simple quantum system in detail. It is a particle stuck in a pipe of square cross section. While relatively simple, this case describes some of the quantum effects encountered in nanotechnology. In later chapters, it will be found that this case also provides a basic model for such systems as valence electrons in metals, molecules in ideal gases, nucleons in nuclei, and much more.




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