This article is a ‘‘sampler,’’ which shows how quantum mechanics may be presented to students in a way that makes apparent how natural quantum mechanics is as a description of the world. The mathematical machinery of Hilbert space, the idea of representing observables by operators, the Schrödinger equation, and the position-momentum uncertainty relation all follow from natural assumptions that students can readily accept. The basic ideas of quantum mechanics are developed from intuitive first principles to the point where one can connect with more traditional treatments of quantum mechanics.
This article is a ‘‘sampler,’’ which shows how quantum mechanics may be presented to students in a way that makes apparent how natural quantum mechanics is as a description of the world. The mathematical machinery of Hilbert space, the idea of representing observables by operators, the Schrödinger equation, and the position-momentum uncertainty relation all follow from natural assumptions that students can readily accept. The basic ideas of quantum mechanics are developed from intuitive first principles to the point where one can connect with more traditional treatments of quantum mechanics.
A type of mechanics will be presented that possesses some distinctive properties. On the one hand, its physical description & rules of operation are readily comprehensible & intuitively clear. On the other, it fully satisfies all observable predictions of non-relativistic quantum mechanics. Within it, particles exist at points in space, follow continuous, piecewise differentiable paths, and their linear momentum is equal to their mass times their velocity along their path. Yet the probabilities for position and momentum, conditioned on the state of the particle's environment, follow the rules of quantum theory. Indeed, all observable consequences of quantum theory are satisfied; particles can be entangled, have intrinsic spin, this spin is not local to the particle, particle identity can effect probabilities, and so forth. All the rules of quantum mechanics are obeyed, and all arise in a straightforward fashion. After this is established, connections will be drawn out between this type of mechanics and other types of quantum worlds; those that obey Bohmian mechanics, stochastic mechanics, the many worlds interpretation, and physical collapse. In the final section, a relativistic version of the mechanics will be presented.
Abstract It can easily be argued that a fully mature and complete knowledge of quantum mechanics should include historical, axiomatic, formal mathematical, and even philosophical background to the subject. However, for a student approaching quantum theory for the first time in a serious way, it can be the case that an approach utilizing his or her existing knowledge of, and intuition for, classical physics (including mechanics, wave physics, and electricity and magnetism) as well as emphasizing connections to experimental results can be the most pro- ductive. That, at least, is the point of view adopted in this text and can be illustrated by a focus on the following general topics:
Most engineers need a basic understanding of quantum mechanics, and for this the typical college introduction is enough. Such an introduction tends to be axiomatic but physically unenlightening. The rules of quantum mechanics (QM) are a fait accompli; justified because they work. This is a good beginning, but those needing to learn more face a special quantum barrier: more advanced texts often continue in the axiomatic tradition. Infinite-dimensional spaces, amplitude vectors, matrices, and operators in lieu of momentum and energy-these and other concepts may be introduced with little rationalization or intuition. To an inquiring student, QM can quickly become bizarre; something to be manipulated but not understood. This need not be so. Here the authors present one teaching solution. They show that many of QM's key ideas can be reasoned out from just a few physical assumptions with mathematics that is low in dimension, linear and accessible to many. Some results include a simple reason for believing probability amplitudes are more natural to use than probabilities, the duality of functions as vectors and vice versa, an example of how a matrix, an operator and a physical value might be related, the Heisenberg uncertainty principle and Schrodinger's equation.
It is shown that the assumption that nature acts to make optimum estimates of position maximally in error leads to the Schrödinger (energy) wave equation (SWE). In this way, the SWE follows from a simple statement of uncertainty. The approach grows out of probability estimation theory, in particular the principle of minimum Fisher information (or maximum Cramer–Rao bound). The minimized Fisher information turns out to be proportional to the mean kinetic energy of the particle. This approach is an attractive supplement to conventional ways of introducing quantum mechanics to students, since it avoids the use of imperfect physical models (such as vibrating strings) or the immediate need for complex momentum operators, and follows from a plausible assumption as to how nature works.
Quantum mechanics is our most successful physical theory. It underlies our very detailed understanding of atomic physics, chemistry, and nuclear physics, and the many technologies to which physical systems in these regimes give rise. Additionally, relativistic quantum mechanics is the basis for the standard model of elementary particles, which very successfully gives a partial unification of the forces operating at the atomic, nuclear, and subnuclear levels. However, from its inception the probabilistic nature of quantum mechanics, and the fact that “quantum measurements” in the orthodox formulation appear to require the intervention of non-quantum mechanical “classical systems,” have led to speculations by many physicists, mathematicians, and philosophers of science that quantum mechanics may be incomplete. Among the Founding Fathers of quantum theory, Einstein and Schrödinger were both of the opinion that quantum mechanics is in some way unsatisfactory, and this view has been amplified in more recent profound work of John Bell, among others. In an opposing camp, many others in the physics, mathematics, and philosophy communities have attempted to provide an interpretational foundation in which quantum mechanics remains a complete and self-contained system. Among the Founding Fathers, Bohr, Born, and Heisenberg maintained that quantum mechanics is a complete system, and a number of recent proposals have been made to improve upon or to provide alternatives to their “Copenhagen Interpretation.” The debate continues, and has spawned an enormous literature.
I have been interested in quantum mechanics or, synonymously, quantum physics since the very days when I first learned about them as a student. The initial impression was that of an immense mathematical beauty of the formalism of quantum mechanics. The second impression was the fantastic agreement of the theoretical predictions of quantum physics for experiments, and the third and somewhat surprising impression was that there was widespread disagreement about the interpretation of the new theory.