At this point in our course we have developed our formal formulation of quantum mechanics directly from the de Broglie Hypothesis, verified that our formulation is consistent will all previously known experimental information, such as F=ma from classical mechanics, and even gone on to make predictions of new phenomena using our formulation. It is important for the student to understand that our formulation of quantum mechanics is not unique, that there is more than one valid formulation of quantum mechanics. In fact there are three distinct formulations of quantum mechanics in common use today.
Each of these formulations is quite different in terms of the physics which it emphasizes. Given problems are usually most directly attacked within one of the formulations. Although these three formulations are quite different, they are all known to be equivalent. All three formulations include the following features a) events are viewed as probabilistic, b) the probability for the occurrence of an event is given by the complex magnitude squared of the corresponding quantum amplitude, and c) a prescription is given for computing the quantum amplitude associated with a given event. It is only in how the amplitudes are computed that the formulations differ. We know that all three formulations are equivalent because we can show that they all give the same quantum amplitudes.
The formulation which we have built up in this course is known as Wave Mechanics. This formulation, where amplitudes are computed by solving differential equations is due to Schrödinger. This formulation is most convenient when working with potentials of arbitrary form in one dimension. The second formulation is Matrix Mechanics, and is due to Heisenberg. We have had some exposure to this formulation in our treatment of the simple harmonic oscillator (SHO) where we used operator algebra rather than differential equations to determine the allowed states. In practice research physicists do not use matrix mechanics as originally formulated by Heisenberg. Rather, we usually work with operator algebra using a special notation developed by Dirac, known as ``bra-ket'' notation. This operator/bra-ket formulation of quantum mechanics is most convenient when working with problems with special symmetries such as the Simple Harmonic Oscillator, problems involving angular momentum and problems involving the spin of particles, and when dealing with multiple particle systems. In 8.05 and 8.059, you will be exposed heavily to this formulation. The third and final of the three equivalent formulations is due to Feynman. In this formulation, one computes amplitudes for events by summing over all ways in which the event may happen. In practice, this is done using Feynman diagrams. This formulation is extremely powerful and is mostly used in quantum field theory and in the study of many-particle systems such as those found in condensed matter physics. You will use this formulation in 8.323, 8.512 and 8.334. Table 1 summaries the basic features of the three formulations.
Table 1: Independent, equivalent Formulations of Quantum
Mechanics