Matrix Mechanics - Further Discussion

Further Discussion

When it was introduced by Werner Heisenberg, Max Born and Pascual Jordan in 1925, matrix mechanics was not immediately accepted and was a source of great controversy. Schrödinger's later introduction of wave mechanics was favored.

Part of the reason was that Heisenberg's formulation was in a strange new mathematical language, while Schrödinger's formulation was based on familiar wave equations. But there was also a deeper sociological reason. Quantum mechanics had been developing by two paths, one under the direction of Einstein and the other under the direction of Bohr. Einstein emphasized wave-particle duality, while Bohr emphasized the discrete energy states and quantum jumps. DeBroglie had shown how to reproduce the discrete energy states in Einstein's framework--- the quantum condition is the standing wave condition, and this gave hope to those in the Einstein school that all the discrete aspects of quantum mechanics would be subsumed into a continuous wave mechanics.

Matrix mechanics, on the other hand, came from the Bohr school, which was concerned with discrete energy states and quantum jumps. Bohr's followers did not appreciate physical models which pictured electrons as waves, or as anything at all. They preferred to focus on the quantities which were directly connected to experiments.

In atomic physics, spectroscopy gave observational data on atomic transitions arising from the interactions of atoms with light quanta. The Bohr school required that only those quantities which were in principle measurable by spectroscopy should appear in the theory. These quantities include the energy levels and their intensities but they do not include the exact location of a particle in its Bohr orbit. It is very hard to imagine an experiment which could determine whether an electron in the ground state of a hydrogen atom is to the right or to the left of the nucleus. It was a deep conviction that such questions did not have an answer.

The matrix formulation was built on the premise that all physical observables are represented by matrices whose elements are indexed by two different energy levels. The set of eigenvalues of the matrix were eventually understood to be the set of all possible values that the observable can have. Since Heisenberg's matrices are Hermitian, the eigenvalues are real.

If an observable is measured and the result is a certain eigenvalue, the corresponding eigenvector is the state of the system immediately after the measurement. The act of measurement in matrix mechanics 'collapses' the state of the system. If you measure two observables simultaneously, the state of the system should collapse to a common eigenvector of the two observables. Since most matrices don't have any eigenvectors in common, most observables can never be measured precisely at the same time. This is the uncertainty principle.

If two matrices share their eigenvectors, they can be simultaneously diagonalized. In the basis where they are both diagonal, it is clear that their product does not depend on their order because multiplication of diagonal matrices is just multiplication of numbers. The Uncertainty Principle then is a consequence of the fact that two matrices A and B do not always commute, i.e., that AB − BA does not necessarily equal 0. The famous commutation relation of matrix mechanics:

shows that there are no states which simultaneously have a definite position and momentum. But the principle of uncertainty (also called complementarity by Bohr) holds for most other pairs of observables too. For example, the energy does not commute with the position either, so it is impossible to precisely determine the position and energy of an electron in an atom.