Heisenberg's matrix mechanics equivalent to Schrödinger's wave mechanics from "summary" of The Principles of Quantum Mechanics by P. A. M. Dirac
In the development of the quantum theory, two different formulations have been given, one by Heisenberg and the other by Schrödinger. These two formulations are connected with each other by a general principle of duality. Heisenberg's matrix mechanics is based on the use of matrices to represent physical quantities. In this formulation, the state of a system is described by a state vector, and the observables of the system are represented by matrices. The evolution of the system is governed by matrix equations, which give the time evolution of the state vector. On the other hand, Schrödinger's wave mechanics uses wave functions to describe the state of a system. In this formulation, the observables of the system are represented by differential operators, and the evolution of the system is governed by the Schrödinger equation. Despite the apparent differences between the two formulations, it has been shown that Heisenberg's matrix mechanics is equivalent to Schrödinger's wave mechanics. The equivalence between the two formulations is established by showing that the matrix elements of observables in Heisenberg's formalism can be related to the expectation values of corresponding operators in Schrödinger's formalism. This equivalence allows us to translate results obtained in one formulation to the other. For example, the time evolution of a state vector in Heisenberg's formalism can be related to the time evolution of a wave function in Schrödinger's formalism. The connection between Heisenberg's matrix mechanics and Schrödinger's wave mechanics highlights the underlying unity of quantum theory. Despite the different mathematical formalisms used, both formulations describe the same physical reality. The choice between the two formulations is often a matter of convenience, depending on the problem at hand. In practice, physicists often use a combination of both formulations to tackle different aspects of a quantum system. This flexibility reflects the richness and complexity of quantum theory, which continues to be a fruitful area of research and discovery in modern physics.
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