Operators corresponding to physical quantities from "summary" of The Principles of Quantum Mechanics by P. A. M. Dirac
Let us consider the concept of operators corresponding to physical quantities. In quantum mechanics, physical quantities such as position, momentum, energy, and angular momentum are represented by operators. These operators act on wave functions to yield the values of these physical quantities when measured. For example, the position operator corresponds to the physical quantity of position. When this operator acts on a wave function, it gives the position of the particle in space. Similarly, the momentum operator corresponds to the physical quantity of momentum. When this operator acts on a wave function, it gives the momentum of the particle. The eigenvalues of these operators represent the possible values that can be measured for the corresponding physical quantity. For instance, the eigenvalues of the position operator represent the possible positions that a particle can occupy. The eigenvalues of the momentum operator represent the possible momenta that a particle can have. Operators in quantum mechanics are represented by matrices or differential operators, depending on the context. These operators have specific properties that govern their behavior, such as commutation relations and Hermiticity. The commutation relation between two operators determines whether they can be measured simultaneously with arbitrary precision. If two operators commute, they can be measured simultaneously with arbitrary precision. If two operators do not commute, there is a limit to how precisely they can be measured simultaneously. Hermiticity is a property of operators that ensures that their eigenvalues are real and that their eigenvectors form an orthonormal basis. This property is crucial for the interpretation of quantum mechanical measurements.- Operators in quantum mechanics correspond to physical quantities and play a central role in the theory. They allow us to calculate the values of physical quantities and make predictions about the behavior of quantum systems. By understanding the properties of these operators, we can gain insight into the fundamental principles of quantum mechanics.
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