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black_kitty
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What for do we need operators in QM. Where is the complete state description of a quantum object?
black_kitty said:What for do we need operators in QM. Where is the complete state description of a quantum object?
Operators in quantum mechanics are mathematical objects that represent physical observables, such as position, momentum, energy, and spin. They act on quantum states and produce measurable values when applied to those states.
Operators are essential in providing a complete description of a quantum state. Each operator has a corresponding set of eigenstates, and the eigenvalues of the operator represent the possible outcomes of a measurement on the state. By combining the eigenstates and eigenvalues of all relevant operators, a complete state description can be obtained.
Hermitian operators are self-adjoint, meaning that their eigenstates form an orthogonal set and their eigenvalues are real. Non-Hermitian operators do not have these properties, and their eigenvalues may be complex. In quantum mechanics, observables are represented by Hermitian operators while non-Hermitian operators are used to describe time evolution and other mathematical operations.
Yes, operators in quantum mechanics can be represented using matrices. The size of the matrix depends on the dimension of the Hilbert space in which the operator acts. For example, a position operator in one dimension would be represented by a 1x1 matrix, while a spin operator in three dimensions would be represented by a 3x3 matrix.
In classical mechanics, observables are represented by functions and operations are performed using simple algebra. In contrast, operators in quantum mechanics are represented by matrices and operations are performed using linear algebra. Additionally, the values of observables in classical mechanics are certain, while in quantum mechanics they are probabilistic and described by the eigenvalues of the corresponding operator.