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In quantum mechanics, conjugate variables are realized as pairs of observables whose operators do not commute. In conventional terminology, they are said to be incompatible observables . Consider, as an example, the measurable quantities given by position ( x ) {\displaystyle \left(x\right)} and momentum ( p ) {\displaystyle \left(p\right)} .
For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value. Physically meaningful observables must also satisfy transformation laws that relate observations performed by different observers in different frames of reference .
In quantum mechanics, a complete set of commuting observables (CSCO) is a set of commuting operators whose common eigenvectors can be used as a basis to express any quantum state. In the case of operators with discrete spectra, a CSCO is a set of commuting observables whose simultaneous eigenspaces span the Hilbert space and are linearly ...
[1]: 17 These observables play the role of measurable quantities familiar from classical physics: position, momentum, energy, angular momentum and so on. The dimension of the Hilbert space may be infinite, as it is for the space of square-integrable functions on a line, which is used to define the quantum physics of a continuous degree of freedom.
A certain number of good quantum numbers can be used to specify uniquely a certain quantum state only when the observables corresponding to the good quantum numbers form a CSCO. If the observables commute, but don't form a CSCO, then their good quantum numbers refer to a set of states. In this case they don't refer to a state uniquely.
a way of representing a state by several numbers, which corresponds to a complete set of commuting observables. A common example of quantum numbers is the possible state of an electron in a central potential: (,,,), which corresponds to the eigenstate of observables (in terms of ), (magnitude of angular momentum), (angular momentum in ...
A = observables (eigenvalues of operator) ... j = total angular momentum quantum number; m j = total angular momentum magnetic quantum number; Spin: ...
Some examples of two-dimensional electron systems achieved experimentally include MOSFET, two-dimensional superlattices of Helium, Neon, Argon, Xenon etc. and surface of liquid Helium. The presence of degenerate energy levels is studied in the cases of Particle in a box and two-dimensional harmonic oscillator , which act as useful mathematical ...