Complete set of commuting observables
Encyclopedia
In quantum mechanics
, a complete set of commuting observables (CSCO) is a set of commuting operators whose eigenvalues completely specify the state of a system .
For example, in the case of the hydrogen atom
, the Hamiltonian
H, the angular momentum
L and its projection
Lz along any arbitrary z axis form a CSCO (if one ignores the spin
of the proton
and of the electron
as well as the movement of the center of mass
).
Since each pair of observables in the set commutes, the observables are all compatible so that the measurement of one observable has no effect on the result of measuring another observable in the set. It is therefore not necessary to specify the order in which the different observables are measured. Measurement of the complete set of observables constitutes a complete measurement, in the sense that it projects the quantum state of the system onto a unique and known vector in the basis defined by the set of operators. This specifies the conditions for the preparation of a well-defined quantum state.
Also, the states which form the basis of the CSCO are completely defined by specifying the eigenvalues
, 
, ..., associated with each observable 
, 
, ... of the CSCO. In the above example of the hydrogen atom, the basis states are defined by the quantum numbers
, and it is not necessary to include the wave function or the atomic orbital
.
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, a complete set of commuting observables (CSCO) is a set of commuting operators whose eigenvalues completely specify the state of a system .
For example, in the case of the hydrogen atom
Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...
, the Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
H, the angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
L and its projection
Projection (mathematics)
Generally speaking, in mathematics, a projection is a mapping of a set which is idempotent, which means that a projection is equal to its composition with itself. A projection may also refer to a mapping which has a left inverse. Bot notions are strongly related, as follows...
Lz along any arbitrary z axis form a CSCO (if one ignores the spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
of the proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....
and of the electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
as well as the movement of the center of mass
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
).
Since each pair of observables in the set commutes, the observables are all compatible so that the measurement of one observable has no effect on the result of measuring another observable in the set. It is therefore not necessary to specify the order in which the different observables are measured. Measurement of the complete set of observables constitutes a complete measurement, in the sense that it projects the quantum state of the system onto a unique and known vector in the basis defined by the set of operators. This specifies the conditions for the preparation of a well-defined quantum state.
Also, the states which form the basis of the CSCO are completely defined by specifying the eigenvalues
, , ..., associated with each observable , , ... of the CSCO. In the above example of the hydrogen atom, the basis states are defined by the quantum numbers, and it is not necessary to include the wave function or the atomic orbitalAtomic orbital
An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...
.