Complete set of commuting observables

A complete set of commuting observables (vSkO) is a term from quantum mechanics , in which measurands such as energy , position or momentum are represented by operators and called observables . Measurements that can be precisely determined at the same time are called commuting observables ; they have the property that their operators interchange with one another.

Such behavior is rather the exception in quantum mechanics. Most pairs of observables can not be measured at the same time with any precision, which is a consequence of Heisenberg's uncertainty principle . One then speaks of complementary observables .

In order to uniquely characterize a quantum mechanical state , several observables are often necessary. For example, with the hydrogen atom it is not sufficient to just state the energy (by means of the main quantum number ), but two further observables are necessary: the amount of the angular momentum (quantum number ) and the component of the angular momentum (quantum number ). These three quantities then form a complete set of commuting observables. ${\ displaystyle n}$ ${\ displaystyle l}$ ${\ displaystyle z}$ ${\ displaystyle m}$

definition

A lot of observables , , , ... forms a vSkO when an orthonormal basis of the state space from common eigenvectors exist of the observables, and this base (up to a phase factor) is unique. ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle C}$

An equivalent formulation is:

A lot of observables , , , ... forms a vSkO if and only if: ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle C}$

swap all observables with each other in pairs , and

the specification of the eigenvalues of all these operators is sufficient to uniquely determine a common eigenvector (except for one factor).

meaning

In order to solve a quantum mechanical problem one tries to find a set of observables that describe the system and form a vSkO. By specifying the measured values of the observables (these are the eigenvalues of the observables) it is possible to clearly determine the state of a system. Conversely, this means that a measurement must be extended to a complete set of commuting observables in order to uniquely determine the state of the system after the measurement by specifying the measured values.

construction

Let an observable be given whose eigenvectors form a basis of the state space. If all of these are not degenerate , then the state of the system can be clearly characterized by specifying the eigenvalue belonging to an eigenvector. then forms a vSkO "for itself". If the eigenvectors are degenerate in some form, however, another observable is added, which is swapped with and whose eigenvectors in turn form a basis of the state space. The non-degenerate ones are now chosen from both sets of eigenvectors. Make these provide a basis of the state space and a vSkO. If not, one takes as long as other observables , can, ... which will in pairs swap with other observables, until a basis of eigenvectors to non-degenerate eigenvalues construct. ${\ displaystyle A}$ ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle A}$ ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle C}$ ${\ displaystyle D}$

Examples

An observable with non-degenerate eigenvalues, i.e. a non-degenerate spectrum, forms a vSkO “for itself”. An example for such a case is the Hamilton operator of the infinitely high potential well in one dimension.

The position operator , and the momentum operator respectively form "by itself" a vSkO of the state space of a spin -free particle.

For a spinless particle in a central potential , the Hamilton operator , the square of the angular momentum operator , and any component of the angular momentum operator (where ) form a vSkO. The eigenvalues of the three observables correspond to the principal quantum number , the angular momentum quantum number and the magnetic quantum number (see quantum number ). The indication of the triple clearly describes a quantum mechanical state (e.g. for the hydrogen atom ). ${\ displaystyle H}$ ${\ displaystyle L ^ {2}}$ ${\ displaystyle L_ {i}}$ ${\ displaystyle i = x, y, z}$ ${\ displaystyle n}$ ${\ displaystyle l}$ ${\ displaystyle m}$ ${\ displaystyle (n, l, m)}$

literature

Claude Cohen-Tannoudji , Bernard Diu, Franck Laloë: Quantum Mechanics . 2nd Edition. De Gruyter, Berlin 1999, ISBN 3-11-016458-2 .
Wolfgang Nolting: Basic Course Theoretical Physics, Volume 5/1, Quantum Mechanics: Fundamentals . 3. Edition. Vieweg, Braunschweig 1996, ISBN 3-528-06935-X .
Franz Schwabl: Quantum Mechanics An Introduction . 6th edition, Springer, Berlin, Heidelberg, New York 2002, ISBN 3-540-43106-3 .