Matrix element (physics)

This article was registered in the quality assurance of the physics editorial team . If you are familiar with the topic, you are welcome to participate in the review and possible improvement of the article. The exchange of views about this is currently not taking place on the article discussion page , but on the quality assurance side of physics.

In quantum mechanics , physical quantities or processes are represented by operators , with each operator in the Hilbert space of state vectors corresponding to a linear mapping . The letter for the physical measurand is often used as a symbol for the operator and given a circumflex , e.g. B. for the x-coordinate the symbol . Prominent exception is for the Hamiltonian , the energy E represents. ${\displaystyle {\hat {x}}}$ ${\displaystyle {\hat {H}}}$

Are state vectors with given that an orthonormal base form of the Hilbert space, then the operator can be completely represented by a matrix with the elements . The matrix element states the component with which the basis vector is contained in the vector that was created by applying to the basis vector . ${\displaystyle \vert \phi _{n}\rangle }$ ${\displaystyle (n=1,2,\dots )}$ ${\displaystyle {\hat {O}}}$ ${\displaystyle O_{mn}:=\langle \phi _{m}\vert {\hat {O}}\vert \phi _{n}\rangle }$ ${\displaystyle O_{mn}}$ ${\displaystyle \vert \phi _{m}\rangle }$ ${\displaystyle {\hat {O}}}$ ${\displaystyle \vert \phi _{n}\rangle }$

In addition, the term matrix element is generally used when two state vectors and the size are used. In Fermi's Golden Rule e.g. B. the initial state and the observed final state of a certain process are chosen for the two states, whereby neither a whole basis is specified nor the two states have to be orthogonal at all. ${\displaystyle \vert \phi \rangle }$ ${\displaystyle \vert \psi \rangle }$ ${\displaystyle \langle \phi \vert {\hat {O}}\vert \psi \rangle }$

Individual evidence

1. ^ Georg Joos: Textbook of Theoretical Physics. 15th edition. AULA-Verlag, Wiesbaden 1989, ISBN 3-89104-462-3 , p. 576.