Hermitian operator

Hermitian operators , named after Charles Hermite , are operators considered in mathematics that play a central role in the mathematical structure of quantum mechanics . The term Hermitian operator is defined inconsistently in the literature, in this article the physical view and notation are used in particular.

Different conventions

The term Hermitian operator is defined inconsistently in the literature. In some mathematical representations, the term “Hermitian operator” does not appear at all; instead, so-called symmetric , symmetric, tightly defined, essentially self-adjoint and self-adjoint operators are considered. Differences only appear in the infinite-dimensional spaces that are important for physical applications.

In the physical literature, on the other hand, the term “symmetrical operator” is usually not used at all: instead, one speaks from the outset of Hermitian operators (more precisely one would have to say: “Hermitian in the narrower sense”) in order to emphasize that one does not has to do with real, rather than complex Hilbert spaces. In the later chapters of the standard physics textbooks, “Hermitian operators” (more precisely: “Hermitian in the broader sense”) are usually used to denote self-adjoint operators (the somewhat subtle difference is often neglected or by synonyms such as “hypermaximal Hermitian operators” instead of the "Self-adjoint operators" simplified).

The representation chosen here is a compromise in that the term “symmetrical operator” is also used for the complex-valued Hilbert spaces of physics, but otherwise the conventions of physicists with the identification “Hermitesch = self-adjoint” are used, as in textbooks, for example by Albert Messiah can be found. The presentation given here is aimed primarily at physically interested readers, which is why the Bra-Ket notation, which goes back to Dirac , is used, which lets certain mathematical subtleties recede into the background. This is discussed in the section on Mathematical Remarks .

Definitions

operator

Let be an operator on a Hilbert space , that is, a map on this Hilbert space in itself. The elements of this Hilbert space are written as ket vectors and often represent functions from spaces , e.g. B. the wave function of a quantum mechanical state . Such an operator transforms one vector into another: ${\displaystyle A}$ ${\displaystyle |\varphi \rangle }$ ${\displaystyle L^{2}}$ ${\displaystyle \varphi ({\vec {r}},t)}$

${\displaystyle |\psi \rangle =A|\varphi \rangle .}$

It is not required that a different vector is assigned to each vector; such an assignment is often only successful for vectors in a dense subspace. For example, if the Hilbert space is a -space and the operator is a derivative operator, then it can only act on differentiable functions. ${\displaystyle L^{2}}$

This operation should be linear in order to ensure the physically relevant superposition principle . The complex number , i.e. the scalar product of with a Bra vector of another state, is consistently referred to in physics as the matrix element of . ${\displaystyle \langle \chi |\psi \rangle =\langle \chi |\left(A|\varphi \rangle \right)}$ ${\displaystyle |\psi \rangle =A|\varphi \rangle }$ ${\displaystyle \langle \chi |}$ ${\displaystyle \,A}$

Sometimes a roof is placed over the operator symbol in order to distinguish the effect of the operator on a vector from the multiplication of the vector by a complex number . But this is only necessary if you want to denote operators and their eigenvalues ​​with the same letter; you can then write an eigenvalue equation . No use is made of this in this article. ${\displaystyle {\hat {A}}}$ ${\displaystyle |\varphi \rangle }$ ${\displaystyle A}$ ${\displaystyle {\hat {A}}|\varphi \rangle =A|\varphi \rangle }$

Adjoint operator

→ Main article : Adjoint operator

The operator to be adjoint   is defined by the fact that its matrix elements are the complex conjugate numbers of the matrix elements of if Bra and Ket are exchanged: ${\displaystyle A}$ ${\displaystyle A^{\dagger }}$ ${\displaystyle \,A}$

${\displaystyle \langle \chi |\left(A^{\dagger }|\varphi \right)\rangle =\left[\;\langle \varphi |\left(A|\chi \right)\rangle \;\right]^{*}.}$

Therefore, the bra is assigned to the ket , and the round brackets can be dispensed with in the matrix element without risk of confusion. ${\displaystyle \langle \varphi |A^{\dagger }}$ ${\displaystyle A|\varphi \rangle }$

Is , the matrix element is called the expected value of in the state . ${\displaystyle |\varphi \rangle =|\chi \rangle }$ ${\displaystyle \,A}$ ${\displaystyle |\varphi \rangle }$

Hermitian operator

${\displaystyle A}$is formally called self-adjoint (or, in physical parlance, Hermitian ), if . ${\displaystyle A=A^{\dagger }}$

Then applies to its matrix elements . All expectation values ​​are then real, because applies to every vector from the domain . Then every eigenvalue is also real, because the eigenvalues ​​are the expected values ​​for the respective normalized eigenvectors. Since in quantum mechanics all measurable quantities ( observables ) are represented by expectation or eigenvalues ​​of operators, these must be Hermitian operators so that the predicted measurement results are real. ${\displaystyle \langle \chi |A|\varphi \rangle =\left[\;\langle \varphi |A|\chi \rangle \;\right]^{*}}$ ${\displaystyle |\varphi \rangle }$ ${\displaystyle \langle \varphi |A|\varphi \rangle =\left[\;\langle \varphi |A|\varphi \rangle \;\right]^{*}}$ ${\displaystyle \,A}$

Examples

X coordinate

In the spatial representation one considers the space of all square-integrable functions on the three-dimensional visual space . Typical Hermitian operators are, for example, multiplication by the -coordinate to measure the -coordinate of the location of a particle, ${\displaystyle L^{2}(\mathbb {R} ^{3})}$ ${\displaystyle x}$ ${\displaystyle x}$

${\displaystyle A|\psi (x,y,z)\rangle =|x\,\psi (x,y,z)\rangle }$

or the Hamilton operator to determine the energy ${\displaystyle -{\frac {\hbar ^{2}}{2m}}\Delta +V(x,y,z)}$

${\displaystyle A|\psi (x,y,z)\rangle =|-{\frac {\hbar ^{2}}{2m}}\left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}\right)\psi (x,y,z)+V(x,y,z)\,\psi (x,y,z)\rangle }$,

where stands for the scalar potential of a field under whose influence the particle moves. Other important examples are the momentum operator or the angular momentum operators . ${\displaystyle V(x,y,z)}$

Finite-dimensional examples

A two-dimensional example is obtained by treating the spin . The space is created by the two vectors “spin up” and “spin down” . The operators on two-dimensional spaces are matrices, for example the Hermitian Pauli matrices . ${\displaystyle (|{\mathord {\uparrow }}\rangle )}$ ${\displaystyle (|{\mathord {\downarrow }}\rangle )}$ ${\displaystyle 2\times 2}$

A finite-dimensional Hermitian operator (a Hermitian matrix ) with the elements is adjoint as follows: ${\displaystyle A}$ ${\displaystyle m_{ij}\in \mathbb {C} }$

${\displaystyle A^{\dagger }={\begin{pmatrix}m_{11}&&m_{12}&&\cdots &&m_{1n}\\m_{21}&&m_{22}&&\cdots &&m_{2n}\\\vdots &&\vdots &&\ddots &&\vdots \\m_{n1}&&m_{n2}&&\cdots &&m_{nn}\end{pmatrix}}^{\dagger }={\begin{pmatrix}m_{11}^{*}&&m_{21}^{*}&&\cdots &&m_{n1}^{*}\\m_{12}^{*}&&m_{22}^{*}&&\cdots &&m_{n2}^{*}\\\vdots &&\vdots &&\ddots &&\vdots \\m_{1n}^{*}&&m_{2n}^{*}&&\cdots &&m_{nn}^{*}\end{pmatrix}},}$

where is the complex conjugation of . It is therefore true , that is, the -th component of the adjoint is the complex conjugation of the -th component of the starting matrix. ${\displaystyle m_{ji}^{*}}$ ${\displaystyle m_{ji}}$ ${\displaystyle m_{ij}^{\dagger }=m_{ji}^{*}}$ ${\displaystyle (i,j)}$ ${\displaystyle (j,i)}$

Momentum operator

The following simple example clearly shows the difference between symmetric and Hermitian (= self-adjoint) operators. We consider the momentum operator in -direction . More precisely, you will define a finite (or infinite) interval as the domain for the function , for example , and you will initially only want to specify that the function is integrable with the square of the specified interval. The question then remains, which boundary conditions should be required for. At first one is inclined to assume that should be; because then - as one can easily show by means of partial integration - the "symmetry" is given: ${\displaystyle x}$ ${\displaystyle P_{x}=-\mathrm {i} \hbar {\frac {\partial }{\partial x}}}$ ${\displaystyle \psi (x)}$ ${\displaystyle x\in [0,1]}$ ${\displaystyle P_{x}\psi (x)}$ ${\displaystyle \psi (x)}$ ${\displaystyle \psi (0)=\psi (1){\stackrel {!}{=}}0}$

${\displaystyle {\begin{aligned}\langle \psi (x)|P_{x}\psi (x)\rangle &=\int _{0}^{1}\,\mathrm {d} x\,\psi (x)^{*}\,(P_{x}\psi (x))\\&=(-i\hbar )\{\psi ^{*}(1)\psi (1)-\psi ^{*}(0)\psi (0)\}+\int _{0}^{1}\,\mathrm {d} x\,(P_{x}\psi (x))^{*}\psi (x)\\&=\langle P_{x}\psi (x)|\psi (x)\rangle \end{aligned}}}$

because when “rolling over” the derivative from right to left through the partial integration under the integral, the last term before the outermost equality sign, a minus sign is created, which is compensated by the term and the transition to the conjugate complex , while the boundary terms in the Integration explicitly result in zero. The boundary terms also compensate to zero if one only demands that the function should fulfill the condition . ${\displaystyle -i}$ ${\displaystyle -i^{*}=+i}$ ${\displaystyle \psi (x)}$ ${\displaystyle \psi (0)\equiv \psi (1)}$

With the second, the weakened boundary condition, the system is not just “symmetrical”, as with the first boundary condition, but is even self-adjoint. This is not only relevant mathematically but also physically: this is the only way to obtain measurability and a complete system of eigenfunctions. These can be specifically named here, with the whole numbers running through. On the other hand, not a single one of these functions would match the first-mentioned boundary condition, because they all have values ​​other than zero at the decisive point ,,. ${\displaystyle \psi _{n}(x)\sim \exp(i2\pi n\,x)\,,}$ ${\displaystyle n}$ ${\displaystyle x=0}$

The first-mentioned boundary condition is therefore unphysical and can only be approximately realized with special non-trivial potentials, while free electrons and vanishing potential can be assumed for the second boundary condition.

Mathematical remarks

The above examples already show that the quantum mechanical operators cannot be applied to all ket vectors. The result of the multiplication with the coordinate is generally no longer in the Hilbert space of the ket vectors, and in the case of the Hamilton operator some functions lack differentiability properties. A generalization to distribution derivatives is of no use here either, since not all such derivatives are again in the space of the ket vectors. One is therefore forced to restrict the operators in their area of ​​action to a subspace , which is at least a dense subset in the space of all ket vectors. If "all" ket vectors are mentioned in physical representations for an operator equation, then all of the domain of definition of the operators involved are always meant. ${\displaystyle x}$ ${\displaystyle L^{2}(\mathbb {R} ^{3})}$ ${\displaystyle L^{2}}$

The restriction to a subspace has the consequence that the adjoint operator is not defined everywhere either. The requirement that for all ket vectors (from the domain of ) is real then means that is an extension of the operator , that is, the domain of includes that of and both operators agree on the latter. Such operators are called symmetric . ${\displaystyle \langle \varphi |A|\varphi \rangle }$ ${\displaystyle A}$ ${\displaystyle A^{\dagger }}$ ${\displaystyle A}$ ${\displaystyle A^{\dagger }}$ ${\displaystyle A}$

Symmetric operators are generally not self-adjoint , that is, generally does not apply , since the domains of both operators would have to match. The physically relevant operators that describe measurable quantities are, however, self-adjoint, because only then is the full spectral set available (in the language of physics: the “ development theorem ”). You need this u. a. in the axiomatic treatment of the quantum mechanical measuring process (see e.g. quantum mechanical state ) and in the concrete calculation of functions of operators , such as that made possible by the unlimited Borel functional calculus . Often one can extend symmetric operators to self-adjoint operators by means of certain closing operations . This is especially true for downwardly bounded operators, as they occur with Hamilton operators, because energies are downwardly bounded, see Friedrichs extension . ${\displaystyle A=A^{\dagger }}$

Many physics textbooks do not place great emphasis on this difference. On the one hand, considering the domains of definition usually does not provide any deeper physical insights into the system under consideration, and furthermore one can usually rely on the fact that the use of the "correct" operators, paired with the "correct" physical intuition, leads to "correct" ones “Results Leads. Furthermore, all physically relevant functions to which the operators apply, namely the eigenfunctions, are always in the domain of definition. Caution is also advisable with the so-called "improper eigenfunctions" (e.g. with Dirac functions or with monochromatic waves), because these cannot be normalized and are therefore not in the space of the ket vectors.

Dirac's notation supports the pragmatic approach of physicists. A mathematically complete representation of quantum mechanics up to and including the solution of the hydrogen problem can be found in the textbook by Hans Triebel given below .

Individual evidence

1. ↑ In the finite-dimensional vector spaces of linear algebra , all terms are identical.
2. ↑ The definition of an operator with the property "Hermitian in the narrower sense" (here: "symmetrical") is for all states or from the domain of the operator in a complex Hilbert space with a scalar product . As "Hermitian in the wider sense" (or rather "self-adjoint") is known, however operators for which applies, while only implies "Hermitian in the narrow sense" that an extension of is, ${\displaystyle A}$ ${\displaystyle \langle \psi _{a}|A\psi _{b}\rangle =\langle A\psi _{a}|\psi _{b}\rangle }$ ${\displaystyle \psi _{a}}$ ${\displaystyle \psi _{b}}$ ${\displaystyle A}$ ${\displaystyle \langle .|.\rangle }$ ${\displaystyle A^{\dagger }=A}$ ${\displaystyle A^{\dagger }}$ ${\displaystyle A}$ ${\displaystyle A^{\dagger }\supseteq {A}\,.}$
3. ^ Siegfried Großmann: Funktionanalysis , Akademische Verlagsgesellschaft, Vol. 2, p. 189 describes self-adjoint-restricted operators as Hermitian
4. ↑ Werner Döring: Introduction to Quantum Mechanics , Göttingen 1962, on the other hand, uses the term "hypermaximal Hermitian" as a synonym for "self-adjoint" (this synonym originally comes from John von Neumann ) and the term "Hermitian" instead of "symmetrical"
5. ↑ In the textbook by Michael Reed and Barry Simon: Methods of Mathematical Physics , Volume 1, Academic Press, 1980, p. 255 defines "Hermitian" synonymously with "symmetrically and densely defined"
6. ^ Albert Messiah: Quantum Mechanics , 2 volumes, de Gruyter 1976, 1991, Vol. 1 ISBN 3-11-011452-6 , Vol. 2 ISBN 3-11-012669-9 , French original: Mécanique quantique , Dunod, Paris 1959, 1964, 1969, English translation: Quantum Mechanics , New York, Interscience and Amsterdam, North Holland, 1961/62
7. ↑ Mathematicians often write for the adjoint operator ${\displaystyle A^{\ast }}$
8. ^ Hans Triebel: Höhere Analysis , Berlin, Deutscher Verlag der Wissenschaften 1972, 2nd edition, Harri Deutsch 1980, (English Higher Analysis, Barth 1992)