Quantum observable

The quanta observable ( observable by the quantic system , sometimes simply observable ) is a linear self-adjoint operator acting on the separable (complex) Hilbert space of pure states of a quantum system. In the intuitive physical understanding, the norm of the observable operator is the largest absolute value of the measured numerical value of a physical quantity.

Sometimes instead of the term "observable" they use "dynamic quantity", "physical quantity". However, temperature and time are physical quantities , but are not observable in quantum mechanics .

The fact that linear operators are compared to quantum observables poses the problem of the connection of these mathematical objects with experimental data, which are real numbers. Experimental measures are real numerical values corresponding to the observable in a given state. The most important characteristics of the distribution of numerical values on the real line are the average value $\langle A\rangle$ ${\ displaystyle \ langle A \ rangle}$ $\ langle A \ rangle$ observable and variance $D(A)$ ${\ displaystyle D (A)}$ $D (A)$ observable.

It is usually postulated that the possible numerical values of a quantum observable that can be measured experimentally are the eigenvalues of the operator of this observable.

It is said that the observed $A$ ${\ displaystyle A}$ $A$ capable of $\rho$ ${\ displaystyle \ rho}$ $\ rho$ is accurate if the variance $A$ ${\ displaystyle A}$ $A$ equals zero $D(A)=0$ ${\ displaystyle D (A) = 0}$ $D (A) = 0$ .

Another definition of a quantum observable: the observed quantum systems are self-adjoining elements. $C^{*}$ ${\ displaystyle C ^ {*}}$ $C ^ {*}$ algebras

Use of structure $C^{*}$ ${\ displaystyle C ^ {*}}$ $C ^ {*}$ -algebras allow formulating classical mechanics in the same way as quantum mechanics. Moreover, for noncommutative $C^{*}$ ${\ displaystyle C ^ {*}}$ $C ^ {*}$ -algebras describing quantum observables, the Gelfand-Naimark theorem holds : any $C^{*}$ ${\ displaystyle C ^ {*}}$ $C ^ {*}$ -algebra can be realized by the algebra of bounded operators acting in a certain Hilbert space. For commutative $C^{*}$ ${\ displaystyle C ^ {*}}$ $C ^ {*}$ -algebras describing classical observables, we have the following theorem: every commutative $C^{*}$ ${\ displaystyle C ^ {*}}$ $C ^ {*}$ -algebra $M$ ${\ displaystyle M}$ $M$ is isomorphic to the algebra of continuous functions defined on the compact set of maximal ideals of the algebra $M$ ${\ displaystyle M}$ $M$ .

The following statement is often postulated in quantum mechanics. Each pair of observed $A$ ${\ displaystyle A}$ $A$ and $B$ ${\ displaystyle B}$ $B$ corresponds to the observed $C$ ${\ displaystyle C}$ $C$ setting the lower bound of simultaneous (for the same state) measurability $A$ ${\ displaystyle A}$ $A$ and $B$ ${\ displaystyle B}$ $B$ in the sense that $D(A)D(B)\geq \langle C\rangle ^{2}$ ${\ displaystyle D (A) D (B) \ geq \ langle C \ rangle ^ {2}}$ $D (A) D (B) \ geq \ langle C \ rangle ^ {2}$ where $D(A)$ ${\ displaystyle D (A)}$ $D (A)$ - the variance of the observed, equal to $\langle A^{2}\rangle -\langle A\rangle ^{2}$ ${\ displaystyle \ langle A ^ {2} \ rangle - \ langle A \ rangle ^ {2}}$ $\ langle A ^ {2} \ rangle - \ langle A \ rangle ^ {2}$ . This statement, called the uncertainty principle, is performed automatically if $A$ ${\ displaystyle A}$ $A$ and $B$ ${\ displaystyle B}$ $B$ are self-conjugate elements $C^{*}$ ${\ displaystyle C ^ {*}}$ $C ^ {*}$ algebras Moreover, the uncertainty principle takes its usual form, where $C=i[A,B]$ ${\ displaystyle C = i [A, B]}$ $C = i [A, B]$ .

The concepts of a quantum observable and a quantum state are additional, dual. This duality is due to the fact that in the experiment only the average values of the observables are determined, and this concept includes both the concept of the observable and the concept of state.

If the evolution of a quantum system in time is completely characterized by its Hamiltonian, then the observed equation of evolution is the Heisenberg equation. The Heisenberg equation describes the change in a quantum observable Hamiltonian system over time.

In classical mechanics, a real smooth function defined on a smooth real manifold describing the pure states of a classical system is called observable.

There is a relationship between classical and quantum observables. It is usually assumed that to define a quantization procedure means to establish a rule according to which each observable classical system, that is, a function on a smooth manifold, is assigned a certain quantum observable. In quantum mechanics, operators in a Hilbert space are considered observable. As a Hilbert space, usually choose a complex infinite-dimensional separable Hilbert space. The function itself that corresponds to this operator is called the operator symbol.

Literature

F.A. Berezin, M.A. Shubin, “The Schrödinger Equation” M .: MGU, 1983. 392 p.
Bom D. “Quantum mechanics: fundamentals and applications”. Translated from English. M .: Mir, 1990. - 720s.
U. Bratelli, D. Robinson "Operator Algebras and Quantum Statistical Mechanics" M .: Mir, 1982. - 512s.
Jet Nestruev, “Smooth varieties and observables,” ICNMO, Moscow, 2000–300c.
Fadeev, LD, Yakubovsky, O. A. “Lectures on quantum mechanics for students of mathematics” L .: Izd-vo Leningrad State University, 1980. - 200 p.
Emh J. “Algebraic methods in statistical mechanics and quantum field theory” M .: Mir, 1976. 424с.

Quantum observable

See also

Literature

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