Observable

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In

and eventually reading a value.

Physically meaningful observables must also satisfy

transformations

that preserve certain mathematical properties of the space in question.

Quantum mechanics

In

As a consequence, only certain measurements can determine the value of an observable for some state of a quantum system. In classical mechanics, any measurement can be made to determine the value of an observable.

The relation between the state of a quantum system and the value of an observable requires some

V. Two vectors v and w are considered to specify the same state if and only if ${\displaystyle \mathbf {w} =c\mathbf {v} }$ for some non-zero ${\displaystyle c\in \mathbb {C} }$. Observables are given by self-adjoint operators on V. Not every self-adjoint operator corresponds to a physically meaningful observable.^[5][6][7][8] Also, not all physical observables are associated with non-trivial self-adjoint operators. For example, in quantum theory, mass appears as a parameter in the Hamiltonian, not as a non-trivial operator.^[9]

In the case of transformation laws in quantum mechanics, the requisite automorphisms are

, the mathematics of frames of reference is particularly simple, considerably restricting the set of physically meaningful observables.

In quantum mechanics, measurement of observables exhibits some seemingly unintuitive properties. Specifically, if a system is in a state described by a vector in a

of the state of the larger system.

In quantum mechanics, dynamical variables ${\displaystyle A}$ such as position, translational (linear)

${\displaystyle {\hat {A}}}$ that acts on the

eigenvalues

of operator ${\displaystyle {\hat {A}}}$ correspond to the possible values that the dynamical variable can be observed as having. For example, suppose ${\displaystyle |\psi _{a}\rangle }$ is an eigenket (

eigenvector

) of the observable ${\displaystyle {\hat {A}}}$, with eigenvalue ${\displaystyle a}$, and exists in a Hilbert space. Then

This eigenket equation says that if a measurement of the observable ${\displaystyle {\hat {A}}}$ is made while the system of interest is in the state ${\displaystyle |\psi _{a}\rangle }$, then the observed value of that particular measurement must return the eigenvalue ${\displaystyle a}$ with certainty. However, if the system of interest is in the general state ${\displaystyle |\phi \rangle \in {\mathcal {H}}}$ (and ${\displaystyle |\phi \rangle }$ and ${\displaystyle |\psi _{a}\rangle }$ are

of ${\displaystyle a}$ is one-dimensional), then the eigenvalue ${\displaystyle a}$ is returned with probability ${\displaystyle |\langle \psi _{a}|\phi \rangle |^{2}}$, by the Born rule.

Compatible and incompatible observables in quantum mechanics

A crucial difference between classical quantities and quantum mechanical observables is that some pairs of quantum observables may not be simultaneously measurable, a property referred to as

This inequality expresses a dependence of measurement results on the order in which measurements of observables ${\displaystyle {\hat {A}}}$ and ${\displaystyle {\hat {B}}}$ are performed. A measurement of ${\displaystyle {\hat {A}}}$ alters the quantum state in a way that is incompatible with the subsequent measurement of ${\displaystyle {\hat {B}}}$ and vice versa.

Observables corresponding to commuting operators are called compatible observables. For example, momentum along say the ${\displaystyle x}$ and ${\displaystyle y}$ axis are compatible. Observables corresponding to non-commuting operators are called incompatible observables or complementary variables. For example, the position and momentum along the same axis are incompatible.^[10]: 155

Incompatible observables cannot have a complete set of common eigenfunctions. Note that there can be some simultaneous eigenvectors of ${\displaystyle {\hat {A}}}$ and ${\displaystyle {\hat {B}}}$, but not enough in number to constitute a complete

See also

• Measure (physics)
• Observable universe
• Observer (quantum physics)
• Table of QM operators
• Unobservable

References

Further reading

• Auyang, Sunny Y. (1995). How is quantum field theory possible?. New York, N.Y.: Oxford University Press.
  ISBN 978-0195093452
  .
• Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (2019). Quantum Mechanics, Volume 1. Weinheim: John Wiley & Sons.
  ISBN 978-3-527-34553-3
  .
• de la Madrid Modino, R. (2001). Quantum mechanics in rigged Hilbert space language (PhD thesis). Universidad de Valladolid.
• Teschl, G. (2014). Mathematical Methods in Quantum Mechanics. Providence (R.I): American Mathematical Soc.
  ISBN 978-1-4704-1704-8
  .
• von Neumann, John (1996). Mathematical foundations of quantum mechanics. Translated by Robert T. Beyer (12. print., 1. paperback print. ed.). Princeton, N.J.: Princeton Univ. Press.
  ISBN 978-0691028934
  .
• Varadarajan, V.S. (2007). Geometry of quantum theory (2nd ed.). New York: Springer.
  ISBN 9780387493862
  .
• ISBN 9780691141206
  .
• Moretti, Valter (2017). Spectral Theory and Quantum Mechanics: Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation (2 ed.). Springer.
  ISBN 978-3319707068
  .
• Moretti, Valter (2019). Fundamental Mathematical Structures of Quantum Theory: Spectral Theory, Foundational Issues, Symmetries, Algebraic Formulation. Springer.
  ISBN 978-3030183462
  .