# What is observables: Definition and 114 Discussions

In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum physics, it is an operator, or gauge, where the property of the quantum state can be determined by some sequence of operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value.
Physically meaningful observables must also satisfy transformation laws that relate observations performed by different observers in different frames of reference. These transformation laws are automorphisms of the state space, that is bijective transformations that preserve certain mathematical properties of the space in question.

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1. ### A Complex potential/ observable

I have an observable denoted by C, related to a complex potential B by : ## C= \bar{B}B ,## where ##B## is a complex potential. I know that ## \left. C \right|_0 =C_0 ##, a known constant, where the evaluation at ##_0## denotes an equilibrium \ reference state. From this, I can not make any...
2. ### Relation between Mutual information and Expectation Values

I've make progress in obtaining the values for the mutual information using the following: $I(\rho_A:\rho_B) = S(\rho_A) +S(\rho_B) - S(\rho_{AB}) = 1 + 1 - 0 = 2.$ I would like to compute the expectation but I'm facing a problem in the case of $\langle\psi |\mathcal{O}_A|\psi \rangle$ since...
3. ### Two-level Quantum System - initial state

TL;DR Summary: Find the initial state of a two-level quantum system, given the probability of measurements for two observables and the expected value of an operator. Dear PFer's, I have been struggling with the following problem. It was assigned at an exam last year. Problem Statement For a...
4. ### A States & Observables: Are They Really Different?

Usually states and observables are treated as fundamentally different entities in quantum theory. But are they really different? A state can always be represented by a density "matrix", which is really a hermitian (or self-adjoint) operator. Since observables are also hermitian (or self-adjoint)...
5. ### I Interpretation of observables

So generally in most literature observables are represented via self-adjont (or equivalently real-valued) linear operators in QT. But that definition leaves it open for a wide variety of operators that can be view as observables. I was always a bit uncertain how to understand this freedom, so i...
6. ### I Are commuting observables necessary but not sufficient for causality?

A couple of the sites that say that the vanishing commutator defines causality: https://www.physics.umass.edu/events/2020-02-07-causal-uncertainty-quantum-gravity https://phys.org/news/2019-11-aspects-causality-quantum-field-theory.html Is "define" too strong? Or, to put it another way: cannot...
7. ### I Observables on the "3 polarizers experiment"

Observables on the "3 polarizers experiment" Hi guys, I was analyzing the 3 polarizers experiment. This one: (first 2 minutes -> ) Doing the math (https://faculty.csbsju.edu/frioux/polarize/POLAR-sup.pdf) I realized that the process is similar to the Stern-Gerlach' experiment. Using spins...
8. ### Commutation and Measurement of Observables

Hello there, I am having trouble with part b. of this problem. I've solved part a. by calculating the commutator of the two observables and found it to be non-zero, which should mean that ##\hat B## and ##\hat C## do not have common eigenvectors. Although calculating the eigenvectors for each...
9. E

### B Simple question about compatible observables

Hey, sorry if this is a dumb question but I wondered if someone could clarify. If you have two Cartesian coordinate systems ##\mathcal{F}## and ##\mathcal{F}'## whose origins coincide except their ##x## axes point in opposite directions (i.e. ##\hat{x} = -\hat{x}'##), then the spin operators...
10. ### A Barbero-Irmizi and LQG observables?

Hello, around 2008-2009 I self-studied a little bit of LQG, and become disillusioned by it because I understood that Barbero-Immirzi parameter was related to time parametrization (in a way similar to a scale factor, like $c$), and the dependence of observable mean values was in conflict with...
11. ### A Stone's theorem on one-parameter unitary groups and new observables?

I have been following the proof of the Stone's theorem on one-parameter unitary groups. The question is if the current list of self-adjoint operators used in quantum mechanics, including position and momentum operators, is exhaustive or not? Put it another way, can we say that there is no...
12. ### I Von Neumann entropy for "similar" pvm observables

The von Neumann entropy for an observable can be written ##s=-\sum\lambda\log\lambda##, where the ##\lambda##'s are its eigenvalues. So suppose you have two different pvm observables, say ##A## and ##B##, that both represent the same resolution of the identity, but simply have different...
13. ### I Compatible observables and commutator

Is commutator being zero for two operators the same statement as the two observables are compatible?
14. ### I How to derive Born's rule for arbitrary observables from Bohmian mechanics?

Where can I find a reference to a derivation of Born's rule for arbitrary observables from Bohmian mechanics?
15. ### I Why observables are represented as operators in QM?

Can somebody provide an explanation why the dynamical variables/observables are represented in QM as linear operators with the measured values being eigenvalues of these operators? For energy this is probably trivially and directly follows from the stationary Shrodinger equation which solutions...
16. ### A Observables when the symmetry is not broken?

Hi, Let be a scalar field φ that permeates all space. The quantum of the field has a mass m. The field is at the minimum of its potential. When this minimum is for φ≠0 (a broken symmetry), the quantum may be observed by exciting the field, as with the Higgs boson. But if the symmetry is not...
17. ### A Observables in condensed matter (QFT)

Quantum field theory is a powerful tool to calculate observables given the amplitude of some process. I only know the application to high energy physics: you have a Lagrangian with an interaction term between some fields, and you can calculate the amplitude of some process. Once you have this...
18. ### How Do Compatible Observables Determine Common Eigenstates?

Homework Statement So-called compatible observables correspond to operators which commute, i.e. [A, B] = 0, where [A, B] stands for AB − BA. a) [3pt] Suppose Hermitian operators A and B represent two compatible observables, and all eigenvalues of A are different. Show that eigenstates of A...
19. ### I Why are spinors not observables?

Hello! I am reading some QFT and it is a part about how causality implies spin-statistic theorem. In general, one needs 2 observables to commute outside the light-cone. For scalars, we have $$[\phi(x),\phi(y)]=0$$ outside the light-cone, and by using the operator form of the field you get that...
20. ### I Commuting observables for Fermion fields?

In nonrelativistic QM, we usually describe the Hilbert space by choosing a complete set of commuting observables, so that the set of states that are eigenstates of all the observables can be used as a basis. For instance, the "wavefunction" is the state as expressed in terms of "states" with...
21. ### A Renormalisation: what are the physical observables?

I'm trying to understand renormalisation properly, however, I've run into a few stumbling blocks. To set the scene, I've been reading Matthew Schwartz's "Quantum Field Theory & the Standard Model", in particular the section on mass renormalisation in QED. As I understand it, in order to tame the...
22. ### I Do the SU(n) generators represent any observables?

Hey there, I've recently been trying to get my head around Yang-Mills gauge theory and was just wandering: do the Pauli matrices for su(2), Gell-Mann matrices for su(3), etc. represent any important observable quantities? After all, they are Hermitian operators and act on the doublets and...
23. ### I Noncommuting observables and indeterminacy principle

Hi all, What is the link between noncommuting observables {\displaystyle [{\hat {A}},{\hat {B}}]={\hat {A}}{\hat {B}}-{\hat {B}}{\hat {A}} \neq 0} and indeterminacy principle (which is about inequality relation of standard deviation of the expectation value of observables A and B ) ? If the...
24. ### Mathematical Quantum Field Theory - Observables - Comments

Greg Bernhardt submitted a new PF Insights post Mathematical Quantum Field Theory - Observables Continue reading the Original PF Insights Post.
25. ### I Commuting observables vs. exchanging measurements

Hi. I'm afraid I might just be discovering quite a big misunderstanding of mine concerning the meaning of the expectation value of a commutator for a given state. I somehow thought that if the expectation value of the commutator of two observables ##A, B## is zero for a given state...
26. ### Repeating measurement of observables?

I have a question on my homework set and I'm not sure the principle behind it: 1. Alice measured an observable F (a matrix) and passed the measured system immediately to another experimentalist, Bob, who is going to measure another observable G. Alice claims that she can deduce the experiment...
27. ### B What is the origin of Observables?

Why does an Observable have to be Hermitian, and why do the eigenstates and eigenvalues have to respresent the possible measured values? Is is by definition? What is the origin of this convention?
28. ### I How Do Quantum Observables Influence State Collapse?

I'm trying to grasp the subtleties of quantum mechanics, and that's not easy :). Let H be a two dimensional complex Hilbert space of quantum states. An observable is a self adjoint operator A on H. Let ##(|\phi_1\rangle,|\phi_2\rangle)## be an orthonormal eigenbasis to A, with corresponding...
29. ### Quantum Measurements of L2 in a Wave Function: Calculations and Probabilities

Homework Statement A particle is in a state described by a wave function of the form ψ(r) = (x+y+z)f(r). What are the values that a measurement of L2 can yield? What is probability for all these results? Homework EquationsThe Attempt at a Solution I feel this problem shouldn't be too hard but...
30. ### Degeneracy removed when commuting observables are specified?

The following is a proof that two commuting operators ##A## and ##B## possesses a complete set of common eigenfunctions. The issue I have with the proof is the claim that the eigenvalue ##a_n## together with the eigenvalue ##b_n## completely specify a particular simultaneous eigenfunction...
31. ### Pairs of observables in uncertainty principle

Observables are paired up in the uncertainty principle such that we can't measure both to a high degree of accuracy. Specifically, ## \sigma_x \sigma_y>\frac{\hbar}{2} ## where ## \sigma_x ## and ## \sigma_y ## are the standard deviations of our measurements. I've got two lines of questions...
32. ### How does the wavefunction "know" the observables?

The thread title is probably confusing, but I couldn't really think of a better one. There is a basic feature of quantum mechanics that I have been puzzled by for a long time. My guess is that my issue stems from some fundamental misunderstanding of the theory, so I would appreciate any efforts...
33. ### Characterization of a gauge theory in terms of observables

Although I have a good understanding of how to do calculations in gauge field theory, I am still dissatisfied with my understanding of why we use them in the first place. From a philosophical point, it should be possible to characterize a gauge theory in terms of observables only. I suppose one...
34. ### Question about quantum observables

Homework Statement Given this three operators in the same orthonormal base, [H]=[{0,1,0},{1,0,0},{0,0,1}], [A]=[{1,0,0},{0,1,0},{0,0,1}] and =[{1,1,1},{1,1,1},{1,1,1}], tell which of these observables form a complete set of compatible observables: [H], [H,A], [H,B] or [H,A,B] Homework...
35. ### Insights Higher Prequantum Geometry V: The Local Observables - Lie Theoretically - Comments

Urs Schreiber submitted a new PF Insights post Higher Prequantum Geometry V: The Local Observables - Lie Theoretically Continue reading the Original PF Insights Post.
36. ### Do observables for polarizers at different angles commute?

Hi. We can write a polarised photon as ##\left|\alpha\right\rangle=\cos(\alpha)\left|\updownarrow\right\rangle+\sin(\alpha)\left|\leftrightarrow\right\rangle##. Trigonometry gives us $$\left\langle\alpha | \beta\right\rangle=\cos(\alpha)\cos(\beta)+\sin(\alpha)\sin(\beta)=\cos(\alpha-\beta)$$...
37. ### Relationship between commutators and observables

Homework Statement Suppose A^ and B^ are linear quantum operators representing two observables A and B of a physical system. What must be true of the commutator [A^,B^] so that the system can have definite values of A and B simultaneously? Homework Equations I will use the bra-ket notation for...
38. ### Observables vs. continuum and metric?

Space in quantum mechanics seems to be modeled as a triplet of real numbers, i.e. a continuum. Same happens in special relativity. General relativity I do not know (nor field theories). And then we apply the Pythagorean theorem and triangle inequality and so forth... I have a few general...
39. ### Motivations for the C*-algebra of observables?

As far as I understand, the Hilbert space formalism can be derived using functional analysis and representation theory (not familiar with those) from the requirement that observables (their mathematical models) form a C*-algebra and the possible states of a system map the members of the algebra...
40. ### Quantum Mechanics: Three Spin Observables

When trying to solve ##\mathbb{S}^2 =\hbar^2s(s+1)\mathbb{I},## I got that ##\mathbb{S}^2 = \mathbb{S}^2 _x+\mathbb{S}^2_y+\mathbb{S}^2_z = \frac{3\hbar^2}{4} \left[\begin{array}{ c c }1 & 0\\0 & 1\end{array} \right] = \frac{3\hbar^2}{4}\mathbb{I},## but how does ##\frac{3\hbar^2}{4} =...
41. ### Are observables like position emergent properties?

Title basically says it all. I'm a physics undergrad trying to wrap my head around quantum physics, and I was hoping people here could help. My question comes from something in one of my textbooks. It tries to explain particle-wave duality through a piece of string, which I'll quickly go over as...
42. ### Variant of Bohm Mechanics for Other Observables

This seems like an obvious thing to ask, so I assume it has been asked and answered, but I'd like to know what is special about position in the Bohm-DeBroglie interpretation of quantum mechanics (called "B-DB" in the following). Here's a way of thinking about B-DB: Because not all observables...
43. ### Non Commuting Observables not Representing a Complete Set?

I am going through James Binney's course on Quantum Mechanics. I love all of the little misconceptions he points out along the way. One thing he mentions in his text and the lectures is found on page 20 and 21 starting with the heading "Commutators" eq. 2.21. He states that non commuting...
44. ### Sequential measurements of conjugate observables

There is an argument that accurate sequential measurement of conjugate observables A and B on the same state is possible if the state is an eigenstate of one of the observables. When the state is an eigenstate of A, an accurate measurement of A will not disturb the state, so B can then be...
45. ### Commutators, mutual eigenkets, and observables

I have two quick questions: 1. Why if say [x,y] = 0, it implies that there is a mutual complete set of eigenkets? where x and y can be anything, like momentum, position operators. 2. If an operator is not hermitian, why isn't it an observable? (More specifically, why isn't its...
46. ### What are some examples of non-commutating observables in quantum mechanics?

The Wikipedia entry on uncertainty says, "Since the Robertson and Schrödinger relations are for general operators, the relations can be applied to any two observables to obtain specific uncertainty relations. A few of the most common relations found in the literature are given below." It...
47. ### Prediction of observables from wavefunction

Homework Statement The wavefunction of a particle is given as $$u(r,\theta,\phi) = AR(r)f(\theta)\cos(2\phi),$$ where ##f## is an unknown function of ##\theta##. What can be predicted about the results of measuring a) The z component of angular momentum b)The square of the angular momentum...
48. ### How to understand operators representing observables are Hermitian?

As we know, all operators representing observables are Hermitian. In my undersatanding, this statement means that all operators representing observables are Hermitian if the system can be described by a wavefunction or a vector in L2. For example, the momentum operator p is Herminitian...
49. ### Simultaneous measurement of noncommuting observables

It is often said that non-commuting observables cannot be simultaneously and precisely measured. The rough idea is that the procedures to measure each observable are different. Is the statement strictly correct? If so, what is the mathematical argument? Is it possible to use Ozawa's...
50. ### Set of Commuting Observables for pures states 2p-1,2px and 2s

Hi, Here I have a question, apparently easy, but that I think it is a bit tricky. Homework Statement Indicate how can a hydrogen atom be prepared in the pure states corresponding to the state vectors ψ2p-1 and ψ2px and ψ2s. It is assumed that spin-related observables are not...