Questions tagged [observables]

A quantum observable is a measurable operator whose corresponding property of the state can be determined by some sequence of physical operations ("observation"), such as submitting the system to various electromagnetic fields and eventually reading a value. In systems governed by classical mechanics, any experimentally observable value can be shown to be given by a real-valued function on the set of all possible system states.

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Sequential Stern-Gerlach devices - realizable experiment or teaching aid?

At least one textbook [1] uses sequential Stern-Gerlach devices to introduce to students that the components of angular momentum are incompatible observables. Viz., the $z$-up beam from a SG device ...
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218 views

What is the physical meaning of the sum of two non-commuting observables?

Scenario: ${\mathcal A}$ and ${\mathcal B}$ are two observables. Mathematically we model them by two Hermitian operators $A\colon H \to H$ and $B\colon H \to H$ on a separable Hilbert space. ...
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Where does a fermionic coherent state live (which Hilbert space)?

There have been a couple of questions on fermionic coherent states, but I didn't find any that covered the following question: If I define a coherent fermionic state in the 2-level-system spanned by $...
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How are local observables encoded in this formulation of quantum field theory as a functor?

I've recently begun trying to understand a formulation of quantum field theory as a functor from a category of spacetimes-with-boundaries (bordisms) to a category of Hilbert spaces, as reviewed in [1]....
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Measurement formalisms - POVM formalism vs Hermitian observables

I am thinking in following way of thinking about measurements in quantum mechanics. Please correct any false statements I may be making below. We start with POVMs. Let our POVM be a set of positive ...
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Measuring the Dirac field

If the Dirac field $\psi(x)$ is to the electron as the Electromagnetic field is to the photon, why is it that we can measure the Electromagnetic field, whereas the Dirac field we cannot?
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The sum of two observables: Can its interpretation depend on more than just those two observables?

In the context of quantum theory, suppose we have two models $M_1$ and $M_2$ formulated on the same Hilbert space. Suppose that the operator $A$ is an observable in both models, with the "same" ...
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215 views

Tracing out an observable vs integrating over unitaries

Let $O$ be an observable on a Hilbert space $\mathcal{H}$, and let $B$ be a subset of the spins composing $\mathcal{H}$, and let $\bar{B}$ be its complement. Now define $$\displaystyle O_B = \frac{1}{...
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What significance do field-operators have, if they don't correspond to observables because of non-hermicity?

Since field-operators are not always hermitian (for example in case of a complex scalar field, or the dirac-field), they don't (in the quantum-mechanical sense) correspond to observables. Does that ...
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77 views

Physical significance of the canonical energy-momentum tensor

I have a question regarding the physical significance of the canonical energy momentum tensor $T_\nu ^\mu$ in the context of classical field theory. It is defined as $T_\nu ^\mu = \frac{\partial \...
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90 views

$C^*$ algebra of observables for a particle in a ring

It is known that for a free particle in $\mathbb{R}$, the $C^*$ algebra of the observables is given by the Heisenberg algebra i.e. generated by $p,q$ such that $[q,p]= i$ ($h=1$). For technical ...
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What is the Hamiltonian of this number theoretic system?

Background Let us have the following orthonormal basis such that: $$ \langle m | n \rangle = \delta_{mn}$$ Consider the following operators defined as: $$ \hat 1 = | 1 \rangle \langle 1 | + | 2 \...
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Is the vanishing commutator of observables outside the light cone only a necessary or also a sufficient condition for causality?

The equal-time commutator of observables in QFT has to vanish outside the light cone in order to ensure causality. Mathematically spoken, $[ \bar{\psi}(x)\Gamma_1\psi(x),\bar{\psi}(y)\Gamma_2\psi(y)]|...
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Is there a physical significance to non-normal states of the algebra of observables?

Quantum theory may be formalized in several different ways. Generally, the physical discussion of different states of a quantum system distinguishes pure and mixed states, and then subsumes both in a ...
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335 views

Hamiltonian symmetry Lie algebra

What is the connection between complete set of commuting observables and generators of the Lie group? I have a Hamiltonian written down in second quantized formalism and I also checked that it ...
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Observables labelling one-particle states in Quantum Field Theory

I'm studying introductory QFT using the first volume of Weinberg's series, and i'm having problems in understanding how single particle states of the free theory are labelled, i.e. what observables ...
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How to define an Operator Product Expansion (OPE) on arbitrary Riemann surface for a CFT?

Whenever we define the OPE of a 2D CFT, we do so (at least in the literature that I have come across) on the complex plane. Similarly, the commutation relations between conformal generators $L_n$ and ...
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Quantum Observation

Bear with me if I present a lack of knowledge - QM is not my field. There's a common notion in QM that until a particle is observed (measured), its properties are not definite, but rather are spread ...
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What does “contextuality” mean in the context of partial rather than complete quantum measurements?

The Kochen-Specker theorem is often described as ruling out "noncontextual" classical hidden variable theories. I understand the math behind the theorem, but I'm a little unclear on the exact, ...
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What happens when one "observes' a quantum field, and how do particles get involved?

I've recently begun my journey to understand QFT. I apologize in advance for the length of the post, but there are gaps in my understanding of how I, as an experiementalist, interact with fields to ...
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301 views

Simultaneous measurement of non-commuting observables without uncertainty

A pair of non-commuting Observables $\hat{X}$ and $\hat{P}$ does not have a common set of eigenfunctions, i.e., it can not be measured simultaneously. Let us for the sake of simplicity assume that $[\...
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Have Witten-type TQFT's nonconservation of energy and momentum in interactions?

Witten-type topological quantum field theories are based on cohomology theories. Every observable must lie in a cohomology class. May be $G$ a geometric field. Then every observable expectation value ...
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What are you studying when you study a Harmonic Oscillator in QM?

This probably is a naive question - so please forgive a self-studier. In the text I am studying, one builds a HO by placing a particle in a potential that increases quadratically from the origin. The ...
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What is a continuous superselection sector?

I'm studying the terrible subject of continuous superselection rules and I faced with the following problem. Usually (continuous or discrete) superselection rules are defined involving a direct ...
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How does linearity of a measurement imply that the commutator of all measured observables are $c$-numbers?

I really don't understand with the linearity conditions I have where this comes from.
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Expectation value of a ladder operator

I am going back over old Q.M simple harmonic motion material and, as I can't see an answer on the web, I would like to confirm the validity of an assumption. Using the ladder operators: $$ {\...
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What is the intuitivity about C*-Algebras being used as the fundamental objects in physics?

While asking about operators on this site, many answers mentioned "C*-algebras" to be the fundamental mathematical element corresponding to an observable (in QFT and QM at least), and choosing a ...
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Quantum Mechanics “inside out”

Let us assume that we know only some basic QM notions which are part of the Heisenberg picture of quantum mechanics and Dirac quantization Physical observables are represented by Hermitian operators $...
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Good quantum numbers from a given hamiltonian

The primary reason asking this question to understand good quantum number from a giver Hamiltonian. Is there any good approach that we can identify them? For example: We have a square and in that ...
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Conservation laws and Gauge transformations

I am studying gauge transformations, and my professor asked me: "Can the potentials obtained by the Lorenz gauge be considered physical quantities?" I assumed that "physical quantity" is ...
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129 views

What are the orthogonal eigenstates of the field operator?

In Peskin & Schroeder section 9.2, they derive the two-point function in the path integral formalism: $$\langle \Omega | \mathcal{T} \left\{ \hat{\phi}(x_1)\hat{\phi}(x_2)\right\} | \Omega \...
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For any two unitarily equivalent observables, can both be measured by the same experimental apparatus?

If we have an observable $A$, and a unitary operator $\hat U$, one can easily show that both $\hat A$ and $\hat U \hat A \hat U^{\dagger}$ have the same spectrum - in fact, they are called unitarily ...
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What is the variance of |S| in Bell's inequality (CHSH inequality)

Sorry that this isn't a quick question but I didn't know how to make it shorter. I am struggling with this for quite a long time and I would appreciate every help that I can get. I could not find a ...
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2answers
69 views

Is the period a physical observable in General Relativity?

I am currently seeing the classical tests of GR. To justify the introduction of a test based on the Doppler effect, the professor says that the previous test ( Shapiro and echo-radar test ) is based ...
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70 views

Relationship between the Galilei Group and the Phase Space

This question is kind of a follow up question to my last question on the need for canonical commutation relations and conjugate observables. A comment from Valter Moretti suggested that, given a ...
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128 views

Physical quantity related to the parity operator

There is a statement in quantum mechanics that for every physical quantity, there exists a Hermitian operator. The converse is also true. So the question is, what physical quantity is related to the ...
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Can we measure without collapsing (too much) the wave function, according to decoherence theory?

According to decoherence theory, the collapse of the wave function is a continuous process due to interaction with environment. In a measure, there are interactions with photons (for example). Can we ...
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1answer
141 views

Domain space of compatible and incompatible operators (observables)

Sakurai (Modern Quantum Mechanics, by J.J. Sakurai) states in the section on compatible operators: Let us first consider the case of compatible observables A and B. As usual, we assume that the ket ...
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Uncertainty Principle with the corresponding operators

Why does the corresponding operator do not commute if there is uncertainty related to two observables A and B that states $\Delta A\,\Delta B > 0 $ ?
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How to measure $\mathbb{L}^2$ and $L_z $ simultaneously

What does an experiment look like, in which both quantities are measured simultanously?
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“Independent simultaneous eigenbras” in Dirac's book 'Principles of Quantum Mechanics'

I've been puzzling through this book off and on and can usually work out what is going on via other external references on the Intertubes. But, this paragraph from pages 55 and 56 has me a bit ...
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Fields: Fundamental and Physical, yet Unobservable?

I'm currently working through Robert Klauber's Student Friendly Quantum Field Theory, which by the way is much more accessible than other texts like, say, Peskin and Schroeder, for others also coming ...
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What is the Hamiltonian operator, and is it unique?

$$\hat V=\sum_i v_i |v_i\rangle \langle v_i| $$ An observable in quantum mechanics is defined as above, with {$| v_i \rangle$} being an orthonormal basis, so the observable $\hat V$ is a Hermitian ...
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Eigenvalues of Unitary Matrices

I am considering the standard equation for a unitary transformation $\alpha^* = U \alpha U^{-1}$, where $\alpha$ is an arbitrary linear operator and $U$ is a unitary matrix. Since in quantum ...
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Why are we only interested in unitary/anti-unitary transformations of the underlying Hilbert space when considering symmetries in quantum mechanics?

Background to question: We briefly looked at 'symmetries' in my quantum mechanics course. I was dissatisfied with the fact that we only considered unitary (touched on antiunitary) operators when ...
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What are some examples of classical observables that change with observation?

I was reading H. Moysés Nussenzveig "A course in basic physics, Volume IV" and in chapter 8, he is introducing the basic ideas of quantum mechanics, where he states that: In quantum mechanics, one ...
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805 views

Eigenvalues of time dependent Hamiltonian

Consider the Schrödinger equation $H\Psi=i\hbar\frac{\partial \Psi}{\partial t}({\bf r},t)$. The hamiltonian $H$ is: \begin{equation} H=-\frac{\hbar^2}{2m}\nabla^2+V({\bf r},t) \end{equation} And the ...
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The origin of hamiltonians of physical systems

How do we decide on the Hamiltonian for different physical systems in quantum mechanics, for example for a spinning charged particle, we define the magentic dipole moment as $$\vec{\mu} = \gamma \vec{...
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2answers
257 views

Prove conservation law in quantum mechanics

I major in Math, and I am studying Quantum Mechanics (QM). I see the conservation law in QM as a mathematical theorem. Please check if my understanding is right, and help me to prove the theorem? ...
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Representing states of spin of multi-particle systems in QM

I'm starting to learn about quantum spin so this might be a trivial question. From a section explaining Clebsch-Gordan Coefficients it states that generally we have $$|sm\rangle = \sum_{m_1 +m_2 = m}C^...