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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Born rule and expectation value integral for $L = x \times p$

Physical interpretation of the wave function $\psi (x,t)$ is a probability amplitude for location $x$ and the Fourier transform of $\psi (x,t)$ can be interpreted as a probability amplitude for ...
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Spectrum of an operator

Why does the mathematical definition of a spectrum of an operator namely the set of complex numbers without the resolvent set, agreed with the real physical spectrum of an observable? is this a ...
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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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Why does the time-evolution operator $U(t)$ depend explicitly on time in the Schrodinger picture?

Schrodinger's picture is that operators are time-independent. But time evolution operator $U(t)$ is time-dependent. Why is that?
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Uncertainty Principle: Commutators [duplicate]

How are commutators the mathematical basis for uncertainty principle? What makes one say that commutators imply uncertainty principle or vice-versa?
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Commutativity vs Compatibility

As far as I know, two compatible observables have a complete set of common eigenvectors, and using this fact, one can prove that their corresponding operators are commutative. Well now is the converse ...
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In what sense (if any) is Action a physical observable?

Is there any sense in which we can consider Action a physical observable? What would experiments measuring it even look like? I am interested in answers both in classical and quantum mechanics. I ...
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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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Can particles be in positional eigenstate in reality?

Do i understand quantum right in the following description? what we observed as particle is just the phenomenon they have some kind of quality corresponding to the macroscopic stones in experiments ...
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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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understanding resolving power in QM

Consider the definition of resolving power of two states in quantum mechanics as being the absolute value of the difference between the probabilities of two states following the measurement averaged ...
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Paradox about the Stern-Gerlach experiment with $B=0$, $\nabla B\ne 0$

In a modern interpretation of the historical Stern-Gerlach experiment, a beam of neutral silver atoms, each with spin 1/2, was sent in the z direction through a nonuniform magnetic field having both a ...
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Common eigenkets of spherically symmetric Hamiltonian

In a QM text it states: "Consider a spinless particle subjected to a spherical symmetrical potential. The wave equation is known to be separable coordinates, and the energy eigenfunctions can be ...
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Kinetic Energy operator in Quantum Mechanics

Firstly, I'd like to point out clearly that I'm not a physicist but I'm a nano engineer studying quantum mechanics so that I understand my work on surface sciences better, so please don't presume my ...
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Is it obvious that the Hamiltonian observable in Quantum Mechanics should also be the Energy observable?

In Quantum Mechanics, the Hamiltonian observable is defined as the generator of time translations. It's easy to show that if we take this to be the definition of the Hamiltonian, then it is of the ...
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Are eigenvectors always orthogonal each other?

When an observable/selfadjoint operator $\hat{A}$ has only discrete eigenvalues, the eigenvectors are orthogonal each other. Similarly, when an observable $\hat{A}$ has only continuous eigenvalues, ...
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From uncertainty to commutation relations

Consider the famous problem of measuring both the position and momentum of an electron. We start with two photographs at different times, then worry about the momentum of the photon, the wavelength of ...
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284 views

Diffeomorphism invariance and correlation functions

Consider the following paragraph taken from page 15 of Thomas Hartman's lecture notes on Quantum Gravity: In gravity, local diffeomorphisms are gauge symmetries. They are redundancies. This means ...
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Gauge transformations at infinity

Consider the following paragraph taken from page 15 of Thomas Hartman's lecture notes on Quantum Gravity: In an ordinary quantum field theory without gravity, in flat spacetime, there two types of ...
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Analogy expectation of an observable / random variable

I'm trying to figure out the analogies between the expectation of a random variable $X$ and the expectation of an observable of a quantum mechanical system $A$ (using this wikipedia article). The ...
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1answer
922 views

Significance of imaginary momentum in the finite potential well

In the case of the finite potential well, when the particle enters the classically forbidden region its wavenumber, $k$ is imaginary. However $k=\frac{2\pi}{\lambda}=\frac{p}{\hbar}$ so we get an ...
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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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QM: Find expectation value of measurements

We have an observable $$O \mapsto \begin{pmatrix} 0 & 2 \\ 2 & 4 \end{pmatrix}$$ We find the eigenvalues and eigenvectors by O$\psi$ = o$\psi$. The eigenvalues will give us the possible ...
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Lie Algebra of Classical Observables under Poisson Bracket

I am confused with understanding the fundaments of classical mechanics. All classical observables commute since they are represented by regular functions on phase space. All classical observables form ...
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What exactly is the relationship between the algebraic formulation of Quantum Mechanics and the geometric formulation of Classical Mechanics?

Okay so if we consider a particular physical system, the classical description of the system starts by first introducing a symplectic manifold, which is the cotangent bundle of a configuration ...
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Do measurements of commuting obervables really commute?

Say I have two commuting operators $A$ and $B$, with joint eigenvectors $|n\rangle$. Say I have a state $S = \sum_n a_n | n \rangle$. If I measure $A$ first, I pick out say an eigenstate $|k\rangle$ ...
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Basic question about wave funtions: using the Born rule

I have a textbook giving an example of what the probability is of observing system $Ψ = a|A⟩ + b|B⟩$ in states $a|A⟩$ and $b|B⟩$. I'm not sure I understand it fully. How do I use the Born rule to know ...
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In what types of QFTs are the Wilson loops of interest?

I have a very basic question about Wilson lines (WL). This is what I know about the WL: WL help us to learn about the important properties of gauge fields (treated as connections on the space of ...
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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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Average momentum in quantum mechanics over some finite interval of space

Why can't the expectation value of momentum be computed over some finite interval of space? Something like, $$ \int_a^b \psi^* \hat{p}\psi ~\mathrm{d}x.\tag{1}$$ I understand that usually we compute ...
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Hermiticity of the Lagrangian in QFT

Why must the Lagrangian (density) of a given quantum field theory (QFT) be Hermitian? It's something that is mentioned, but not really explained (as far as I can tell) in Srednicki's QFT book, ...
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Do imaginary measurements exists? [duplicate]

I'm sorry if this question is too metaphysical, but I will give it a try. My textbook in introductory quantum mechanics is basing a lot of its proof and derivations on the fact that the value of the ...
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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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Why can operators be represented as matrices in quantum mechanics?

I am studying introductory quantum mechanics in our undergraduate course. I saw that operators can be represented as matrices too. I can't figure out the proper reason. My attempt is: As operators ...
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101 views

Can Schrodinger equation be defined for observable of another quantity?

Hamiltonian, being a hermitian operator, is an observable for energy. I wonder if Schrodinger equation can be defined for any $K$, also a hermitian operator, but an observable for another quantity, i....
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What is the correct *first* interpretation of the time derivative of some measurable quantity?

For example, take the position function $x(t)$. When I take $(d/dt)(x(t))$, I know that I must ultimately conclude that the result is the velocity function $v(t)$. But this feels like a ''jump ahead'' ...
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351 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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Is hermiticity a basis-dependent concept?

I have looked in wikipedia: Hermitian matrix and Self-adjoint operator, but I still am confused about this. Is the equation: $$ \langle Ay | x \rangle = \langle y | A x \rangle \text{ for all } x ...
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360 views

Symmetric, (essentially) self-adjoint operators and the spectral theorem

At the moment I got a bit confused about the notation in some QM textbooks. Some say the operators should be symmetric, some say they should be essentially self-adjoint, others that they has to be ...
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How many physical quantities could be computed in a generic Quantum Field Theory? [duplicate]

Given a generic Quantum Field Theory, I am able to compute only two physical quantities: the decay rate of particles and cross sections of interactions. Does exist other physical observables which can ...
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Representations of Quantum States

I am trying to get understand how different representations of a quantum state are equivalent: For example if we have our quantum state $$| \psi \rangle = \sqrt{\frac{1}{3}}|R_{21}Y_{1}^{0} \rangle \...
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Hermitian operators: observables and dynamical variables

In the postulates of quantum mechanics, it is sometimes said that dynamical variables are associated with hermitian operators. And sometimes, it is said that observables are associated with hermitian ...
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What are Hermitian operators in QFT?

In this answer, Lubos explains that in quantum field theory there are linear hermitian operators representing observables. Quantum field theories are a subset of quantum mechanical theories. So ...
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351 views

System of combined observables presented as tensor products

A state can be written as $$| \psi \rangle = \sum c_n | \psi_{nlm} \rangle$$ where $| \psi_{nlm \rangle}$ is the stationary states or eigenstates of the Hamiltonian in three dimensions (spherical ...
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Positional probability density for combined spin and position states

In one dimension, given a particle in a quantum state $| \psi\rangle$, the probability density of position is given as $| \psi(x) |^2 = \psi^*(x) \psi(x) =\langle x | \psi \rangle\langle \psi | x \...
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Eigenstates of momentum and energy of a free particle

Given the momentum operator $\hat{P}:= \frac{\hbar}{i}\frac{d}{dx}$, as I understand, the eigenvalue equations are $$\hat{P}f_{p}(x)= \frac{\hbar}{i}\frac{d}{dx}f_{p}(x) = p f_{p}(x)$$ and the ...
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Regarding Energy and Momentum in QM

I've been trying to learn Quantum Mechanics for a few months now, and there's a pretty fundamental thing I never quite understand: What is energy and momentum in quantum mechanics? I've been ...
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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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Probability associated with observables of multi-particle systems

Consider the two cases outlined in an introductory text of QM: Case 1: Two particles (of spin 2 and spin 1) are in a box and the total spin is 3 and its $z$ component is $0$, then a measurement of $...

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