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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What does it mean for two compatible observables to be a "coarse-graining" of a third?
In reading about quantum contextuality, I've encountered the statement that
if [A,B] = 0, then there exists another observable C such that the
spectral projections of A and B are a coarse-graining ...
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Can this shape of matrix elements in the path integral formalism be linked to some sort of expectation value?
This question is about expressions of the form
$$
\langle x_f, t_i | \hat{x}(t) | x_i, t_i \rangle = \frac{1}{N} \int_{x(t_i) = x_i}^{x(t_f) = x_f} \mathcal{D} x~x(t)e^{i S[x]}.
$$
In the following ...
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Common eigenstate of incompatible observables
In many resources I have seen that incompatible observables cannot have a common eigenbasis set, but may share one or few eigen states. I followed the thread Can incompatible observables share an ...
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Intuition for momentum operator in position space
The derivation of the momentum operator in position space. But, several assumptions are usually made that a) we are dealing with the particle in free space or b) that the two representations are ...
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Repeating observations in quantum theory
Suppose we prepare a state $\psi$ in a quantum system, represented in some Hilbert space, and suppose $A$ is an observable represented by the matrix $A$ (which possibly has infinite order).
QUESTION A
...
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Observables from boson correlation functions
I am studying the formalism of quantum optics in the approximation of a two-level system coupled to a reservoir made of boson in thermal equilibrium. As usual, the latter subsystem is described in ...
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What is the distribution for a function of different quantum observables?
Suppose we have a quantum mechanical particle prepared in a pure state $\psi$, and an apparatus that can measure the orbital angular momentum of the particle along a specified orthogonal axis ($x$, $y$...
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What is the fundamental observable in casual set theory?
In ordinary quantum theory or string theory, the fundamental observables are correlation functions or scattering amplitude that can be measured by particle physics experiments .
In loop quantum ...
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Why can we assign both energy and particle number to each state in grand canonical emsemble?
The system in grand canonical emsemble together with the surrounding reservoir is isolated, thus have conserved particle number $N$. However, the system itself only has fixed average $\langle N\rangle$...
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Canonical momentum not observable vs energy is observable
I have seen explanations that canonical momentum for charged particles $p = mv + qA/c$ is not a measurable quantity/observable because it is not gauge invariant. However, there are many quantities ...
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What are the physical state invariants of loop quantum gravity?
What are the "physical state" invariants for Loop Quantum Gravity?
The Wikipedia page talks about "physical states" being invariant, diffeomorphism invariance, "quantities ...
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What are possible experimental (optical/microscopic/spectroscopic) observables for identifying and differentiating superconductivity?
In a system one is analysing for superconductivity signatures, what are possible experimental (optical/microscopic/spectroscopic) observables for identifying and differentiating superconductivity?
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Can there exist energy eigenstates that cannot be labelled by the good quantum numbers?
I'm trying to visually understand good quantum numbers for the example Hamiltonian of a composite system $$H = \lambda J_{1}.J_{2}$$
As I understand it, the energy of the system (assuming fixed ...
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Quantum observables in nonstandard Hilbert space
Consider a Hermitian $(n \times n)$-matrix $A$, and a Hilbert space $\mathbb{C}^n$, foreseen with a nonstandard inner product. (An inner product $s(\cdot,\cdot)$ is standard if for any two vectors $x =...
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Heterodyne detector
In which sense an heterodyne detector does not measure an observable? I mean, its POVM is proportional to the projector on coherent states, and the coherent states are an overcomplete set for the ...
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Measuring the observable
I am working through this quantum mechanics homework question and I am a little confused on what I am being asked to do. I have come to one of two possible answers but I don't think either is correct. ...
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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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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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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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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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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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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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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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What is the observable for the optical field?
Typically, observables in quantum mechanics are associated with Hermitian operators. However, Glauber argues in 1963 ([1]) that the electric field operator $\hat{\mathbf{E}}(x,t)$ is not the relevant ...
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References on obtaining experimental observables from band structure
I've recently been watching this lecture series in Condensed Matter physics. We have covered second quantization, used it to obtain the tight-binding model and then studied the band structure of ...
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Equivalence of gauge-invariance and physical observable
This is somewhat philosophical than physics.
In gauge theories, it is true (more like the first principle) that
\begin{equation}
\text{ physical observable } \Rightarrow \text{gauge invariant}
\end{...
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Algebra of observables in Quantum Mechanics
When reading books about Quantum Mechanics, it is generally stated (in a kind of axiomatic way) that in Quantum Mechanics, the state of the system is represented by a vector in some Hilbert space $H$, ...
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Showing that time ordering does not matter for the measurement of commuting observables
Suppose I have two observables $R$ and $S$ who are represented by operator $R$ and $S$ which commute (I will hereafter ignore the distinction between observables and the operators representing them), ...
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Relationship between anti-commutators and correlation
Ballentine (in his solution at the back of the book to his Problem 8.10) writes that
$$[Tr(\rho \{A,B\}/2)]^2$$
is related to the correlation between the observables represented by $A,B$, but gives no ...
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Is this the Format of the Observable Universe?
The way I have it is: the Observable Universe looks as follows. In some ball, all the galaxy clusters exist, then in a bigger concentric ball the dark ages exist (no galaxies), then on the surface of ...
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Showing that interaction and Schrödinger pictures are equivalent -- does the result hold without commutativity?
I am trying to show that the interaction picture represents a valid picture. We use this picture when our Hamiltonian takes the form $H = H_0 +H_1$. "Valid" means that it should reproduce ...
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From classical to quantum observables
In classical mechanics, usual observables of a system are its position $x(t)$ and its momentum $p(t)$, which are just symbols holding the unique position/momentum value at a given time.
I wonder why ...
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Is there a quantum number associated with linear momentum?
Energy and angular momentum have a quantum number associated with them:
$$ E\rightarrow n $$
$$ |\vec{L}|\rightarrow l $$
$$ L_z\rightarrow m_l $$
Is there also a quantum number associated with linear ...
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Can Quantum observables be modeled as real functions on the phase space?
In the context of justifying the failure of modeling quantum observables in the 'more natural' way as real functions on the phase space (i.e. similar to the mathematical image modeling classical ...
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Difference between transformations written as $TOT^{\dagger}$ and as $TOT^{-1}$ in quantum matrix mechanics?
When we write quantum operators in matrix form, we perform transformations. I have seen that at some points the transformation $T$ of a matrix $O$ is written as, for example,
$O\to TOT^{\dagger}$
$O\...
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What defines the gauge group for pure gravity?
Asymptotic symmetry of space-time corresponds to diffeomorphism transformation of physical space-time, which manifests as isometry near Conformal Boundary. Asymptotic symmetry can be defined using ...
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How to Multiply a Wave Function by a Pauli Matrix?
Let's say I have a random wave function of $|\psi\rangle=(2+i)|0\rangle+(4-i)|1\rangle$ and I want to find the average value of the Pauli matrix with this function. How would I go about that? I tried $...
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Do we care about the distance between quantum observables?
In quantum mechanics, we care about the spectrum of quantum observable, the eigenvalues of observables, as they give the measurement results. I wonder if we have to care about the "distance&...
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Unmeasurable observables in quantum mechanics
Let's consider a single particle in 1D harmonic oscillator for definiteness. In standard QM, we say that any Hermitian operator on the Hilbert space is an "observable". It seems that (in ...
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What does energy, momentum, etc mean in quantum mechanics?
I see all those nice operators that are used to give the expectation values of dynamical varaibeles, but what does it actually mean to measure kinetic energy, momentum, etc of a particle? How is it ...
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"Length" of spin vector
Quantum mechanics ,McIntyre,pg 58
For the case of spin $1 / 2$, note that the expectation value of the operator $S^{2}$ is
$$
\left\langle\mathbf{S}^{2}\right\rangle=\frac{3}{4} h^{2}
$$
which would ...
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Complete set of commuting observables for an electron in a periodic potential (crystal)
In the case of an electron in a crystal,
lets say that $\hat{T}(a_1)$, $\hat{T}(a_2)$ and $\hat{T}(a_3)$ are the traslation operators.
We know that $[\hat{H}, \hat{T}(a_i)] = 0$, is this a complete ...
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Is an orthonormal eigenbase of an observable Hamel or Schauder?
Suppose $\mathcal{H}$ is an infinite-dimensional Hilbert space, and let $A$ be an observable. The quantum system is prepared in a state $\vert \Psi \rangle$. After measuring the observable the state $\...
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Regarding the physical significance of the eigenvalues of the permutation operator
Is the permutation operator an observable? I know that it is Hermitian* and unitary. If yes, what is the physical quantity that corresponds to the eigenvalues of this operator?
If we apply the ...
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Sequential Stern-Gerlach experiments without an intermediate measurement
What happens if we pass a particle through several Stern-Gerlach devices (say, oriented along x- and z-axes) without measuring the state of the system between them? My initial assumption has been that ...
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