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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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2 votes
1 answer
775 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 ...
8 votes
2 answers
357 views

In quantum mechanics, can we measure anything else than position?

In the basic quantum mechanics lectures, we learn that we can measure any observable. That means mathematically, all Hermitian operators correspond to a physically measurable quantity. In strong ...
0 votes
3 answers
111 views

How is Zig-zag Motion Observable in Quantum Mechanics Given Wave Function Collapse?

I'm puzzled by a concept I read about in a physics text concerning quantum measurement. The text describes the potential to observe a "zig-zag" motion if one could capture images of an ...
2 votes
0 answers
36 views

What is the interpretation of the covariance of two quantum observables?

I have been studying the covariance matrix in continuous variable quantum systems and I am struggling to understand the interpretation of this object. In statistics the covariance measures the joint ...
1 vote
1 answer
264 views

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 ...
0 votes
0 answers
38 views

If an electron is inside an atom, does the expected value of spin measurements also depend on the orbital wavefunction?

The total quantum state of an electron in an atom can be written as the product of the orbital wavefunction and a spinor representing its spin state, $\Psi = \psi(r,\theta,\phi) \otimes \chi$. Say you ...
10 votes
4 answers
1k views

What is the description of measurement in the Heisenberg picture?

In all the books I've read this picture is presented only briefly, by essentially saying that in the HP the whole time dependence is assigned to the operators (representing observables), whereas the ...
2 votes
1 answer
265 views

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 ...
1 vote
1 answer
52 views

Is Number Operator a Generalized Momentum?

In superconducting circuit, the number operator, $\hat{n}$, and phase operator, $\hat{\varphi}$ are conjugate pairs. Is $\hat{n}$ the canonical momentum, conjugate momentum, and also generalized ...
2 votes
1 answer
338 views

If every physical quantity corresponds to a Hermitian operator, what does the parity correspond to?

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 ...
1 vote
1 answer
34 views

Wavefunction with determinate momentum

In page 100 Griffiths' Introduction to Quantum Mechanics, Griffiths states that the eigenvector of $\hat{p}$ in the position basis is $\frac{1}{\sqrt {2\pi\hbar}}e^{\frac{ipx}{\hbar}}$ and states that ...
0 votes
2 answers
59 views

Closed expression for expected values of $\hat{p}\,\,^{2j}$ for the vacuum state

I am wondering if there is a closed expression for the expected value $\left<0\lvert \hat{p}\,\,^{2j}\lvert 0\right>$ with $j\in\mathbb{N}$, where $\left|0\right>$ is the vacuum state of the ...
1 vote
0 answers
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Probability for a quantum many particle system of temperature $T$ to be found at temperature $T'$

Suppose we have a system of $N$ particles (let's postpone the question re statistics) in thermal equilibrium, described via a density operator $$ \rho_\beta = Z^{-1} e^{-\beta H} $$ $$ Z = \text{tr} \,...
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1 answer
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How can $L_x$ be an observable?

I'm working with the orbital angular momentum operator $L_x$, and I don't quite understand how it can represent an observable. Using ladder operators, I can write $L_x$ as: $$L_x=\frac{1}{2}(L_++L_-)$$...
3 votes
0 answers
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Is Torsion Observable?

Are there, have there been, or could there be any experiments that might detect any torsion in our corner of the universe? Any results? Or is torsion an unobservable?
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0 answers
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On the choice of observables in linear response theory

For linear response theory I need two observables. The idea is to see how the change in one observable changes the other under weak perturbations in equilibrium state. Suppose I want to see the change ...
10 votes
3 answers
11k views

Susceptibilities and response functions

It is often confusing whether a susceptibility is the same as a response function, specially that often they are used interchangeably, in the context of statistical mechanics and thermodynamics. Very ...
1 vote
0 answers
65 views

What is the Taylor series of the expectation value of an observable in quantum mechanics? [closed]

I recently came across a form of the expectation value of an observable, but a Taylor series (I think?) was taken up to second-order: \begin{equation} \langle O\rangle = \langle O\rangle_{C=0} + \...
1 vote
1 answer
84 views

Is the annihilation operator an observable (it is non-Hermitian)?

In most treatments of quantum mechanics that I have seen, observables of a quantum system are defined using Hermitian operators. The most intuitive reason for this is that Hermitian operators have ...
1 vote
1 answer
42 views

Why can't we measure different properties of an electron in accordance with its frequency?

sorry if this is a stupid question but its my first one.. Why cant we just observe and electron in accordance to its frequency. Like during one frequency peak we could observe position, next could be ...
0 votes
1 answer
84 views

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 ...
2 votes
1 answer
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Why does the Pauli objection not disqualify the existence of the position operator?

According to the Pauli objection (see for example here or the answer to this question) there can be no time operator $\hat{T}$ canonically conjugate to the Hamiltonian $\hat{H}$ of a physical system ...
1 vote
1 answer
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Observation without interaction thought experiment [closed]

Here I am going to talk about a thought experiment that I have thought There is some isolated place in the universe where there is no EM field other than the field created by a moving point charge ...
-1 votes
2 answers
111 views

Is there a physical cause of uncertainty? [closed]

The uncertainty principle is confusing me. Considering this image from the article: Is the particle believed to be physically moving with similar capriciousness in real space; and if so, what ...
2 votes
0 answers
56 views

Observing a particle [duplicate]

We say that there is uncertainty in position when we observe a particle. But first i want to know in detail about how do we observe a particles position and momentum. Suppose we know that an electron ...
2 votes
2 answers
119 views

Dictionary between interpretations of field operators

For now, let $\hat{\phi}(x)$ be a quantization of a classical, real scalar field $\phi(x)$. My understanding is that, for fixed $x$, there are three ways to regard the operator $\hat{\phi}(x)$: The ...
2 votes
3 answers
143 views

How to represent the state of this system after the measurement?

This is a question my professor gave us in the test. Consider the spin-$\frac{1}{2}$. Alice measured the system along the $z$ direction, and she observed the outcome as $+1$. After Alice's measurement,...
3 votes
2 answers
201 views

Why are all observed particles on-shell?

I've been trying to self-learn how to do basic QFT calculations and I'm a little bit confused as to what's considered "an interaction". If I want to model an electron releasing a photon I ...
2 votes
1 answer
224 views

Why does a Hermitian operator have a basis of its own eigenvectors?

Suppose I have a hermitian operator $\Omega$. The proof of the existence of a orthonormal eigenbasis as given in Shankar is given. What I don't understand is why the second eigenvector $\left| \...
2 votes
0 answers
84 views

Introduction of symmetries in quantum mechanics

The (Italian) book that I am currently reading introduces the topic of symmetries in quantum mechanics in the following way: Let O and O' be two distinct observers and let $A$ and $B$ be two ...
1 vote
1 answer
101 views

Showing the Variance of an observable in a determinate state is always zero

I am working through Introduction to Quantum Mechanics by David J. Griffiths, and part 3.2.2 shows that the standard deviation of an obervable, $Q$, is always $0$ but I do not understand the steps ...
0 votes
0 answers
29 views

How to prove that the spin operator commutes with the position operator? [duplicate]

In the lecture notes on Quantum Mechanics I'm reading, the author claims that the position operator $\hat{q}$, the square spin operator $\hat{s}^2$ and the spin operator component $\hat{s}_0$ (in a ...
2 votes
1 answer
78 views

Why reasonable observables are made of an even number of fermion fields?

On Michele Maggiore book on QFT (page 91) is stated, out of nothing, that "observables are made of an even number of fermionic operator" and similar sentences is in Peskin book (page 56). Is ...
5 votes
1 answer
347 views

What is the set of observables of a quantum system?

This is a question I am wondering about because the answer to it seems to have some interesting - but perhaps already long considered and dismissed because it's been settled - implications for the ...
4 votes
2 answers
558 views

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 ...
0 votes
1 answer
263 views

Density of observable is expected value of Dirac delta

I am currently studying Statistical Mechanics and already have a background in probability and statistics. However, there are still things that remain unclear to me. So far I understand that time ...
1 vote
1 answer
146 views

Equation 7.16 Susskind Theoretical Minumum Quantum Mechanics

Could someone explain/expand equation \eqref{7.16} from Susskind's "Quantum Mechanics-The Theoretical Minimum" in particular, what are the indexes $a',a$ in the operator $\,\mathbf L\,$ ...
36 votes
5 answers
6k views

Why do we observe particles, not quantum fields?

My understanding is that, in the context of quantum field theory, particles arise as a computational tool. We perform an expansion in the path integral in some parameter, and the terms in these ...
-1 votes
3 answers
264 views

Post-measurement $\psi$ in quantum mechanics

I have a question regarding the wave function after a measurement. Everything I found online says that this is the following formula: $\psi = \frac{M_m\psi}{\sqrt{P(m)}}$ Where $P(m)$ is the ...
0 votes
1 answer
40 views

Questions regarding measurement of a qubit

I am trying to understand the basic concept of observable and state. Say that we have a state $|\psi>=|0>$ which is $(1\ \ \ 0)^T$, we have an observable $\sigma_x= \begin{bmatrix}0&1\\1&...
0 votes
1 answer
92 views

What kind of physical process would correspond to an operator that doesn’t result in an eigenvalue equation: $ \hat{A}ψ=a ψ$?

I'm studying quantum mechanics and I'm trying to understand the concept of operators. They can be represented in general by the equation: $$ \hat{A}ψ=ψ'. $$ Here the wavefunction is changed to $ψ'$ ...
-3 votes
1 answer
65 views

How to practically check if a Hermitian matrix is an observable?

I get that in QM an observable corresponds to a Hermitian operator. And I also get that not all Hermitian operators will correspond to an observable - with all that $C^*algebra$ stuff. Is there a ...
1 vote
1 answer
71 views

Does 2-form curvature $\Omega \in \Omega^2(P,\mathfrak{g})$ represent a physical quantity in gauge theory?

In gauge theory, all measurable physical quantities remain invariant under a gauge transformation. I have always seen that the curvature 2-form $\Omega \in \Omega^2(P,\mathfrak{g})$ associated to a ...
1 vote
1 answer
79 views

Why is it essential in quantum mechanics for the eigenvectors of an observable to compose a basis spanning the entire state space?

After going through the questions and answers, I still have a question lingering in my mind. So, an observable is defined as a Hermitian operator whose eigenvectors make up a basis for the state space....
3 votes
1 answer
45 views

Counterexample to the observable algebra of a region and its causal completion being the same

I was reading a paper by Ed Witten called "Algebras, Regions and Observers". It can be found here: https://arxiv.org/abs/2303.02837 A major theme is theorems relating the algebra of ...
3 votes
3 answers
3k views

Is there a difference between a Hermitian operator and an observable? [duplicate]

My poorly written lecture notes say that any Hermitian operator does have a complete set of orthogonal eigenstates with real corresponding eigenvalues and is therefore an observable. In the article ...
1 vote
0 answers
61 views

Eigenvalues of Hermitian Operators [duplicate]

In quantum mechanics, it's well-known that observables are associated as the eigenvalue of a Hermitian operator. My question is, is the converse also true? i.e. the eigenvalue of a Hermitian operator (...
0 votes
0 answers
54 views

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 ...
1 vote
0 answers
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Observables expressed by the lesser Green's function $\tilde{\mathcal{G}}^{<}(k, E)$

I am reading the paper PRB 106, 035102 (2022). In the supplementary materials, it says, within the Green’s function formalism, the thermal and quantum average of an observable θ is expressed as, $$\...
0 votes
1 answer
51 views

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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