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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Why is this the exact shape of expectation values in the path integral formalism?

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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Resolution of the identity of operator with mixed spectrum

In most quantum mechanics text books, the resolution of the identity or completeness relation is stated in the following (or similar) form $$ \mathbb I_\mathcal H = \sum\limits_n |\lambda_n\rangle \...
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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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Can one measure two components of spin exactly by measuring two components of entangled, say, electrons?

Of a single electron, two different components of spin can't be have simultaneously well defined values. But what if we entangle two of them and we measure, say, $(S_1)_z$ and $(S_2)_x$ simultaneously....
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Path-integral and measurements [duplicate]

I did a question a couple days ago and I didn't express it correctly. What happened is that i have a field obtained by solving the equations of motion of the $\lambda\phi^{4}$ theory. It was solved in ...
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How to get the weight of an eigenstate inside the state of the system without knowing the state?

Let us suppose we have a system in a state $\Psi$, with: $\Psi = \sum_m c_m \psi_m$ Let us further suppose that we don't know what $\Psi$ or the $c_m$ are, but that we know what the $\psi_m$ are since ...
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What is the physical meaning of the eigenstates of an operator in quantum mechanics?

Let us suppose that we have an Hamiltonian that describes a quantum system. If one would like to know all of the possible values that the energy of the system described by that hamiltonian, one has to ...
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Help with path integral formalism for quantum field theory [duplicate]

Recently i was doing some work with quantization of fields. I learned to quantize using the canonical method, writing in terms of the ladder operators. Then I saw that there was a more powerful method,...
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Why aren't the thermodynamic suceptibilities zero in the thermodynamical limit?

As it is explained in this answer (and nicely so!), the second derivative of a thermodynamic potential to an intensive quantity, for example pressure or magnetic field strength (or temperature) will ...
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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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Minimal variance for an operator

For an operator $Q$, the variance is given by: $$\left\langle Q^2 \right\rangle - \left\langle Q \right\rangle^2$$ If the eigenvalue equation for the operator is given by $$\hat{Q} |q_i \rangle = q_i |...
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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 it mean to "quantise" a system?

Suppose we have a physical system, let's say a ring of $N$ atoms held together by elastic force. (This is just an example, we could have picked any physical system) Classically we can easily find the ...
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What are measurable physical quantities?

The book "Fundamentals of many body physics" by Wolfgang Nolting at the beginning of chapter 3 says: Measurable physical quantities are: the eigenvalues of observables the expectation ...
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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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Valid Intuition? - Why observables are represented by eigenstates/eigenvalues

So I've been frustrated with the usual presentation of the operator formalism being presented as an axiom, and have been after a more intuitive explanation. Would the following intuition be considered ...
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Can any eigenstate of a degenerate quantum mechanical system be expressed as a linear superposition the totality of its possible eigenfunctions?

Sorry for the terribly long question title. In a non-degenerate quantum mechanical system, the totality of the linearly independent eigenfunctions (chosen to be orthonormal), $(\phi_1,\phi_2,...,\...
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How does the Fourier picture relate to non-commutativity?

A compelling video by 3Blue1Brown visualizes the uncertainty principle with Fourier transforms. The gap I'm trying to bridge is between this "Fourier picture" and the matrix-based statement $...
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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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Is the uncertainity principle explained by disturbances or only by the Fourier picture?

Qualitatively, the tradeoff in uncertainty between two non-commuting observables $\hat{x}$ and $\hat{y}$, could be explained by... the Fourier picture where the more one variable is defined (i.e., ...
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Cross Product of two Hermitian Operators

The operator for linear momentum $\mathbf{p}$ and the operator for orbital angular momentum $\mathbf{L}$ ($\mathbf{L} = \mathbf{r} \times \mathbf{p}$) are Hermitian. Is the cross product between $\...
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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 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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Why did Heisenberg (in his 1925 paper) assume his observables had the following time dependence? [closed]

Many of the assumptions made by Heisenberg in his revolutionary 1925 paper could be justified in some form or another (although they are not by any means obvious), like for example his matrix ...
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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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Symmetries in QM

I have the following question; if we an operator corresponding to a spacetime translation: $$ \hat{\Omega}$$ and a hermitian operator, $$\hat{A}$$ commutes with this translation: $$[\hat{\Omega},\hat{...
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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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$\Delta E \Delta t \geq \hbar /2$ and energy conservation

As I understand it, time is not an observable in quantum mechanics, therefore $\Delta E \Delta t \geq \hbar/2$ is just a way to say that, given any operator $\hat{A}$, $$\sigma_H\sigma_A\geq\frac{\...
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In lay terms, what are the real world consequence of the gauge invariances/symmetries upon which the Standard Model is built?

We learn that the SM is based on gauge invariance. Gauge invariance in turn is a consequence of symmetries (as I understand it) - meaning that a gauge theory having a symmetry is what makes it a gauge ...
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For a generic two-state quantum system, are there interpretations for the observables corresponding to all Hermitian operators?

The simplest non-trivial system is a two-level system. Classically, it is a system which can be in one state labelled $H$ or another state labelled $T$. There is no necessary reference to any ...
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Does the generator of the group need to be physical observables?

In the Class, We have been told that the compact groups like $SO(3)$ have hermitian generators. Now in $SO(3)$, these generators turn out to be angular momentum components apart from some dimensions. ...
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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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Observables in QM, are they the eigenvalues or the elements of the spectrum?

The question is rather simple. We all hear all the time that an observable of a quantum mechanical system (for some observable that is given by a some self-adjoint operator, let's call it $H$) is ...
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What would go wrong if quantum observables were not represented by linear operators? [closed]

If quantum mechanical operators corresponding to physical observables were not hermitian, the corresponding eigenvalues may not be real. Since the eigenvalues are the outcomes of measurements of ...
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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| \...
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How to simply explain 'quantum state' to a beginner?

While explaining 'quantum state' to a beginner, is it scientifically accurate to say that "just like '$v$' represents velocity and '$p$' represents the momentum of an object, $|ψ\rangle$ ...
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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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Trying to understand post-measurement density matrices in a state that spans 2 Hilbert spaces

What I would like to understand mathematically is the following situation: Prepare a quantum state that spans two Hilbert spaces Operate on one space with observable operator $\hat{O}$. Obtain ...
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Can an Hermitian unitary matrix in a Hilbert-rigged space be 3-dimensional?

While studying a couple of concepts, I've understood the following premises to be true: Hermitian unitary matrices eigenvalues are unimodal (that's $\pm1$). In physics, operators/observables are ...
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Angle operator $\hat{\phi}$ doesn't exist when doing quantum mechanics on the circle $S^1$?

$\newcommand{\ket}[1]{|#1\rangle}$When doing quantum mechanics on the circle $S^1$, it is well documented (yet seemingly controversial) that a self-adjoint "angle/position operator" $\hat{\...
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Can local supersymmetry be characterized entirely in terms of observables?

Global symmetries can be defined through their effect on observables. In contrast, quantum theories are often constructed with the help of symmetries that leave observables invariant, like the ...
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Inserting a position operator in the path integral in QFT

With the usual path integral description, we have the formula $$\langle q''t''|q't'\rangle =\int\mathcal{D}q \exp{(iS)}$$ where $S=\int_{t'}^{t''}L(q,\dot{q})$ is the action evaluated for $t\in (t',t''...
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Do states with infinite average energy make sense?

Do states with infinite average energy make sense? For the sake of concreteness consider a harmonic oscillator with the Hamiltonian $H=a^\dagger a$ and eigenstates $H|n\rangle=n|n\rangle$, $\langle n|...
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Compatible observables for a quantum harmonic oscillator 3D

In quantum mechanics, from what I understand from the theory, if any operators commute then they have a complete set of simultaneous eigenvectors. This means that any vector of space can be expressed ...
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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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Is the number of possible values of a quantum observable finite, countably infinite or uncountably infinite?

Do we know, or have a theory about, whether the number of possible values any fundamental quantum property can assume upon observation is finite, countably infinite or uncountably infinite?
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Observables that encode all the information in a wavefunction

Let's consider the position space representation of the single particle Hilbert space and for simplicity let's stick to one dimension: $L^2(\mathbb{R})$. Let's say a collection of observables $O_1,...,...
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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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Measurement and Born rule for observable with a degenerate eigenspectrum after von Neumann

Suppose $A$ is an observable with a degenerate eigenspectrum, coming with a Hilbert space $H$. Let $E$ be the eigenspace corresponding to the eigenvalue $\lambda$ of $A$, and suppose the dimension is ...
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Do I run into trouble if I interpret the fermionic field operator as a linear combination of a real and an imaginary part?

As some other questions on this website suggest, I have a really hard time with the fermionic field operator $\psi(x)$. I'd like to come to terms with this blockade. It serves as the smallest building ...
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