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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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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560 views

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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205 views

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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234 views

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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131 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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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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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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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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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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139 views

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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106 views

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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783 views

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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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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535 views

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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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^...
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Spin operators on two spin-$\frac{1}{2}$ particle system

In a text I am using (Introduction to Quantum Mechanics by Griffiths), this is an introductory text hence much of rigor is omitted. The following is stated: Suppose that we have two spin-$\frac{1}{2}$...
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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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Is the expectation value the same as the expectation value of the operator?

I was reading the book Introduction to Quantum Mechanics by Daniel Griffith, and also following Brant Carlson's videos. He basically makes videos about parts of the book. The book was discussing $\...
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Average value of an observable

I got confused by the concept of the average value of an observable. I know that when we measure a physical quantity $A$ in a specific state described by $\psi_1$, we only get the eigenvalue $a$ of ...
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“Any physical law which can be expressed as a variational principle describes a self-adjoint operator”

The title is a Wikipedia quote but has no citation associated to it on Wikipedia except a reference to Cornelius Lanczos. I would like to clarify the statement, and explore the relevant concepts in ...
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306 views

How to think of matrices as observables?

I'm reading Nielsen and Chuang. In one of the early chapters, they introduce some matrices such as $$X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}.$$ They interperet this as a gate that ...
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Interpretation of observable of quantum state equation

In quantum mechanics, my understanding of operators related to observables are as follows: By a postulate of QM, every observable can be represented as an Hermitian operator. The eigenfunctions of ...
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Expectation values of $x$, $y$, $z$ in hydrogen?

The expectation value of $r=\sqrt{x^2 + y^2 + z^2}$ for the electron in the ground state in hydrogen is $\frac{3a}{2}$ where a is the bohr radius. I can easily see from the integration that the ...
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Simultaneous eigenbasis of the energy and momentum operators of a particle in a 1-dimensional box

Does there exist a simultaneous eigenbasis of the energy operator $T=\hat{p}^2/2m$ and the momentum operator $\hat{p}=-i\hbar\, d/dx$, for a particle in a 1-dimensional box of unit length? Actually, ...
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What is difference between operating wave function with operator of an observable and measuring for an observable?

People say operator of an observable helps in measuring for an observable. We also know that measuring leads to collapse of wave function. But operator on wave function gives a number times same wave ...
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Bras and kets of continuous spectrum

Does anyone know why in quantum mechanics the second statement is always true? "When the spectrum of an operator $A$ has a continuous part, we associate a bra $\langle a|$ and a ket $|a \rangle$ ...
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Unbounded operators defined only on dense subdomain of Hilbert space in QM?

I am relatively new to quantum mechanics. In a set of notes I am using, the following is a description of an aspect of some operators corresponding to observables. The notes state the following: "...
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Does the outcome depend on the phase arbitrariness in the definition of the eigenstates?

Imagine you have a Hamiltonian that is a real symmetric matrix in some basis. From general theory we know that the eigenvalues $E_n$ are real but also eigenvectors can be chosen purely real $|n\rangle$...
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Time travel backwards, would anything be observable? [closed]

I'm by no means a physicist theoretical or otherwise, so please do excuse any things that I may be ignorant of (likely a very long list). If by whatever mechanism a traveller were able to travel back ...
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Commutator and Order of Measurement

I was going through Prof. Leonard Susskind's lectures on Quantum Field Theory (Lec 2). Professor said that the commutator of two observables $AB-BA$, has nothing to do with the 'measurement'- B ...
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Expectation value of position of a particle

As you might be able to tell from my question that I'm just starting to learn about quantum mechanics. My question is the following: With a wave function like this: $\displaystyle{\Psi (x, t) = ...
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2answers
819 views

Hermiticity of Momentum Operator (matrix) Represented in Position Basis

I read that the momentum operator, $\hat P$ should be Hermitian (some would say by QM postulate). That makes perfect sense when it is represented in the momentum (p) basis: $\hat P =\int_{-\infty}^{\...
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3answers
446 views

Collapse of wave function

Suppose a quantum system is initially at a state $\psi_0$ and that a measurement of an observable $f$ is performed. Immediately after the measurement, the system will be in a state that is an ...
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What information does the mean value of a matrix give us? [closed]

Last time I found that question and even I spent a lot of time I didn't find any answer. By mean value of matrix, I mean ...