Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

13
votes
4answers
1k views

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 ...
6
votes
2answers
304 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 ...
3
votes
1answer
135 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 ...
1
vote
1answer
102 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 \...
2
votes
2answers
363 views

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 ...
3
votes
1answer
731 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 ...
0
votes
2answers
259 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 ...
4
votes
3answers
489 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 \...
3
votes
2answers
3k views

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 ...
1
vote
2answers
219 views

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 ...
2
votes
0answers
137 views

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)]|...
0
votes
1answer
190 views

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 $...
0
votes
0answers
162 views

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^...
2
votes
1answer
364 views

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-$\frac{1}{2}$...
2
votes
1answer
2k views

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: $$ {\...
-1
votes
1answer
257 views

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 $\...
3
votes
3answers
3k views

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 ...
8
votes
1answer
320 views

“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 ...
2
votes
1answer
274 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 ...
2
votes
2answers
123 views

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 ...
1
vote
1answer
1k views

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 ...
5
votes
1answer
898 views

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, ...
1
vote
2answers
493 views

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 ...
2
votes
1answer
189 views

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 $|a \rangle$ ...
5
votes
2answers
357 views

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: "...
2
votes
1answer
58 views

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 $|n\rangle$...
-1
votes
1answer
105 views

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 ...
2
votes
2answers
539 views

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 ...
0
votes
1answer
42 views

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) = ...
2
votes
2answers
718 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}^{\...
3
votes
3answers
434 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 ...
-6
votes
1answer
183 views

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 ...
1
vote
1answer
1k views

Definitions of position operator in QM

We define the position operator $\hat{X}$ by $$\hat{X}|\psi\rangle := \bigg(\int dx |x \rangle x \langle x | \bigg) | \psi \rangle \tag{1}$$ for some state vector $| \psi \rangle \in \mathcal{H}$. ...
2
votes
1answer
133 views

Quantum states after real world measurements

Regarding measurements of an observable in a quantum system. My understanding, from the postulates of quantum mechanics, is that when we measure an observable quantity, the state of the system ...
8
votes
0answers
325 views

Where does a fermionic coherent state live (which Hilbert space)?

There have been a couple of questions on fermionic coherent states, but I didn't find any that covered the following question: If I define a coherent fermionic state in the 2-level-system spanned by $...
3
votes
2answers
268 views

What happens to Pauli's argument (that says that there is no time operator) when applied to $X$ operator for some simple systems?

An argument by Pauli is usually referred to in the literature when it is stated that there cannot be a time operator in quantum mechanics. This argument can be found as a footnote to P63 of W. Pauli, ...
2
votes
1answer
407 views

Lagrangian gauge theory with physically observable local degrees of freedom

In my answer at What, in simplest terms, is gauge invariance?, I mentioned that in certain contexts there can be a "gauge theory" with a local symmetry that leave the Lagrangian/Hamiltonian invariant ...
2
votes
1answer
189 views

Why are general wave functions expressed in terms of energy eigenfunctions?

I have read that the eigenfunctions of any hermitian operator can be used as a basis to express any function, but I have only ever really seen the eigenfunctions of the Hamiltonian used. Why is this? ...
2
votes
2answers
139 views

How do I get observables to calculate uncertainty?

Given an infinite potential square well with $0<x<L$, I need to calculate the uncertainties of position and momentum. The eigenstates in the position basis are $$\lvert E_n\rangle\to \psi_n(x)=\...
10
votes
1answer
791 views

In quantum mechanics, how exactly do we associate Hermitian operators to classical observables? [duplicate]

In a first course on quantum mechanics, everybody learns some version of the following statement: Postulate: To every classical observable $A$ of a physical system, there corresponds a Hermitian ...
7
votes
2answers
2k views

Why hermitian, after all? [duplicate]

This question is going to look a lot like a duplicate, but I've read dozens of related posts and they don't touch the subject. Here we go. Why are observables represented by hermitian operators? ...
0
votes
1answer
1k views

Bohr frequency of an expectation value?

Consider a two-state system with a Hamiltonian defined as \begin{bmatrix} E_1 &0 \\ 0 & E_2 \end{bmatrix} Another observable, $A$, is given (in the same basis) by \begin{bmatrix} 0 &a \...
0
votes
1answer
69 views

Interpretation of two different observables, both with the same resolution of the identity

Suppose you have a resolution of the identity $\hat{\mathbb{1}}=\sum_i\hat{p_i}$ (pairwise othogonal), and construct two (non-degenerate) pvm observables, $\hat{B}=\sum_ib_i\hat{p_i}$ and $\hat{C}=\...
19
votes
7answers
2k views

Why is a Hermitian operator a “quantum random variable”?

To me, as a stupid mathematician, a random variable is a measurable function from some probability space $(\Omega, \sigma, \mu)$ to $(\Bbb{R}, B(\Bbb{R}))$. This makes sense. You have outcomes, events,...
5
votes
1answer
205 views

Our choice of basis surely cannot effect possible outcomes of a measurement?

Common sense says that, of course, the outcome of a measurement on a quantum system cannot be affected by what base we choose to represent it in. However, while studying QM text, it seems like they ...
0
votes
1answer
72 views

Gauge Bosons at Finite Temperature

I was reading a paper¹, and it states: " Therefore, the gauge fields themselves cannot be entities of the physical reality, as any observations should be independent of the chosen gauge" I'm trying ...
0
votes
1answer
209 views

Does Heisenberg's uncertainty hold for any two quantum measurements?

Heisenberg's uncertainty principle is most commonly expressed in terms of the uncertainty in measurement of position and momentum of a particle, $$\Delta x\Delta p \geq \hbar$$and uncertainty in ...
1
vote
0answers
68 views

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 $ ?
5
votes
1answer
351 views

Heisenberg's uncertainty principle derivation in a ring [duplicate]

The standard derivation But now suppose the space is a ring of length $L$, it seems the derivation could work out exactly the same and we get $$\Delta p \Delta x \geq \hbar/2.$$ But since $\Delta x$ ...
3
votes
2answers
2k views

How to recognize a Complete Set of Commuting Operators (CSCO)

A question about 'completeness'. These two operators are commuting, but I want to know more about their completeness. How do you know if {H}, {B}, {H,B} and/or {$H^2$,B} are forming (a) Complete Set(...