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 the Expectation Value of a unnormalised state $|\psi\rangle$ is $\langle A\rangle_\psi=\frac{\langle\psi|A|\psi\rangle}{\langle\psi|\psi\rangle}$? [closed]
The typical expectation value formula given most places is
$$\langle A \rangle_\psi = \langle\psi|A|\psi\rangle.$$
this assumes that the state is normalised.
For unnormalised states the formula is
$$\...
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Direct observables in observers hand
Observation of a measurable is the event of interaction between the observer and the universe after which a value of that measurable is obtained.Since all observations occur solely by interactions and ...
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Can counterfactual measurements be used to measure incompatible observables?
Imagine we have a qubit system prepared by choosing one of two states as measured in one of two possible, incompatible bases, i.e. there are four total possible states, such as e.g. polarizations of 0,...
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Physical meaning of a complete set of compatible observables
I am a Math student, new at Quantum Mechanics, and I am having some troubles understanding the physical meaning of the notion of “complete set of compatible observables". I know its mathematical ...
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Why are the principal quantum number $n$, orbital angular momentum quantum number $l$, and magnetic quantum number $m$ so important for hydrogen atom?
They commute with each other and are therefore good quantum numbers. Furthermore, the energy is the eigenvalue in the time-independent Schrödinger equation. But are there other reasons I missed?
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Observing $L_x$ of a particle which is present in an eigenstate of $L_z,L^2$
I wonder what will be the probability of finding the particle in a particular $L_x,L^2$ state after we know its $L_z,L^2$. Definitely,the $L^2$ value will not change. Here is what I tried:
First of ...
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Does an electron have an indeterminate time like its position before measurement?
Consider the case of quantum entity :
If electron is present in region of high curvature of spacetime geometry
Wavefunction of electron is spread over large region such that difference in time ...
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Commutator relation and its acurate meaning
Suppose there are 2 sets of complete basis states $S_1,S_2$. That is each state in $S_1$ are orthogonal to the other states in $S_1$. Orthogonal states are linearly independent . Meaning ,it is ...
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Difficulty while reading Quantum Statistics [duplicate]
]1
While reading the following topics from Kerson-Huang statistical mechanics, I encountered this . Here $\psi$ is the wavefunction of the system plus surroundings . There is interaction between the ...
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What generates translation in a qubit measurement?
I have been working with quantum theory for a while, but have stumbled across a concept I can't wrap my head around.
My question was initially inspired in reading the Wikipedia article on Bells ...
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Do quantum entities lack definite positions because of measurement limits, or is it a fundamental feature of nature? [duplicate]
In quantum mechanics, particles such as electrons do not have a fixed position until they are measured
Questions:
Is this indeterminacy simply due to the limitations of our current measurement ...
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Why is the calculation of variance using operators in quantum mechanics an expectation value?
Variance of an eigenvalue q from operator $\hat{Q}$ is defined as $$\delta q^2 = \int_{-\infty}^{\infty}\psi^*(\hat{Q} - \langle Q\rangle)^2\psi\ dT,$$ where this is an integral over the entire space ...
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Defining or Listing Measurable/Observable quantities [duplicate]
In Quantum Mechanics, certain quantities are considered to be "measurable" or "observable". We demand the following properties of these quantities:
They must be real
They must be ...
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Why are expectation values in quantum mechanics calculated the way they are?
To calculate expectation value of some variable $Q$, we take:
$$\langle Q\rangle = \int_{-\infty}^{\infty}(\psi^*\hat{Q}\psi)\mathrm{d}x.$$
This is however assumed from the concept of $|\psi(x, t)|^2$ ...
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How does quantum tunneling conserve Energy?
Picture a particle in a square well where:
\begin{align}
-\frac{\hbar^2}{2m}\nabla^2 \psi = E\psi &\qquad \text{if $|x|<L$} \\
\left ( -\frac{\hbar^2}{2m}\nabla^2 + V_0\right) \psi = E\psi &...
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Measuring quantum velocity by measuring position at two instant separated by $dt$. Final state is in position eigenstate, not momentum...?
My question is closely related to this one, but it asks a more precise/different confusion I have.
I consider a free quantum particule starting in the position eigenstate $|x_0\rangle$, at time $t_0$.
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Why is the expectation value given by $\langle\Psi\!\! |A\,|\Psi\rangle$ in quantum mechanics?
Suppose a particle is in the quantum state $\vert \Psi \rangle$. Then the expectation value for an observable $A$ is given by:
$$\langle \Psi \vert A \vert \Psi \rangle.$$
But why is this the case? ...
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Examples of simultaneous eigenstates for non-commuting observables [duplicate]
In Dirac's textbook The Principles of Quantum Mechanics, Ch. 2 part 13, he explicitly discusses the relationship between the commutability and compatibility of observables. In one part of the text, he ...
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Particle phase as part of a phase space
In quantum mechanics observables tend to come with conjugates, e.g. position and momentum. Suppose that a particle has an observable phase, an $e^{i\omega t}$ variation* which can interfere ...
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Relations between compatible observables [duplicate]
Let's say we have three observables $A$, $B$ and $C$. If $A$ is a compatible observable with $B$, and $B$ is a compatible observable with $C$, then is it true that $A$ is compatible with $C$? I've ...
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Adding weighted operators? [duplicate]
I’m reading Quantum Mechanics: A Paradigms Approach by McIntyre. There is a derivation of the operator for spin of a spin-1/2 particle in an arbitrary direction. This begins by finding the dot product ...
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Meaning of operating on non-eigenvectors? [closed]
I understand that in quantum mechanics we can represent an observable with a matrix that has certain vectors as eigenvectors, and these correspond to observable states. But we already have the ability ...
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What does it actually mean for quantum phase to be "uncertain" in the number–phase uncertainty relation? $\Delta \phi \cdot \Delta N \gtrsim 1$
I’m trying to get a clearer physical and intuitive understanding of the number–phase uncertainty relation in quantum mechanics, especially in quantum optics.
$$\Delta \phi \cdot \Delta N \gtrsim 1$$
I ...
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Why must the eigenstates of an observable form a basis? [duplicate]
I recently asked a similar question but it was from the point of view of axiomatic quantum field theory, so I'd like to ask it in a more general form. Given a quantum theory (either quantum mechanics ...
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In NRQM why are gauge equivalent electromagnetic potentials physically equivalent?
In the NRQM'al theory of a charged particle, if we gauge transform the potentials using some arbitrary gauge function $\Lambda(\textbf{X},t)$ then Schrodinger's equation implies that the quantum state ...
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How to find operators commuting with hamiltonian?
I am studying quantum mechanics and learned that finding operators commute with hamiltonian is quite important to understand the system.
But does the method of finding operator that commutes with ...
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Classical Mechanics in the Mackey's axiomatization of a physical system
G. L. Naber in its Quantum Mechanics gives an axiomatization of a physical system in mathematical terms. The story goes like this:
Definition. A physical system is composed of:
A collection $ \...
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Intuitive understanding for field strength renormalization
Suppose we are calculating some (general) 2-point function $\Sigma_n(p)$ at some order $n$ that contains at least one-loop. Let's say, for simplicity, $\Sigma_n$ diverges with one pole, so
$$\Sigma_n(...
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Why $S(\mathcal{P})^2= 1$ for scalar fields, but only $D(\mathcal{P})^2=\pm 1$ for fermions?
Srednicki in his book "Quantum field theory" on page 141 has mentioned that $$P^{-1}\varphi (x)P=S(\mathcal{P})\varphi (\mathcal{P}x),\tag{23.6+10}$$ where $\mathcal{P}$ is parity, $P\equiv ...
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What is the meaning of the variables in the Lagrangian before renormalization?
I'm currently working through Coleman's lectures on Quantum field theory and I cannot quite get my head around the whole concept of renormalization.
What I am aware of, is that you have some ...
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Why isn't there a four-position operator in QFT? (real scalar field) [duplicate]
I want to give a brief overview as to where I am coming from with this question.
When we go from classical mechanics to QM, one the crucial things we consider is that $\vec x$ and $\vec p$ are ...
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What blocks these causality-violating observables in QFT?
Some questions have been asked on this site [1] [2] about causality violation in relativistic quantum mechanics of a single particle (RQM) vs quantum field theory (QFT), but I have a very specific ...
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What do we mean by coordinate is not physical in GR exactly?
The meaning of this question is: In GR we describe everything with objects defined on manifold directly, and we just need some coordinate system to express these result.
However, suppose in ...
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Is there such a thing as the spectrum of an operator in the path integral formalism?
It is often claimed that the path-integral formalism and the Schrodinger one are equivalent, and the proof usually involves calculating amplitudes in both ways and checking that they are the same, or, ...
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If I measure $Z$ on a state $\rho$ and the outcome is $1$, can the system end up in a different state than $|0\rangle\langle 0|$?
Suppose I perform the Projector Valued Measurement (PVM) of the observable $Z = |0\rangle\langle0|-|1\rangle\langle1|$ on the arbitrary initial state $\rho$ (mixed or pure) and I obtain the outcome $1$...
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The wavefunction and degenerate states
I am trying to learn QM by following MIT 804 online lectures by Prof. Adams. There are two statements mentioned in the beginning lectures which do not make sense to me at all.
In classical mechanics, ...
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Deriving the total angular momentum operator $\hat{\vec{L}}^2$
In Bohmian mechanics, the operators for physical observables can be derived really naturally by calculating the average physical quantity. It is possible for example to derive the average angular ...
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Conditions/Constraints required to know the mass distribution of a system?
Let's say we have a planar mass system (all the patches of area covered lie in the same plane)
If we know the center of mass coordinates, the moment of inertia about an axis at any point perpendicular ...
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Books on QED with an emphasis on measurement?
Most books I've found about QED seem to focus on calculating probabilities without much mention of how the states corresponding to them are measured.
On the other hand, the books which focus the most ...
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In general relativity, how do we know when the coordinates we compute are physical observables?
(One of) the whole point of general relativity, is that the coordinates we mathematically use are just "labels", that can change and live on a curved surface. But at some point we have to ...
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Is the momentum wave function's square amplitude always time-invariant for a free particle?
I have noticed whenever working with free particles that the square amplitude of the momentum wave function $|\Phi(p)|^2$ ends up being time invariant, so I followed this chain of logic supporting the ...
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How to prove the Hermiticity of the angular momentum operator's X-component? [closed]
I understand that the momentum operator is Hermitian (thanks to this proof), as demonstrated by verifying the inner product relation. However, I am unsure how to prove that the angular momentum ...
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Heisenberg's uncertainty isn't about measurement
If we are given two operators $\hat A $ & $\hat B$ corresponding to the physical quantities $A$ & $B$, then for a given wave function $\Psi$ we know that the average values of those quantites ...
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Joint distribution of possible eigenvalues when applying two commutating operators
I'm new to QM and trying to understand the basics of commutators. If you have two operators that commute, then it is said you can measure two observables simultaneously. However, I'm a bit confused:
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Does uncertainty mean fixed, but unmeasurable underlying value, or a measurable value that fundamentally fluctuates? WRT specific question
Consider the following question from Eisberg and Resnick's "Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles," pg 84, #4.34
A boy on top of a ladder of height $H$ is ...
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Does the Heisenberg uncertainty principle only allow location OR momentum to exist?
To elaborate, if you know the exact location of a particle with perfect accuracy, does this mean that the momentum of it still exists but we just don't know what it is/ can't measure it? Or does this ...
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Why we compute expectation value of an operator?
I'm new in quantum mechanics. J.J.Sakurai in his book in chapter 1 defines the expectation value of an oeprator $A$ taken with respect to state $|\alpha \rangle $ as $\langle A\rangle \equiv \langle \...
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Validity of wavefunction being continuous and differentiable after collapse of the wavefunction to a $\delta$-function
Suppose we have a wavefunction of a single particle in a potential. We measure it's position. After collapse the wavefunction collapses to a single eigenstate of position. This means that the ...
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Non-finite expectation values in quantum mechanics [duplicate]
In textbook quantum mechanics, one deals with expectation values of the form
$$\langle O \rangle = \text{tr}(\rho O)$$
where $\rho$ is assumed to be trace-class (in particular, $\text{tr}\rho = 1$). ...
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How do we reproduce QFT's predictions about local field observables, in the low energy limit of predictions of string theory?
I was reading this answer about how string theory may reproduce QFT. It talks about how, in the low energy limit, both theories may have the same scattering amplitudes.
This means that string theory ...
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