Questions tagged [quantization]
Quantization refers to the procedure or methodology for replacing a classical system by a quantum system. If the question is about the quantized or discrete behavior of a phenomenon use the [discrete] tag instead.
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What are the reasons to expect that gravity should be quantized?
What I am interested to see are specific examples/reasons why gravity should be quantized. Something more than "well, everything else is, so why not gravity too". For example, isn't it possible that a ...
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Is a "third quantization" possible?
Classical mechanics: $t\mapsto \vec x(t)$, the world is described by particle trajectories $\vec x(t)$ or $x^\mu(\lambda)$, i.e. the Hilbert vector is the particle coordinate function $\vec x$ (or $x^\...
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What is the "secret " behind canonical quantization?
The way I (and perhaps most students around the world) was taught QM is very weird. There is no intuitive explanations or understanding. Instead we were given a recipe on how to quantize a classical ...
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How general is the Lagrangian quantization approach to field theory?
It is an usual practice that any quantum field theory starts with a suitable Lagrangian density. It has been proved enormously successful. I understand, it automatically ensures valuable symmetries of ...
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How does classical GR concept of space-time emerge from string theory?
First, I'll state some background that lead me to the question.
I was thinking about quantization of space-time on and off for a long time but I never really looked into it any deeper (mainly because ...
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Reason for the discreteness arising in quantum mechanics?
What is the most essential reason that actually leads to the quantization. I am reading the book on quantum mechanics by Griffiths. The quanta in the infinite potential well for e.g. arise due to the ...
user7757
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Why aren't the energy levels of the Earth quantized?
The Hamiltonian of the Earth in the gravity field of the Sun is the same as that of the electron in the hydrogen atom (besides some constants), so why are the energy levels of the Earth not quantized?...
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Operator Ordering Ambiguities
I have been told that $$[\hat x^2,\hat p^2]=2i\hbar (\hat x\hat p+\hat p\hat x)$$ illustrates operator ordering ambiguity.
What does that mean?
I tried googling but to no avail.
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Why does non-commutativity in quantum mechanics require us to use Hilbert spaces?
I am reading Why we do quantum mechanics on Hilbert spaces by Armin Scrinzi. He says on page 13:
What is new in quantum mechanics is non-commutativity. For handling this, the Hilbert space ...
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Is the quantization of gravity necessary for a quantum theory of gravity?
The other day in my string theory class, I asked the professor why we wanted to quantize gravity, in the sense that we want to treat the metric on space-time as a quantum field, as opposed to, for ...
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Central charge in a $d=2$ CFT
I've always been confused by this very VERY basic and important fact about two-dimensional CFTs. I hope I can get a satisfactory explanation here. In a classical CFT, the generators of the conformal ...
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Weyl Ordering Rule
While studying Path Integrals in Quantum Mechanics I have found that [Srednicki: Eqn. no. 6.6] the quantum Hamiltonian $\hat{H}(\hat{P},\hat{Q})$ can be given in terms of the classical Hamiltonian $H(...
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Quantization of a particle on a spherical surface
Suppose we have a particle of mass $m$ confined to the surface of a sphere of radius $R$. The classical Lagrangian of the system is
$$L = \frac{1}{2}mR^2 \dot{\theta}^2 + \frac{1}{2}m R^2 \sin^2 \...
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What is Quantization?
In classical mechanics you construct an action (involving a Lagrangian in arbitrary generalized coordinates, a Hamiltonian in canonical coordinates [to make your EOM more "convenient & symmetric"])...
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Why one-dimensional strings, but not higher-dimensional shells/membranes?
One way that I've seen to sort-of motivate string theory is to 'generalize' the relativistic point particle action, resulting in the Nambu-Goto action. However, once you see how to make this '...
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Difference between discretization and quantization in physics
I am just trying to understand the fundamental difference between these two concepts in physics:
From discreteness of some quantity: one usually interprets it as a quantity being only able to take on ...
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Does Quantum Mechanics assume space and time are continuous?
I was confused when I was listening to a Quantum Mechanics lecture online. Are space and time assumed to be continuous or discrete in Quantum Mechanics?
I can see the question is vague, but this is ...
user403
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The most general procedure for quantization
I recently read the following passage on page 137 in volume I of 'Quantum Fields and Strings: A course for Mathematicians' by Pierre Deligne and others (note that I am no mathematician and have not ...
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Geometric quantization of identical particles
Background:
It is well known that the quantum mechanics of $n$ identical particles living on $\mathbb{R}^3$ can be obtained from the geometric quantization of the cotangent bundle of the manifold $M^...
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Phonons in non-crystalline media
Do sound waves in a gas consist of phonons?
What about a glass? Or other non-crystalline materials such as quasicrystals?
How does the lack of translational symmetry affect the quantization of the ...
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Can symmetry generators be used for quantization?
Take the Poincaré group for example. The conservation of rest-mass $m_0$ is generated by the invariance with respect to $p^2 = -\partial_\mu\partial^\mu$. Now if one simply claims
The state where ...
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Rigorous proof of Bohr-Sommerfeld quantization
Bohr-Sommerfeld quantization provides an approximate recipe for recovering the spectrum of a quantum integrable system. Is there a mathematically rigorous explanation why this recipe works? In ...
Squark
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How does quantization solve the ultraviolet catastrophe?
I understand how classical physics leads to the UV catastrophe. But I cannot understand how quantization solves it.
How can quantization prevent the body from radiating a lot of energy?
I know this ...
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Making sense of the canonical anti-commutation relations for Dirac spinors
When doing scalar QFT one typically imposes the famous 'canonical commutation relations' on the field and canonical momentum: $$[\phi(\vec x),\pi(\vec y)]=i\delta^3 (\vec x-\vec y)$$ at equal times ($...
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Magnetic monopole and electromagnetic field quantization procedure [duplicate]
From the Maxwell's equations point of view, existence of magnetic monopole leads to unsuitability of the introduction of vector potential as $\vec B = \operatorname{rot}\vec A$. As a result, it was ...
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Bohr-Sommerfeld quantization condition from the WKB approximation
How can one prove the Bohr-Sommerfeld quantization condition
$$ \oint p~dq ~=~2\pi n \hbar $$
from the WKB ansatz solution $$\Psi(x)~=~e^{iS(x)/ \hbar}$$ for the Schroedinger equation?
With $S$ the ...
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Physically distinct quantizations
In J. Phys. A: Math. Gen. 22 (1989) 811-822, Crehan considered the classical Hamiltonian,
\begin{align}
H=\frac{p^2}{2}+\frac{q^2}{2}+\lambda(p^2+q^2)^3\,.
\end{align}
Due to the presence of the ...
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First quantization vs second quantization
What is the difference between first quantization and second quantization and where does the name second quantization come from?
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Is the quantization of the harmonic oscillator unique?
To put it a little better:
Is there more than one quantum system, which ends up in the classical harmonic oscillator in the classial limit?
I'm specifically, but not only, interested in an ...
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Dirac's quantization rule
I first recall Dirac's quantization rule, derived under the hypothesis that there would exit somewhere a magnetic charge: $\frac{gq}{4\pi} = \frac{n\hbar}{2} $ with $n$ a natural number.
I am ...
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Why does gravity need to be quantised?
The electroweak and strong forces seem to be completely different types of forces to gravity. The latter is geometric while the former are not (as far as I'm aware!). So why should they all be ...
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What is the rigorous definition of the verb "to quantize"?
I've studied QM and QFT for a couple of years now, so I'm familiar with the tersm "quantize", "quantization" and so on. I'm obviously also familiar with the Lagrangian description ...
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Are there any quantities in the physical world that are inherently rational/algebraic?
Whenever we measure something, it is usually inexact. For example, the mass of a baseball will never be measured exactly on a scale in any unit of measurement besides "mass in baseballs that are ...
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Derivative interaction: $\mathcal{H}_\mathrm{int}\neq - \mathcal{L}_\mathrm{int}$. Question about Feynman Rules
As we known, if there is time derivative interaction in $\mathcal L_\mathrm{int}$, then $\mathcal{H}_\mathrm{int}\neq -\mathcal{L}_\mathrm{int}$. For example, Scalar QED,
$$
\begin{aligned}
\mathcal{...
user153663
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Eigenvalues and eigenfunctions of the exponential potential $ V(x)=\exp(|x|) $
For $a$ being positive what are the quantisation conditions for an exponential potential?
$$ - \frac{d^{2}}{dx^{2}}y(x)+ ae^{|x|}y(x)=E_{n}y(x) $$
with boundary conditions $$ y(0)=0=y(\infty) $$
I ...
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What are the minimal postulates to do quantum mechanics in path-integral formulation without knowing the operator formulation?
I ask this question because many of the books I'm familiar with assumes a familiarity with the operator formulation and then develops the path-integral formulation partly based on a mixture of ...
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Can Fermionic symmetries be fully integrated into geometric deformation complexes or symplectic reduction?
How should a geometer think about quotienting out by a Fermionic symmetry? Is this a formal concept? A strictly linear concept? A sheaf theoretic concept?
How does symplectic reduction work with odd ...
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Connections between classical mechanics and quantum mechanics [duplicate]
I've been studying quantum mechanics and classical mechanics for a little while now, and I still don't feel as though I fully understand the motivation for some of our choices in Heisenberg mechanics. ...
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What fundamental reasons imply quantization?
In classical wave mechanics, quantization can occur simply from a finite potential well.
In quantum mechanics, the quantization is obtained from the Schrödinger equation, which is, to my knowledge, a ...
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Name of concept: Replace classical variables by quantum operators
I feel like there was a name for this sleight of hand approach and I've been unsuccessfully trying to google it for a while.
I think Heisenberg introduced it and it's basically "putting hats on ...
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Virasoro constraints in quantization of the Polyakov action
The generators of the Virasoro algebra (actually two copies thereof) appear as constraints in the classical theory of the Polyakov action (after gauge fixing). However, when quantizing only "half" of ...
Squark
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$\pi$, $\sigma$ - atomic transitions with respect to the magnetic field axis
I am confused about the atomic transition with different polarized lights.
I post the pictures as follows.
There are four cases.
In case 1, the right-handed circular polarized light ...
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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 ...
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Why there is no unique "recipe" for quantization of a classical theory?
I have seen in Wikipedia that different quantization methods exist (see Wiki article with name "Quantization"). Moreover, Wikipedia stated that there is more than one way to quantize a classical ...
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Why do negative norm states break unitarity?
I often hear my teachers say that the negative norm states break unitarity. And I can also read this elsewhere, such as at this place
In this gauge the relation between unitarity and gauge ...
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Gupta-Bleuler and Lorenz Gauge: I don't understand the principle behind Gupta-Bleuler
I would like to make the link between the Gupta-Bleuler Lagrangian and the Lorenz Gauge for Electromagnetism because everything is not clear to me.
I am looking for a simple explanation without too ...
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Quantum gravity at D = 3
Quantization of gravity (general relativity) seems to be impossible for spacetime dimension D >= 4. Instead, quantum gravity is described by string theory which is something more than quantization (...
Squark
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Canonical quantization in supersymmetric quantum mechanics
Suppose you have a theory of maps
$\phi: {\cal T} \to M$
with $M$ some Riemannian manifold,
Lagrangian
$$L~=~ \frac12 g_{ij}\dot\phi^i\dot\phi^j + \frac{i}{2}g_{ij}(\overline{\psi}^i D_t\psi^j-D_t\...
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Equal time commutation relations in canonical quantization of relativistic free fields
Why is equal time commutation relation used in canonical quantization of relativistic free fields? In a relativistic theory, space and time are to be treated on equal footing.
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How to promote algebraic expressions to operators in quantum mechanics?
Okay, I know that in quantum mechanics the quantum observable is obtained from the classical observable by the prescription
$$ X \rightarrow x,\quad P \rightarrow -i\hbar\frac{\partial}{\partial x} $...
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