What is Quantization: Definition and 292 Discussions
Quantization, in mathematics and digital signal processing, is the process of mapping input values from a large set (often a continuous set) to output values in a (countable) smaller set, often with a finite number of elements. Rounding and truncation are typical examples of quantization processes. Quantization is involved to some degree in nearly all digital signal processing, as the process of representing a signal in digital form ordinarily involves rounding. Quantization also forms the core of essentially all lossy compression algorithms.
The difference between an input value and its quantized value (such as round-off error) is referred to as quantization error. A device or algorithmic function that performs quantization is called a quantizer. An analog-to-digital converter is an example of a quantizer.
Today we know that if you make successive Stern-Gerlach measurements the beam of atoms will split according to this formula:
> cos^2 (theta/2)
And this is something people back then could have figured out, they could have done many measurements, plotted the values, and realized it followed...
Hey all,
I am encountering an issue reconciling the choice of prefactors in the canonical quantization of the scalar field between Srednicki and Peskin's books. In Peskin's book (see equation (2.47)), there is a prefactor of ##\frac{1}{\sqrt{2E_{p}}}## whereas in Srednicki's book (see equation...
I noticed that ##V(\phi)## has nonzero minima, therefore I found the stationary points as ##{{\partial{V}}\over{\partial\phi}}=0##, and found the solutions:
$$\phi^0_{1,2}=-{{m}\over{\sqrt{\lambda}}}\quad \phi^0_3={{2m}\over{\sqrt{\lambda}}}$$
of these, only ##\phi^0_3## is a stable minimum...
Is Loop quantum gravity canonical quantization of Ashtekar's new variables
correct ?
if not in principle is there any particular ways to canonical quantization of Ashtekar's new variables ?
are there other methods to quantization of Ashtekar's new variables ?
Modern physics describes matter by real numbers. This means that an absolutely accurate description of any particle requires an infinite amount of information.
Intuitively, it seems that this should not be so, and the model of the Conway's Game of Life looks more close to reality. In this...
This is a question specifically for @A. Neumaier !
At Peter Woit's blog, Arnold commented about his formalism for quantum mechanics, coherent quantization. I left a question but Peter Woit doesn't always let comments through, so, here is the question:
Why aren’t you restricted to unitary...
Hi everyone,
It is about the quantization of the electromagnetic field. The expression of field E and B are defined with:
-the annihilation a- and creation a+ operators, and the frequency ω.
So my question is: how does these fields must be expressed if they where "static"? I mean, how the...
Hello ! I require some guidance on this prove :I normally derive the Hamiltonian for a SHO in Hilbert space with a term of 1/2 hbar omega included. However, I am unsure of how one derives this from Hilbert space to Fock space. I have attached my attempt at it as an image below. Any input will be...
I studied the basics of geometric quantization for a recent work in quantum-classical hybrid systems1. It was an easy application of the method of gometric quantization (prequantization + polarization in ##\mathbb{R}^{3}##).
The whole topic seems interesting since I want to learn more of...
Recall that in the semi-classical Bohr-Sommerfeld quantization scheme from the early days of quantum mechanics, bound orbits were quantized according to the value of the action integral around a single loop of a closed path. Clearly this only makes sense if the orbits in question permit closed...
Are you aware of the 3-article series of Wiesendanger's quantized extension of GR?
This is open access: C Wiesendanger 2019 Class. Quantum Grav. 36 065015 and the two sequels linked to in the PDF. The question is if this work counts as a quantization of a reasonable extension or reformulation...
Hi,
I'm still unclear on the quantization of light.
I watched this 1m video called "Why Light is Quantum" - Why Light is Quantum by minutephysics.
The author says light has the same energy distribution as a gas?
What does this mean?
What is an example of the energy distribution of a gas...
In Quantum Field Theory and the Standard Model by Schwartz, he defines the Hamiltonian for the free electromagnetic field as
(page 20, here's a link to the book). This follows (in my understanding) from the fact that the amplitude of the field at a given point in space oscillates as a simple...
The Wilson-Sommerfeld quantization rule claims (##\hbar=1##)
$$\frac{1}{2\pi} \oint p(x)\,dx=n,\,n=1, 2, ...$$
where integration is done in the classically allowed region. Applying this to a square-well potential with a depth of ##V_0## and width ##a##, we get $$E=\frac{\pi^2 n^2}{2a^2}$$
This...
We know that the energy levels for electrons surrounding nucleus are quantized , only coming in discrete levels.
When I see the standard model of elementary particles table I notice specific masses for each of the particles whether they be quarks or leptons or bosons like the higgs.
I know that...
This is a very remedial question, so thanks in advance for you gentle indulgence :smile: Where do I find the quantization term (the "n") in Planck's Law?
Can anyone explain while calculating $$\left \{ \Psi, \Psi^\dagger \right \} $$, set of equation 5.4 in david tong notes lead us to
$$Σ_s Σ_r [b_p^s u^s(p)e^{ipx} b_q^r†u^r†(q)e^{-iqy}+ b_q^r †u^r†(q)e^{-iqy} b_p^s u^s(p)e^{ipx}].$$
My question is how the above mentioned terms can be written as...
In David Tong's QFT notes (see http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf , page 131, Eq. 6.38) the expression for canonical momentum ##\pi^0## is given by ##\pi^0=-\partial_\rho A^\rho## while my calculation gives ##\pi^\rho=-\partial_0 A^\rho## so that ##\pi^0=-\partial_0 A^0##. Is it...
Hello,
I am freshly retired and enjoy going back to the fundamentals.
I followed the wonderful courses by Alain Aspect on Coursera on Quantum Optics 1 and 2 .
The quantization of Electrodynamics is really easy stuff.
Just follow the correspondence between Poisson brakets and Commutators ... and...
I came to understand that Planck Distribution is necessary to explain UV catastrophe. With that necesity in the background, the distribution equation eventually suggests that the energy emitted by black body has discret values. But I wonder how that's related to E=nhv. I understand that "n" also...
The Schrödinger equation can be derived from the path integral quantization of the Lagrangian of classical, non-relativistic particles.
Can the Klein-Gordon (and maybe the Dirac) equation be derived from the path integral quantization of a given classical (supposedly relativistic) Lagrangian of...
(Simplified version of Baym, Chapter 19, Problem 2)
Calculate, to first order in the inter-particle interaction V(r-r'), the energy of an N+1 particle system of spin-1/2 fermions with on particle of momentum p outside an N-particle Fermi sea (quasiparticle state). The answer should be expressed...
Greetings,
In the scenario of a particle in an infinite potential well, there are discrete energy levels, i.e.##E=\hbar ^2 n^2 \pi ^2/ (2 m L^2)## where L is the width of the potential well, and n takes on positive integers. But what will happen if I put a particle of energy ##E_i## that is not...
Hi, to help further my understanding of the second quantization for one of my modules I would like to show that the following expressions
$$ \hat{H} = \Sigma_{ij} \langle i| \hat{T} | j \rangle \hat{a_i }^{\dagger} \hat{a_j} $$
$$\hat{\psi}(r,t)= \Sigma_k \psi_k(r) \hat{a}_k(t)$$
Obey the...
Hello all,
The second quantization of a general electromagnetic field assumes the energy density integration to be performed inside a box in 3D space. Someone mentioned to me recently that the physical significance of the actual volume used is that it should be chosen based on the detector used...
Homework Statement
For the canonically quantized operators, what are the step in between? how do you get the answer iħ?
[q^,p^]=iħ
q^ is the coordinate and p^ is the momentum.
This thread is a direct shoot-off of this post from the thread Atiyah's arithmetic physics.
Manasson V. 2008, Are Particles Self-Organized Systems?
The author convincingly demonstrates that practically everything known about particle physics, including the SM itself, can be derived from first...
Homework Statement
Show that the radiation field is transverse, ##\vec{\nabla}\cdot\vec{A}=0## and obeys the wave equation ##\nabla^2\vec{A}-\frac{1}{c^2}\partial_t^2\vec{A}=0##. You should start from the expansion of the quantum Electromagnetic field.
Homework Equations
##H=\frac{1}{2}\int...
Homework Statement
Following from \hat{b}^\dagger_j\hat{b}_j(\hat{b}_j
\mid \Psi \rangle
)=(|B_-^j|^2-1)\hat{b}_j
\mid \Psi \rangle
, I want to prove that if I keep applying ##\hat{b}_j##, ## n_j##times, I'll get: (|B_-^j|^2-n_j)\hat{b}_j\hat{b}_j\hat{b}_j ...
\mid \Psi \rangle
.
Homework...
Hi,
I was reading a book about second quantization and there were a few things that I didn't quite understand entirely.
This is what I understood so far:
Given an operator ##\mathcal A## and two orthonormal bases ##|\alpha_i\rangle## and ##|\beta_i\rangle## for the Hilbert space, ##\mathcal...
Coulomb's law states that the force between particles depends on their charge. But protons and electrons have equal but opposite charges. Shouldn't the formula simply have constants with the only changes required being the signs?
I'm self-studying QFT and attempting exercise 2.2 on Peskin & Schroeder. First off, I'm a bit confused on the logic the authors use in the quantization process. They first expand the fields in terms of these ##a_{\vec{p}},a_{\vec{p}}^\dagger## operators which, if I understand correctly, is...
What is the origin of quantization?
For example, electron of hydrogen atom has quantized energy level. But free particle has continuous energy level.
Interaction with another particle or any potential acting on particle makes such quantization?
[Moderator's note: This thread is spun off from a previous thread since it was getting into material too technical for the original thread. The quote at the top of this post is from the previous thread.]
Field quantization doesn't require a photon picture. A measurement device that creates a...
I am puzzled by the fact that a "single-particle" Hamiltonian (in the annihilation and creation operator form) may be used for a multi-particle case (non-interacting particles) or that (only) a "two-particle" Hamiltonian (in the annihilation and creation operator form) may be used for a...
hi guys
i am struggling to understand how and why quantization of energy solves the UV catastrophe and the black body problem ? and how they get to the Rayleigh - jeans equation in the first place ? and why plank modified the equation the way he did ? and why should the harmonic oscillators...
Hello,
Is the quantization of the action the origin of the indetermination?
The reasoning goes like this:
- Just talking abut one example, a single particle with position and momentum - keep it simple -
- The measure of the position could have any value, in principle
- The same for the...
I'd like this issue clarified
I understand that a full nonpertubative quantization of a yang mills gauge theory in 4D is unavailable.
is Dirac quantization of classical theory of gravity e.g GR rewritten Ashtekar Variables
Variables or some variation of the idea, a viable approach to a...
Hello! I read some books on QM and QFT but I didn't really noticed (or I missed it?) a proof for the canonical quantization. For example, for energy and momentum it makes sense to have opposite signs, due to Minkowski metric, be related to the variation of space and time, due to Noether theorem...
Hello,
It is considered that the time is continuous in classical physics, but it sounds paradoxal to me, let me explain.
Let a particle inside a galilean frame of reference. This particle can only be measured either at rest, either in motion, but never simultaneously at rest and in motion...
A new duality between Topological M-theory and Loop Quantum Gravity
Andrea Addazi, Antonino Marciano
(Submitted on 17 Jul 2017)
Inspired by the long wave-length limit of topological M-theory, which re-constructs the theory of 3+1D gravity in the self-dual variables' formulation, we conjecture...
Hi!
I'm having some trouble on understanding how the Hamiltonian of the e-m field in the single mode field quantization is obtained in the formalism proposed by Gerry-Knight in the book "Introductory Quantum Optics".
(see...
I've read up on the history of quantization and I can't seem to find a definitive answer on why Planck actually quantized light. Some sources say that he came up with the idea of energy being quantized in order to solve the blackbody problem, whilst others say that he was trying to create a more...
How can the quatization of energy solve the ultraviolet catastrophe?
I tried explanation on internet and on the book but i found nothing, can you help me?
I am getting started with QFT and I'm having a hard time to understand the quantization procedure for the simples field: the scalar, massless and real Klein-Gordon field.
The approach I'm currently studying is that by Matthew Schwartz. In his QFT book he first solves the classical KG equation...