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.
To simplify, I consider a one-dimensional box of the size L. I need to show in this case that
$$\frac 1 L \sum_{pq} e^{i(px-qy)}[\hat a_p,\hat a^\dagger_q]=\delta(x -y)$$
With the commutation relation above, it becomes
$$\frac 1 L \sum_p e^{ip(x-y)}=\delta(x -y)$$
p is quantized: ##p_m=\frac...
Source: Scully and Zubairy, Quantum Optics, Section 1.1.2 Quantization
Questions:
1. Why are the destruction and creation operators considered a canonical transformations?
2. If these are canonical transformations, does it suggest that we are also canonically transforming the Hamiltonian...
I don’t understand how energy is conserved here. The energy of the atom when n=5 is -.544eV. The energy of the photon is 1.14eV. After release, the energy of the atom is -.544 - 1.14 = 1.68eV. Using this value, I get n = 2.67, not an integer, so n = 3 and the atom has energy = -1.51 eV. I...
OK, so I just want to show ω = E/h = kv, but I keep running into errors, I don't know why.
So, let's start with momentum:
p^2 / 2m = E
p^2 = 2mE
p = sqrt(2mE)
h/λ = sqrt(2mE)
hk = sqrt(2mE)
k = sqrt(2mE)/h
So far so good. Now let's start with conserved Energy
E= ½ mv^2
2E/m = v^2
v =...
In the brilliant.org website talking about quantum properties it is said that neutrons coming from a nuclear oven and passing through two permanent magnets of opposite polarity hit a surface only at the top and the bottom of it (there is no continuity) because the spin property is quantized and...
Please take a look at my first thought experiment:
You have 2 coins to place on the table.
The distance between them may be between 0 (inclusive) and 1 meter (exclusive).
So if you want to store the number 15 you simply set the distance to 0,15m.
You can later read the information by measuring...
A free photon can have any wavelength and energy; no discreteness there. Just because something is quantized, or fundamentally quantum in nature, doesn’t mean everything about it must be discrete.
The idea that space (or space and time, since they’re inextricably linked by Einstein’s theories...
In Quantum Mechanics, the position (or momentum) variable is quantized. I define "quantization" as promoting a variable into a probability distribution.
For example, with the double slit experiment, the classical assumption that the position/path of a particle is "unique" cannot explain...
I am doing a final year project on vortex interactions and have searched for several research articles about quantum hydrodynamics. Most said that ''Any rotational motion of a superfluid is sustained only by quantized vortices.'' Is this something provable from the Gross-Pitaevskii Equation or...
Hi,
I am really new in understanding second quantization formalism. Recently I am reading this journal:
https://dash.harvard.edu/bitstream/handle/1/8403540/Simulation_of_Electronic_Structure.pdf?sequence=1&isAllowed=y
In brief, the molecular Hamiltonian is written as
$$\mathcal{H}=\sum_{ij}...
A long time ago, someone told me that Einstein thought spacetime is "smooth and continuous."
Quantum mechanics is a proven fact. Quantum mechanics exists in spacetime, though.
So can we at least conclude that spacetime is not continuous, and instead discrete? ...quanta of space and time...
Homework Statement
Okay, so the question I'm trying to solve is to find the quantized energies for a particle in the potential:
$$V(x)=V_0 \left ( \frac{b}{x}-\frac{x}{b} \right )^2$$
for some constant b.
The Attempt at a Solution
I am following along with the derivation of the quantized...
Edward Witten's M-theory conjecture is that there is a theory that has 5 10D superstrings and 11D supergravity as its low energy limit, and is defined nonperturbatively.
Now, I know that Urs and Mitchel are not fans of loop quantization program. fine.
Urs and Mitchell do not think that a loop...
Hi all,
Yet another question: if the universe is finite, then linear momentum should be quantized (I assume in a similar manner to an infinite potential well since there are boundary conditions). My question is, then, if one computes a value for ##\Delta p## (momentum variance), is the variance...
Consider the microwave cavity used in a particle accelerator to accelerate particle bunches. Before a bunch of particles enters the microwave cavity can we, if only in principle, quantum mechanically describe the state of the microwave cavity as a proper sum of microwave photon number states...
Hi everyone!
I'm having a problem with calculating the fermionic propagator for the quantized Dirac field as in the attached pdf. The step that puzzles me is the one performed at 5.27 to get 5.28. Why can I take outside (iγ⋅∂+m) if the second term in 5.27 has (iγ⋅∂-m)? And why there's a...
Usually the imaging through lens is described using geometrical optics or wave optics. However, I wish to study some imaging processes through lens where the light intensity is at the single photon level, under which condition I think the correct description should use a quantized...
Is there a relationship between the quantization of an object and its wave function? If an object isn't quantized does it have a wave function? For example, in string theory branes are not quantized, so do they have wave functions?
The common presentation for free field quantization proceeds with the Lorentz and Coulomb (##\phi = 0, \,\nabla \cdot \mathbf{A} = 0 ##) constraints. Then ##A## can be defined
$$\mathbf{A} \propto \iint \frac{d^3 p}{\sqrt{2\omega_p}}\sum_{\lambda} \Big(e^{i\mathbf{p}\cdot...
In most situations in QM we would get a quantized energy basis, that is a countably infinite basis ( I think it's called having a cardinality of aleph 0), In the meanwhile we take the position basis to be continuous ( cardinality of aleph 1?) and I'm pretty sure that there is a theorem stating...
So why do Quarks have fractional non-discrete charge? Wouldn't it just be easier to just define ##\frac{1}{3}e## as +e and vice versa to preserve the discreetness of what we define as e?
I have to say first that I found quite a similar question under https://www.physicsforums.com/threads/poynting-vector-dilemma.331375/ However those thread ended without real answer.There shall be the following simple setup: A battery, and quite at a distance a light bulb. They are connected by...
I entered in physicsforums archive search the wildcards "bohmian why electrons don't lose energy atom" but only got one hit that isn't related to it. In Bohmian mechanics, electron is localized and has trajectory.. why can't it lose energy as it rotates around the nucleus? What wildcard words...
Say you had a crystal lattice box on a micro scale. You can push it with a bigger piston to accelerate it. Say it's lying on a friction-less surface (or this happens in minimal gravity and vacuum).
My question is, would the possible kinetic energy that you can impart on this cube be quantized...
Does the fact that orbital electrons are stable suggest that gravitation is quantized, analagous to the observation that the stability of orbital electrons suggested that electromagnetic energy was quantized?
Quantum Theory perhaps isn't really quantic, as It seems to me there is an underlying continuity.
There are even some reasons to suppose that in Planck scale we will not see the discontinuity most of us expect. Professor's Craig Hogan superinterferometer is opperating in Fermilab; soon we will...
Although I have not been thinking very much about actual physics for while now I suddenly got the urge to read a little bit about it and I decided to start by checking out the birth of quantum mechanics again.
In the book quantum theory by Bohm in chapter 1 the ultra-violet catastrophe and its...
so a quick Q. the equation for charging a capacitor seems to indicate that charge (watts) will always be charging the capacitor, but is it true that as t⇒∞ the charging actually stops and the state of equilibrium is quantized?
Hi, I am currently watching this lecture series, and was wondering about something. At some point the lecturer says that QM is "not quantized", because if you express your solution to the schrodinger equationby introducing a pertubation parameter \epsilon, you will go smoothly from one...
I understand that the result of the hydrogen emission spectrum experiment was that only certain wavelengths of light were emitted and that led to the conclusion that electrons emit light when they relax and that they absorb light when they get excited. How does that prove that the energy for...
Homework Statement
See attached image.
Match the type of system or situation to the appropriate energy level diagram.
1) hadronic (such as +)
2) idealized quantized spring-mass oscillator
3) nuclear (such as the nucleus of a carbon atom)
4) vibrational states of a diatomic molecule such as O2...
Hello everyone, I m currently working on a problem that is freaking me out a bit, suppose I have a second quantized hamiltonian:
\begin{eqnarray}
H=H_{0}+ \epsilon d^{\dagger}d + V(d{\dagger}c_{0} + h.c)
\end{eqnarray}
In terms of some new operators, I would like to rotate the hamiltonian, so...
As to E=hf I understand that a energy can be arbitrary in order of proportional packages to given frequency. But is there any equation in quantum mechanics which imply a freq to be quantized?
Thanks a lot!
Probably an ignorant question, but if gravity is or can be quantized, does that mean that time must be quantized too? I guess I'm thinking of energy- definitely quantized. Gravitational energy... hmmm. Makes me wonder, if space-time truly is smooth (general relativity), then there must be a...
I was reading about zeno's paradox (arrow), Is it like Quantized Spacetime? that we just move from one 'NOW' to the next?
Does zeno paradox can be fully solved?
I have asked this question before on various fora, but I never got a really satisfactory answer. Perhaps physicists simply don't know...
Anyway I'm asking again; an electric field emanating from, say a small pointed metal electrode, is for many practical purposes a continuous field that is...
Is time quantized? I just read an interesting article in scientific american about this subject. The following link should take you there. The most interesting part was about research done by William G. Tifft, a professor of astronomy at the University of Arizona. I was wondering what others...
I was reading a section of a chemistry textbook describing electron energy shells. It compares the electrons to light saying that electrons energies are quantized and so are light energies. Electrons can only jump from one specific energy level to another with no intermediary energy levels. I...
Homework Statement
a+(k) creates particle with wave number vector k, a(k) annihilates the same; then the Klein-Gordon field operators are defined as ψ+(x) = ∑_k f(k) a(k) e^-ikx and ψ-(x) = ∑_k f(k) a+(k) e^ikx; the factor f contains constants and the ω(k). x is a Lorentz four vector, k is a...
The following text is an excerpt from a physics book:
I find this paragraph a bit hard to grasp, so I have some questions about it and I would appreciate if they are clarified.
1) What is meant by "the energy carried off by the radiated electromagnetic waves"? As I interpret this it means...
After reading an article on Planck length, I began to wonder whether or not the theoretical limit implied that position could be quantized in whole integer multiples of Planck length?
To demonstrate what my question is asking mathematically I hope you will scrutinize the equations below:
If...
The smallest amount of energy possible is called quantum of energy and is = hf. But are frequencies also quantized? or are they continuous. If frequencies are continuous, then energy can be continuous, no?
I am thinking that Planck's constant would mean each frequency has its own quantum...
Hey there,
I'm having trouble understanding where two of the formulas for angular momentum of an electron in the hydrogen atom tie together. The first relation L = nh/(2∏) which comes from the de Broglie momentum-wavelength relation and the requirement for the electron wavelength to equal an...
Homework Statement
A classical electron in circular motion with radius r and velocity v.
How would you find the quantum number l that gives quantized angular momentum close to the angular momentum of the classical electron?
Homework Equations
p=mvr
L=(h/2pi)√[l(l+1)]
Can anyone...
Hey guys hope everybody is doing well. Here is my question. From what I understand, according to GR, gravity is present because of the curves in space-time. I also believe, please correct me where I am wrong, that those curves also create ripples in space in time. So that's what einsteins...
according to the Landauer formula, the conductance is proportional to the transmission probability.
But the transmission probability is not quantized generally
So why is the conductance quantized?
What is the mainstream answer? And why is asking if the electric charge is quantized important? What does it mean to have charge that is not quantized?