49
votes
Accepted
How is it possible to differentiate or integrate with respect to discrete time or space?
Let's say space is really a lattice with spacing $\Delta x$. It turns out that this idea has more trouble with experiment than you might think, but we can plow ahead for the purposes of this question.
...
- 55.4k
37
votes
Why, in low energy situations like atomic physics, are massive particles found to be in integer number states?
This is a good question. It turns out to be several different questions.
For example the hydrogen atom always has 1 (and not 1.2 or 1.5) electron quanta
This is not actually true. The hydrogen atom ...
- 371
36
votes
Accepted
Why does energy have to be emitted in quanta?
The book is almost surely referring to the ultraviolet catastrophe.
Classical physics predicts that the spectral energy density $u (\nu,T)$ of a black body at thermal equilibrium follows the Rayleigh-...
- 16.5k
32
votes
Accepted
If energy is quantized, does that mean that there is a largest-possible wavelength?
since energy is quantized
You have a misunderstanding here on what quantization means. At present in our theoretical models of particle interactions all the variables are continuous, both space-time ...
- 235k
23
votes
Accepted
If charges is quantised, how can we use integrals in electrostatics?
You may also wonder why we use the concept of density of a material despite that material is made of molecules, or atoms, or in general quantized entities. This is the basis of hydrodynamics and solid ...
- 5,244
23
votes
When we say a rigid body is a system of particles, what exactly are 'particles' here?
TL;DR Never forget that we are talking about models here. We know that they cover only part of reality, but they are good enough and much easier to handle than "the full story".
A model need ...
- 3,062
21
votes
Accepted
Why do we need to make sense of QED in continuous spacetime anyway?
The premise of the question is flawed - many practitioners of QFT definitely do not really care that people interested in rigor think QFT is lacking. They care that their procedures, whether "...
- 129k
19
votes
Accepted
Are nucleons discrete within a nucleus?
This depends on the energy:
Cold nuclei, low energies (typical order like 1-10 MeV):
Usually, nuclei are not much excited or not at all. We can probe the nucleus experimentally (excite it somewhat) ...
- 1,984
17
votes
Are integers unphysical/unnatural?
There are plenty of exact integers in physics.
Take a wavefunction, for example: $\psi(x) = \langle x | \psi \rangle$. You can count the stationary points and the zero crossings. 1,2,3, ....
In the ...
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16
votes
How is it possible to differentiate or integrate with respect to discrete time or space?
This is a comment, as Andrew's answer is adequate for the problem.
I want to point out , which is not clear in your question, the difference between mathematical modeling and the object modeled.
When ...
- 235k
14
votes
Accepted
How does quantization arise in quantum mechanics?
First and second quantization
Quantization is a misleading term, since it implies discreteness (e.g., of the energy levels), which is not always the case. In practice (first) quantization refers to ...
- 65k
13
votes
Why does energy have to be emitted in quanta?
I think the author is referring to the UV catastrophy, a historical problem in physics which first led physicists to discover that electromagnetic energy was quantized.
The problem basically is this: ...
- 1,187
13
votes
Accepted
Is a magnetic monopole really necessary for charge quantization?
Yes, "quantization of charge" means that all charges are a multiple of some fundamental charge unit $e$.
Of course, everything being made up of electrons and protons with a fixed charge explains ...
- 129k
13
votes
Accepted
Why light coming from distant stars is not discrete?
You are right that single photon detection is a discrete event. But you are under the false assumption that these "rays" are discretely distributed.
Ideally, a photon would have an equal probability ...
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13
votes
Accepted
Generator for parity?
The parity operator does not have a generator in the way that the translation or rotation operators do.
Notice how you gave the translation operators and the rotation operators a parameter, like $\...
- 7,331
12
votes
Generator for parity?
This is a slightly counterintuitive approach, but very easy: take the harmonic oscillator eigenfunctions $\psi_n(x)$. It is known that the functions $\psi_{2k}(x)$ are even under parity and the ...
- 6,926
12
votes
What does it mean for an operator to have both a continuous and discrete spectrum?
When it comes to operators acting on infinite dimensional Hilbert spaces, you need to specify some boundary condition before you find the eigenfunctions and eigenvalues. Playing around with any linear ...
- 3,544
12
votes
Accepted
What does it mean for an operator to have both a continuous and discrete spectrum?
I'll try to give a mathematically rigorous answer which I hope will help you have a clearer understanding of the problem.
The setup
Let us consider the following hamiltonian in $L^2(\mathbb{R}^3)$ $$H ...
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12
votes
Accepted
When we say a rigid body is a system of particles, what exactly are 'particles' here?
In macroscopic mechanics (in other words, "usual" mechanics), a "particule" is a mesoscopic system.
The mesoscopic scale is an intermediary scale between microscopic and ...
- 6,269
11
votes
Accepted
Discrete spacetime: what does it mean for spacetime fields and vacuum?
You should really look into Loop Quantum Gravity for a quantitative example. While unconfirmed and highly speculative, it does offer a toy example for a background independent quantum field theory, ...
- 16.3k
11
votes
Accepted
All periodic phenomena should have quantized energy levels?
"Quantisation" does not actually arise from principle of Quantum Mechanics. Nowhere in the postulates of QM you see that the energy levels are quantised. The quantisation of energy levels ...
- 1,794
10
votes
Accepted
What makes the electron, as an excitation in a field, discrete?
I'll address what I understand to be your question, namely
Where is the discreteness of the number of excitations of QFT coming from, even in free propagation that apparently does not involve a ...
- 66.3k
10
votes
Confusion About Energy Bands
No, they don't. The allowed energies for a quantum mechanical system comprise the spectrum $\sigma(H)$ of the system's Hamiltonian operator $H$. This spectrum may consist of discrete points as in ...
- 71.5k
10
votes
The continuum limit of the path integral & differential operators
There can be no good justification for $\frac{x_{n-1} - x_n}{\Delta t}\to \dot{x}$ because that's not what happens in the actual mathematically rigorous path integral! You can try to do various ...
- 129k
9
votes
If energy is quantized, does that mean that there is a largest-possible wavelength?
You have to keep in mind that the relation
$$
E = hf
$$
holds only true for photons. Photons - generally - can have arbitrary energies, so they can have arbitrary frequencies as well.
When you ...
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9
votes
Are nucleons discrete within a nucleus?
Quasi-elastic scattering has a certain value in this.
The term means (in this context) that you scatter a nucleon out of a nucleus largely without disturbing the remnant (you don't break it up or ...
9
votes
Superposition principle forbids quantisation?
No. It is a good thought but no. In short, you are correct that we can prepare states whose average energy takes on any value between the lowest and highest energy state. But this is different than ...
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9
votes
Is conduction band discrete or continuous?
A useful way to visualize the difference between conductors, insulators and semiconductors is to plot the available energies for electrons in the materials. Instead of having discrete energies as in ...
- 235k
9
votes
Differential charge existing
You're mixing up two descriptions that are, in practice, separate.
$i=dq/dt$ is usually used in macroscopic physics, when it is understood that you don't study actual individual electrons. In fact, ...
- 6,269
9
votes
Accepted
Why discrete gauge fields must be flat?
That's a nice question. The cleanest answer (that I know of) relies on a result of Milnor [1].
But first, let's rephrase things a bit. For a group $G$, let's define a flat $G$-bundle to be a principal ...
- 4,842
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