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Quantum mechanics, is, to the best of our knowledge, the way (almost everything in) the world works. It's not solely about describing "matter waves", although this was fundamental to its inception. It's not solely about describing microscopic phenomena. It's about a fundamental conception of "mechanics" (it's in the name!), an attempt to describe how ...


7

Taking a look at your questions, I would suggest you forget about the wave particle duality being this foundational principle of quantum mechanics. Rather, I would say there are two (at least, these two are the most obvious) important defining characteristics of QM, which are made clear in the Hilbert space formalism: States are represented by vectors, and ...


5

Because a pair of particle and anti-particle can be created from the vacuum, it means that infinite number of pairs of particle and anti-particle can be created from the vacuum. So when you consider relativistic quantum theory it's impossible to only consider a finite number of particles. When you calculate Feynman diagram, you are actually only doing ...


4

A state is something that encodes our knowledge about the system. And that's it. There are many ways to encode a state in quantum mechanics. As a wavefunction ("Schrödinger representation"), as Fock momentum states ("Fock representation"), as a density matrix, as a ray in a Hilbert space, as a linear functional on the $C^*$-algebra of observables, as a ...


4

The fundamental difference between an electron's spin and that of a baseball is that the electron is (as far as we know) a point particle. It therefore cannot rotate in the usual sense, where individual parts move relative to the center of mass; we say that its angular momentum is intrinsic. The magnitude $\lvert\vec{S}\rvert^2$ of a particle's intrinsic ...


4

We don't know what quantum mechanics is about, the theory is formulated in an instrumental way. The postulates of quantum mechanics tell you how to compute the outcome of experiments. When you try to look beyond this, take into account that the experimental apparatus used, the observers etc. are also made out of the atoms and molecules, you are forced to ...


4

No matter what spin measurement Jack performs on his particles, he's going to see half "up" and half "down" (or more precisely, each particle has a 50/50 chance of being up or down). This is so no matter what Sally chooses to do. So there's no way for her to send him a message.


4

To add to ACuriousMind's answer by bringing my personal experience to the table, I too had the background mathematics for many years and understood perfectly all the algebraic machinations in many textbooks but was utterly baffled. My problem was that I was "out" of QM for a long time: I had an elementary exposure to it in undergraduate engineering which was ...


4

this question is really severely damaged: the title (top/bottom quarks) does not match the question being asked (up/down quarks plus tau electrons), and the question literally being asked has a meaningful typo (tau selectrons) which invokes ideas from the still-speculative physics of supersymmetry, which is even crazier. To answer the question you literally ...


3

Essentially, I would like to prove $$ \sum_k f(k) \to \int f(k) \rho dE \tag{1}$$ where $$ \rho = \frac{dk}{dE} \tag{2}$$ is the density of states and $k \to \infty$. As mentioned in the comments, you need to introduce a measure on the LRS to get the dimensions to work out. To put it another way, your $f(k)$ on the LHS can't be the same as your ...


3

Yes. Both universal covers and central extensions incurred during quantization come from the same fundamental concept: Projective representations If $\mathcal{H}$ is our Hilbert space of states, then distinct physical states are not vectors $\psi\in\mathcal{H}$, but rays, since multiplication by a complex number does not change the expectation values given ...


3

You have a wrong understanding of quantum entanglement. What is entanglement? Quantum entanglement emerges naturally from the "only obvious way to do things" at the wavefunction level (distribute a wavefunction over all possibilities of two subsystems), and describes the fact that the general state of these systems cannot be "decoupled" into a pair of ...


3

The basic idea upon which Quantum Mechanics is based is the wave-particle duality. The basic idea is that classically dynamics of configurations fails and needs to be modified. Particle wave duality is ultimately too vague to be a fundamental explanation. So it is seem that particles on these microscopic phenomena behave as waves. Those matter waves ...


3

The short answer is that what you are proposing would be an extraordinarily challenging task! Simulating a single (non-hydrogen) atom accurately in time requires a huge amount of computational power which scales as roughly (simulation resolution)^(3*number of particles)! In computational biology, nearly all simulations are Newtonian based to avoid this. I ...


3

Strictly speaking, you must use the relativistic "momentum". For most objects, it's not simply $mv$. Rather, $$ \lambda=h/p=hc/pc=hc/\sqrt(T^2+2Tmc^2) $$ For an electron, say, even at quite low energy (e.g. 1 eV) the term $ 2Tmc^2 $ is quite high, and so the wavelength still ends up being quite low (~ angstrom scale). In this limit of $T^2 << 2Tmc^2$, ...


3

Due to the quantized energy levels of allowed electron orbitals, single atoms can easily absorb energy around certain narrow wavelengths. A cesium atom has one of these narrow absorption bands at a frequency of 9,192,631,770 Hz. A cesium clock can produce EM radiation in this region (microwaves) and detect how well the cesium atoms are absorbing it. So ...


3

"What really is X" is a tricky question in physics, especially when dealing with such a fundamental and abstract theory as quantum mechanics. So I can give a mathematical definition: Let $\mathcal{H}$ be a complex Hilbert space, that is, a complete vector space over $\mathbb{C}$ equipped with a positive-definite inner product $\langle\, \cdot\, |\, \cdot\, ...


3

So Markovian dynamics usually arises in the field of open quantum systems, where you have some system coupled to a (much larger) reservoir. For instance, consider an atom coupled to the electromagnetic field. In these cases the atom (system) is much smaller than the electromagnetic field (reservoir), and they interact because the atom can absorb and emit ...


3

I) Let us for clarity use a subscript "$S$" (and "$H$") to denote the Schrödinger (Heisenberg) picture, where bras and kets evolve (are unchanged) and operators are unchanged (evolve), respectively. Moreover, let us assume that the two pictures coincide at the instant $t_0$ (which Ref. 1 assumes is $t_0=0$). II) Recall first of all the possibly confusing ...


3

What's an observation? I think your question delves into the nature of "consciousness", a term which has never to my knowledge been satisfactorily defined. The seas observe the moon, and therefore there are tides. Whether or not a "conscious" being observes the seas is of no consequence in physics.


3

For example, the escape velocity of a particle from the galaxy is about 400 km/s and in most conceivable circumstances (unless you are basically on top of the event horizon of a black hole or on the surface of a neutron star), escape velocities will be far, far below relativistic speeds (here defined as $3\times10^4$ km/s). So basically, if a particle has a ...


3

There are apparently several thousand references to "SU(\infty)" on arxiv.org, and some of them are definitely talking about gauge fields or Yang-Mills. I suspect that some of the time, this will just be a way of talking about the large N limit of SU(N), i.e., not referring to a literal SU(∞) field theory, but rather the N→∞ limit of some quantity in SU(N) ...


2

Imagine you have a pair of coins. Whenever you throw them, each of them is fully random, but their outcomes are opposite. Now imagine you throw the two coins. You look at the left coin. When it is head, you discard both coins and start again, when it is tail, you keep it. Since you have never looked at the right coin, it should still be completely random. ...


2

Indeed using a rigged Hilbert Space is almost certainly what you should use and there are tutorials on it on arxiv (I'll add them when/if/I can find them again). However it isn't necessarily going to help you understand what an average physicist is doing on the one hand, and on the other hand there were already many things you were doing that not rigorous ...


2

Dr. Robitaille says that blackbody radiation is not universal, even inside cavities where the surfaces are all at thermal equilibrium. That is highly controversial since the electromagnetic fields in a cavity are usually considered as an additional substance, called a "photon gas" which is also at thermal equilibrium and hence has a temperature. This ...


2

Your final expression is correct except that $V$ should be in front since it does not commute with $Θ_0$ in general. In potential scattering ${Θ_0}V$ is often a compact operator and for large positive imaginary part of $z=E+iℏε$ its norm becomes arbitrarily small so the series converges. In certain cases one can do things a little differently by ...


2

The FourierTransform.com is a website maintained by an enthusiast. The site is not peer reviewed, but it looks as though it might provide helpful explanations. Here's a link which provides some basic introduction to the Fourier transform. And here is another link to class notes provided by Prof. Carlton M. Caves for an introduction to the Fourier ...


2

Comments to the question (v2): The idea to consider the planar large $N_c\to \infty$ limit in $SU(N_c)$ QCD goes back to Ref. 1. In light-cone membrane theory, pioneered in Ref. 2, the group $SU(\infty)$ is naturally identified with area-preserving diffeomorphisms ${\rm SDiff}_0(T^2)$ on the torus $T^2$ connected to the identity. Concretely, OP's proposal ...


2

Things have intrinsic spin. There's no "class of objects" that has spin and another class of objects that doesn't. Everything is a quantum object with a quantum state, and spin is a number that tells you how the state of the object transforms under rotations. It is different from "classical" angular momentum in that spin is not the operator associated to ...


2

The simplest way is to exploit the symmetries here. So, instead of mindlessly going through the algebra, solving equations etc. you just use the fact that you are free to call any direction the x-direction, and set up a right handed coordinate system. In particular, this means that you are free to cyclically permute x, y, z in the equations. The problem of ...



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