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Although it's commonly said that fundamental particles are point particles you need to be clear what this means. To measure the size of the particle to within some experimental error $d$ requires the use of a probe with a wavelength of $\lambda=d$ or less i.e. with an energy of greater than around $hc/\lambda$. When we say particles are pointlike we mean ...

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An animation is worth a million words:

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Because of the Pauli exclusion principle, it's extremely difficult to compress atomic matter beyond a certain density. It's not impossible, because there are always higher-energy electron states available, but there's a very strong force opposing it (called electron degeneracy pressure). This is what it means for space to be full. If you define "empty space"...

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This will be a purely mathematical treatment. It needs to be combined with some practical playing around to really "get" it. Traveling wave Let's start with the description of a harmonic traveling wave in one-dimension. Here "harmonic" just means the mathematical form of the wave is sinusiodal in both time and space. For concreteness we'll using talk ...

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The theory behind the trick is based on the Hellmann-Feynman (HF) theorem $$\frac{dE_{\lambda}}{d\lambda}~=~\langle \psi_{\lambda} | \frac{d\hat{H}_{\lambda}}{d\lambda}| \psi_{\lambda} \rangle,\tag{A}$$ which works with a single derivative, but not with a square of a derivative, cf. OP's failed calculation (5) for the expectation value $\langle\frac{1}{r^2}... 10 Yes, elementary particles such as electrons and quarks (inside protons) are point-like or at least, their internal structure is incomparably smaller than the size of the atom. So the atom is mostly empty space. However, that doesn't mean that atoms may penetrate each other. Matter is impenetrable because of a combination of the uncertainty principle that ... 7 There are two separate issues here (not sure which of the two you mean): The first problem is a severe misconception that is similar to Zeno's paradox of Achilles and the Tortoise: Given a hydrogen atom we have (in principle) an infinite number of shells. However, the gap between the shells gets smaller and smaller. If you would jump from shell to shell, ... 6 I actually think I have figured it out: if we consider the situation when the principle quantum number for, let's say, excited helium is different, then we can have spin-up electron in$1S_1$state and spin-up electron in$2S_1$state. This is going to be a triplet. Please, correct me if I am wrong. 5 how is a standing wave related to the atomic orbit (It is my understanding that the atomic orbit is both a mathematical function that describes the probability of an electron being at a certain place, but it is also the image of this function in terms of real space, i.e. the actual 3 dimensional volume around the nucleus that a particular electron calls "... 5 Regularly spaced light and dark bands far from either anode suggests a simple matter of the electrons picking up the kinetic energy necessary to excite the neutral atoms, giving most of it up when they do so, and then re-accelerating down the tube. We would expect a couple of diagnostic features: The bands nearer the cathode to be more sharply defined ... 4 The orbitals that you are taught about at school are energy eigenstates i.e. eigenfunctions of the Hamiltonian. For a confined system like an electron in an atom the eigenstates of the Hamiltonian have discrete and precisely defined energies so the transition energies are all precisely defined. In the real world electron states are only approximately ... 3 A charged particle like electron maybe is point-like (of radius zero), but it is "long-handed" as it is "felt" far away. In this sense it is not so "point-like". 3 The atomic states win in a landslide. The reason is that the coulomb force is long ranged while the nuclear force is short ranged. That means that the number od discrete nuclear states is finite while the number of discrete atomic states is infinite. Even for a neutral atom where you might think that cancellation of charges would shorten the interaction ... 3 It's simply a Fourier transform relationship. If an atom is in a metastable state, with the latter coupled to the electromagnetic field, then it is fairly straightforward to show that the probability amplitude for decay to happen in the time interval$[t,\,t+\mathrm{d}t]$is: $$\psi(t) = \left\{\begin{array}{ll}\frac{1}{\sqrt{\tau}}\,\exp\left(i\,\omega_0\,... 2 The statement of Martin above: Now, can we "see" atoms? This depends, as I already hinted at, what you mean by "see". If you mean "make a picture in visible light", then you can't do that. is actually not quite true. One can take images using visible light that show single atoms. Here is an example: (1) The reason this works is that this is a ... 2 A standing wave is basically two opposing waves of equal amplitude, as shown in the diagram below (where n is a positive integer): You can see this more clearly if you look at the top line where n=3, and follow it as it goes down, up, down. That's wave one. Then, if you look at the bottom line in the same case as it goes up, down, up, that is the opposing ... 2 There are five relevant quadrupole moment operators, and when labeled by the change \Delta m in angular momentum projection they read$$ \begin{array}{c|ccccc} \Delta m & -2 & -1 & 0 & 1 & 2\\ \hat Q_{2m}& (x-iy)^2 & (x-iy)z & x^2+y^2-2z^2 & (x+iy)z & (x+iy)^2 \end{array}$$These operators arise as (a basis for) all ... 2 In addition to the answer by hsinghal it is worth point out some historical notational quirks. An expression such as$^2P_{3/2}$is called a Term Symbol. The superscript is the multiplicity of the electron spins, i.e. 2$S\$+1 for total spin S. The capital letter, P in this example is the total orbital angular momentum and has letters and values of S=0, P=1, D=...

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"To be at rest" in classical mechanics means "to have definite position and zero momentum", the two properties being (again, in CM) equivalent: if something has definite position, then it must have zero velocity thus zero momentum, and if it has zero momentum, then it must have zero velocity thus definite position. Depending on which of the two properties ...

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Your confusion comes from unfortunate terminology Quantum numbers are supposed to denote every individual orbital quantum numbers denote the state of a quantum system as solution of the Schrödinger equation; in particular they often refer to the eigenvalues of a maximal set of operators used to describe the physics at hand. As an example the hydrogen ...

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Electrons and nuclei both have spin. A spinning charged particle has a magnetic dipole moment. When a magnetic dipole is in a magnetic field, it experiences a force. This oversimplified description gives some brief intuition on the origin of the fine and hyperfine splittings. Fine structure is due to the interaction of the electron's spin with the ...

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The problem with this question is all of the assumptions that go into it. When we're taught physics, we are given analogies that help our understanding, but mislead us when we try to dig deeper. Firstly, charged particles like electrons are always surrounded by an electromagnetic field. Changes in that field propagate through space at the speed of light and ...

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This is a very good book which covers atomic spectra and Laser Physics, with good Physical intuition and fairly rigorous Mathematical proofs. https://www.amazon.com/Atomic-Physics-Oxford-Master-Optical/dp/0198506961

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What we intuitively think of as "solid objects" are actually electromagnetic force-fields repelling each other. So you are correct; atoms are 'empty' in that they contain no solid objects or things. On the other hand, they are 'full' of basic force field which, in the aggregate, on a macro-scale, creates the illusion of 'solidity' that is what we perceive to ...

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As you know that zeeman splitting is due to the phenomena known as spatial quantization. i.e. if there is a fixed or preferred direction in the space (i.e. symmetry of the space is broken by the electric or magnetic field) then the atom can not assume arbitrary orientation. This orientation depends on the angular momentum of the atomic/spectroscopic state. ...

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