# Tag Info

23

One cannot tell by the light spectra. Hydrogen and antihydrogen would give the same lines in the spectrum. The prevalence of matter over antimatter from other evidence indicates matter is predominant in the observable universe, and here is a nice review. How do we really know that the universe is not matter-antimatter symmetric? The Moon: Neil ...

19

There is a rigorous formal analysis which lets you do this. The true problem, of course allows both the proton and the electron to move. The corresponding Schrödinger equation thus has the coordinates of both as variables. To simplify things, one usually transforms those variables to the relative separation and the centre-of-mass position. It turns out that ...

12

I assume you're talking of the hydrogen atom; the hamiltonian of the nucleus + electron system is $$H = \frac{p_e^2}{2 m _e} + \frac{p_n^2}{2 m _n} - \frac{e^2}{|r_e - r_n|}.$$ You can do a change of coordinates (center of mass coordinates) $$\vec{R} = \frac{m_e \vec{r}_e + m_n \vec{r}_n}{m_e+m_n} \\ \vec{r} = r_e -r_n$$ and find the conjugate momenta to ...

10

In a neutral hydrogen atom the ground state has the electron and proton spins anti-parallel i.e. lined up with each other but pointing in opposite directions. The state with the spins parallel and pointing in the same direction has a slightly higher energy, and transitions between these two states produce the notorious 21cm hydrogen line. Since the ...

9

The probability density of the ground state is time independent, so there is no motion in this sense. Yet the expectation value of the kinetic energy is non-zero, so there is movement in this sense. How are these notions of movement reconciled? First off, classically, if we had a particle in a $1/r$ potential and released it from rest, it would indeed bob ...

8

The degeneracy of energy-levels can be traced to the fact that the hydrogen atom possesses an enhanced $SO(4)$ symmetry caused by (among other things) the conservation of the Laplace-Runge-Lenz vector operator, see e.g. this Phys.SE post and Ref. 1. References: G. 't Hooft, Introduction to Lie Groups in Physics, lecture notes, chapter 9. The pdf file is ...

7

This calculation agrees with experimentally measured spectral lines, but why would we expect it to be true, even if we accept that the electron moves according to the Schrodinger equation? After all, there's no particular reason for an electron to be in an eigenstate. Good question! The function $\psi$ does not need to be Hamiltonian eigenfunction. ...

6

The orbitals, which recently have been observed for the hydrogen atom, are probability distributions. These probability orbital distributions have been calculated using quantum mechanical solutions of the Schrodinger equation which give the wave function, and the square of the wave function is the probability distribution for finding the electron at that ...

6

Is the Coulomb potential also used to solve the hydrogen atom in relativistic quantum mechanics? Yes, the Coulomb potential is there in the solution of the hydrogen atom with the Dirac equation, which is formulated in the relativistic framework. Now it is time to specialize to the hydrogen atom for which $$\frac{V}{\hbar c}=-\frac{Z\alpha}{r}$$ ...

6

You say: The Coulomb potential comes from classical electrodynamics but actually the Coulomb potential is predicted by quantum electrodynamics as a low energy limit. Quantum field theory describes the interactions between charged particles as the exchange of virtual particles, and it's not immediately obvious that it would lead to an inverse square ...

4

The time-independent Schrödinger equation for the hydrogen atom is $$-\frac{\hbar^2}{2m}\vec \nabla^2\psi-\frac{e^2}{4\pi \epsilon_0r}\psi=E\psi$$ If your aim is just to verify that the $1s$-wave function $$\psi_{100}=\frac{1}{\sqrt{\pi a^3}}e^{-r/a}\hspace{2cm} a\equiv \frac{4\pi\epsilon_0\hbar^2}{me^2}$$ is indeed an eigenfunction, then your task ...

4

From the Virial Theorem we can say the total energy of the atom is propotional to the potential energy of the atom. The potential energy is given by a coulomb potential and so is it will be roughly proportional to $\frac{1}{\langle r \rangle}$ where $\langle r \rangle$ is the mean radius of the electron's orbital. For a hydrogen atom the energy $E\propto ... 4 The idea here is increasingly complex depending on how deep into modern physics you want to delve, but also key to understanding quantum mechanics. So, I'll give a bit deeper explanation than it seems you've seen, but there's plenty more. It's understood that a photon acts both as a particle and a wave. As a particle it has an amount of energy associated ... 4 This calculation agrees with experimentally measured spectral lines, but why would we expect it to be true, even if we accept that the electron moves according to the Schrodinger equation? Your puzzlement arises because you are putting the cart in-front of the horse. The cart is the theoretical model of quantum mechanics and the horse is the data. As ... 4 With regards your first question: A similar (the same?) question you might reasonably ask is: how can we assume that the proton is stationary, at the centre of the problem, since it is surely going to be attracted by the electron and jiggle about a little? This is a question that would be just as valid directed at a classical system --- say, a planet ... 4 Air is lighter because there are fewer molecules per unit volume compared with a unit volume of liquid water. A mole of water is 18 grams, so a liter of water contains about 55 moles (1000 grams). A mole of air at standard temperature and pressure, however, occupies a volume of 22.4 liters, much more. Dividing a mole of 02 (32 grams) by 22.4, you have ... 3 The numbers are not measured, they are part of a model that explains why the hydrogen atom emits/absorbs only at those wavelengths. The first series of Hydrogen lines to be discovered was the Balmer Series, and nobody knew why they were discrete lines instead of a continuous spectrum. Johann Balmer discovered that the lines all had wavelengths equal to, ... 3 These relations are based on the fact that both the position and the momentum distributions are centred around zero, which is in turn due to the symmetry of the atom. Given that, the width of the position and momentum distributions ($\Delta x$and$\Delta p$) is of the same order as a typical position or momentum within those distributions ($r$and$p$). 3 What you're trying to do is called radiometric calibration. The problem with doing it with an ordinary incandescent light bulb is that the bulb itself would have to be radiometrically calibrated to get a precise spectral calibration of the detector. That's because the bulb's spectral emissivity deviates from an ideal blackbody source with an emissivity ... 3 The electron is the lightest lepton and the proton is the lightest baryon, so it's hard to see what reaction could occur without violating lepton number or baryon number. I suppose if proton decay (to a pion and positron) occurs then there could be a reaction to give a pion and two photons. 3 The symmetries that you're missing are conservation of baryon number$B$and lepton number$L$. We strongly suspect that baryon number is not an exact symmetry, because the universe appears to contain very many baryons and very few antibaryons. Actually, a better metric for the baryon asymmetry of the universe is to compare the baryon density to the density ... 3$r=n^2a_0$for the Bohr model. Similarly, in the Schrodinger model, the most probable radius is$r=n^2a_0$, when$n=l+1$3 Yes, the mean radius grows as$n^2$asymptotically, but to have$1-10^{-20}$for sure you have to add some more ;-). Highly excited atoms are called Rydberg atoms and are dealt with in experiment. 2 The picture of an electron as a little ball that moves around like a billiard ball sometimes works. But it fails enough times that one has to conclude that it's not correct. This is one of those cases when if fails. The wavefunction represents where the electron might be found if an experiment were done to find it. That's not the same as saying that the ... 2 Like garyp says, the electrons are not discrete particles, but rather exist as a smear (a cloud) with the most intensity of their existing in the spaces so described by the wavefunction. Now, all of the electron needs to interact at once, so when an interaction (measurement, chemical reaction, etc.) happens, the wavefunction of the electron changes as well ... 2 If you had a single hydrogen atom, and you watched for a single transition, then yes, you would only see emission at a single frequency. There would be one line in your spectrograph so to speak. But rarely do you have just one atom. And quite often an atom undergoes multiple transitions while you are watching it. When looking at a large ensemble, whether it ... 2 The potential energy function is the same for both. The energy level solutions are the same for both. The key difference is that in (most modern interpretations of) the Schrodinger model the electron of a one-electron atom, rather than traveling in fixed orbits about the nucleus, has a probablity distribution permitting the electron to be at almost all ... 2 No, it is an approach called semiclassical quantum Mechanics. They mix the Schrödinger equation in which quantum particles interact with the classical potentials, to see what the calculations does. And it happens that it gives good results. So even if flawed, it is still predictive enough to be considered good. For instance, it predicts the fine structure of ... 2 Here's another, direct method to go with (and agree with) the Virial theorem method given in By Symmetry's answer. From the Schrödinger equation solution (see the "Mathematical summary of eigenstates of hydrogen atom" heading in the "Hydrogen Atom" Wikipedia page):$\$\psi_{n,\,\ell,\,m}(r,\vartheta,\varphi) = \sqrt {{\left ( \frac{2}{n a_0} \right ...

2

If you assume perfect efficiency, then the energy of dissociation of a liter of water is computed as follows: 1 liter of water, molar mass 18g, => 55.6 moles The energy needed is 237 kJ per mole (from your link - see under "thermodynamics"). 237 * 55.6 => 31.7 MJ of energy for a liter of water. In terms of power, this is 3.67 kW for one hour. This ...

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