Questions tagged [hydrogen]

Could refer to (1) A hydrogen molecule; two hydrogen atoms bonded together or (2) A hydrogen atom; One electron electromagnetically interacting with a nucleus made of a single proton. Hydrogen atoms are the simplest atoms, and they are the only atoms for which we can exactly solve the Schrodinger Equation. Hydrogen atoms are the only atoms which could exist even n a world with fine - structure constant 1.

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Can muonic helium (hydrogen 4.1) be used to (briefly) create ultra heavy water?

Combining oxygen with deuterium creates heavy water and tritium makes super-heavy water. Muonic helium behaves as hydrogen instead of helium, as it has only 1 valence electron. This would be quite ...
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On properties of Hydrogen-like atoms [closed]

Citing Wikipedia: A hydrogen-like atom will have a Bohr radius which primarily scales as ${\displaystyle r_{Z}=a_{0}/Z}$, with ${\displaystyle Z}$ the number of protons in the nucleus. I'm trying to ...
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Where does 13.6 $\rm eV$ of energy go in the process of bringing an electron from infinity to the Bohr ground state?

In the answer to this SE question about the Relation of potential energy and total energy in Bohr Model of the hydrogen atom, it is clearly explained by using the Virial Theorem that: "In the ...
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How to derive the the asymptotic momentum of a classical electron after it is ionized by a strong laser field

I was reading a recent paper, Capture into rydberg states and momentum distributions of ionized electrons. N. I. Shvetsov-Shilovski et al. Laser Physics 19, 1550 (2009). which studies the ionization ...
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Negative energy solutions in Dirac equation for hydrogen atom

Solving Dirac's equation for hydrogen atom $^1$, $$(\boldsymbol{\alpha}\cdot\boldsymbol{p}+\beta m)\Psi=E\Psi$$ after some mathematical machinery we find the condition for energy eigenvalues $$\...
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Amount of atoms in 1000kg of Deuterium per Britannica [closed]

I was recently attempting to calculate the amount of energy released in fusion reactions, and began using as a reference the Britannica article Energy released in fusion reactions. Within that ...
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Why do you use $n_r = n -\ell -1$ as quantum number instead of $n$ for hydrogen atom?

I got two different quantum numbers for the same problem: Hydrogen atom without any interaction. Then, my energy is $$ E_n = -\frac{R_y}{n^2} $$ with the quantum number $n = 1, 2, 3, ....$ In another ...
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If electron fails to get into the lowest energy orbit after neutron decay does it remain bonded to the proton?

I have read this on Wikipedia: A very small minority of neutron decays (about four per million) are so-called "two-body (neutron) decays", in which a proton, electron and antineutrino are ...
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Are the unperturbed hydrogen wavefunctions solutions of the fine structure Schrödinger equation?

The hydrogen atom Hamiltonian, with fine structure effects included, is $$H = \frac {p^2}{2m}-\frac{e^2}{r}-\frac{p^4}{8m^3c^2}+\frac{e^2}{2m^2c^2}\frac{\mathbf{S\cdot L}}{r^3}+\frac{\pi}{2}\frac{e^2\...
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Would metallic hydrogen be powerful enough to fuel a 1g rocket?

I have read that the issue with a 1g rocket is that no fuel in existence is powerful enough to provide a constant acceleration of 9.8 meters per second for years on end, and it would take hundreds of ...
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A nuclear fusion generating cart

In an unrealistic thought experiment, suppose I had a $100$ meter track with a cart ontop that had a "pocket" on the front of the cart. Suppose further that this track and cart were in a ...
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Would isotopes of hydrogen separate in liquid phase?

According to https://www.aqua-calc.com/calculate/volume-to-weight the difference in density of deuterium/hydrogen in liquid state is 70miligrams/cm^3 vs 169miligrams/cm^3. Would this be a sufficient ...
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Connection between Laplace's equation and hydrogenic electron Schrodinger equation

Consider Laplace's equation: $\nabla ^2 V = 0$ This holds for an electric potential $V$ in a region of space where no charges are present. This includes a Coulomb potential of a hydrogen nucleus (...
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Integral of $\exp(-r^2)/r$

I came across the integral $$ \int_0^\infty\frac1r\exp(-2\lambda r^2)\mathrm dr $$ in variational method for hydrogen method when choosing the test wavefuntion as $$ \psi=\exp(-\lambda r^2) $$ since ...
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What is the second excited state of $H^{-}$?

Not considering the electronic spin, the degeneracy of the second excited state of H-atom is 9, while the degeneracy of the second excited state of $H^{-}$ is : The only thing that I need to solve ...
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Why numerical solution of hydrogen Schrodinger equation diverged in higher energy levels?

I am studying Computational Physics by Thijssen. I also read this notebook https://github.com/tamuhey/python_1d_dft/blob/master/numpy_1ddft.ipynb. Then I changed this notebook to solve this ...
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Confusion of spin-orbit coupling in Hydrogen atom

I'm learning the fine structure of hydrogen and feel a little confused about the spin-orbit coupling term in the hamiltonian. The corrected hamiltonian is: $$H = \frac{\vec{P}^2}{2m}+ V(R)-\frac{\vec{...
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What's the hydrogen atom in the string theory? [closed]

It was known that string theory was hard to test. However, I'm wondering how would the string theory do with the existing theories. For example, could the low energy effective field theory of the ...
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How can I find the recombination coefficient of hydrogen for various stars? Which equation can I use to find it? How can I plot it in a bar graph

How can find recombination coefficient of hydrogen for various stars? Which equation can I use to find it? How can I plot it in a bar graph.
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How can find recombination coefficient of hydrogen in various stars

I'm trying to calculate recombination coefficient of hydrogen at different temperatures in various stars. How can I calculate it at different temperatures?
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How to find the expectation value of $P^2$ for the hydrogen atom? [closed]

I tried using spherical coordinates, the wavefunction for the hydrogen atom and the Laplacian in spherical coordinates but I just ended up with a very long integral and I don't know what to do next. I ...
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Fine structure away from the extreme mass ratio limit

In kindergarten, all physicists learn how to find the spectrum of bound states of a hydrogen atom by solving the Schrödinger equation with a Coulomb potential. In particular, we get an exact solution ...
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Perturbation Theory for the Hydrogen Atom

The full Hamiltonian of the hydrogen atom (including fine structure) is $$H = H_0 + H_{rel} + H_{s.o},$$ where $H_0 = \frac{p^2}{2m} -\frac {e^2}r$, $H_{rel}$ (resp. $H_{s.o}$) is the relativistic (...
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How to find the probability current density of hydrogen atom wave function?

Can someone give me a book recommendation or a pdf where I can find how to get the radial, polar and azimuth component of it? I try to do it myself but I can't really figure out what to do next,
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Benefit of Hydrogen Electrolysis

I was considering building a simple solar hydrogen generator for fun and it led me to the question Does the hydrogen produced have more potential energy than the electricity used to get it? Obvious ...
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If water vapour is a green house gas , then how is using hydrogen fuel cell stopping global warming?

Hydrogen fuel cells produce water vapour then aren't they also contributing towards global warming ?
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Natural frequency of electrons and the Hydrogen atom

Natural frequency is the frequency of a system when it vibrates freely without damping. I am in need of clarification about the definition of natural frequency at the atomic and molecular levels. Let'...
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Why is there a non-zero probability density of finding an $l=0$ electron at the origin of a Hydrogen-like atom?

A well known result for the $l=0$ hydrogenic functions is that $$\psi_{nlm_l}=R_{nl}(r)Y_{lm_l}$$ $$|\psi_{n00}|^2=\frac{Z^3}{\pi a_0^3n^3}$$ where $R_{nl}$ and $Y_{lm_l}$ are the radial function and ...
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Hydrogen atom quantum numbers distribution

The wave function of the hydrogen atom is given by $\psi_{n \ell m} \left( r, \theta, \varphi \right) = R_{n \ell} Y_{\ell}^{m} (\theta, \varphi)$ which depend on the quantum numbers $n, \ell, m$. I ...
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Perturbed hydrogen atom transition probabilities

Given the perturbed hydrogen atom having Hamiltonian $$ \hat{H} = \frac{\vec{p}^2}{2m}-\frac{e^2}{|\vec{r}|}+\lambda \vec{S}\cdot \vec{r} $$ (1) say which operators commute with the Hamiltonian among $...
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How to take care of singularities or discontinuities in a numerical Time-Dependent Schrodinger Equation simulation using the Split-Operator method?

I am currently trying to numerically simulate a two-dimensional Hydrogen atom using a split-operator method (elaborated below). I have tested that the code produces a good result for an isotropic ...
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How are Hydrogen Atoms Described in Standard Model? [closed]

Specifically, in order to calculate the quantities like emissivity or the cross section of photon resonant scattering by hydrogen atoms, what logical deduction route should we follow, starting from ...
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Coulomb potential vs Coulomb potential energy

I know that the Coulomb potential in a hydrogen atom is $V(x)=\frac{-e^2}{4 \pi \epsilon_{0} r}$. Usually the potential is given by the potential energy divided by the charge, so I would expect $V(r)=\...
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Inconsistency between the Lamb shift and vacuum polarization

According to Griffiths' Introduction to Elementary Particles, one of the contributions to the Lamb shift is the lack of vacuum polarization. In chapter 2, he explains how vacuum polarization can be ...
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Calculation of the rate of transition of the Hydrogen electron from $|2lm\rangle$ to ground state

This is how it was calculated in my book: We take the electromagnetic field to be part of the system. The initial state of the system is the direct product $|2,l,m\rangle \times |0\rangle$. $|0 \...
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Generating hydrogen from moving water

I am a high school student and I have planned to participate in national innovation award of my country. In this competition, prototype of innovation is needed. I have created a design but I lack ...
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Partition Function of a Hydrogen Atom - How it is derived?

It is well-known that the Partition Function of Hydrogen Atom diverges if we calculate in naive manner. And I could find the partition function named Brillouin-Planck-Larkin partition function which ...
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Can hydrogen be generated from flowing water?

Imagine water (water is made conductive using salt) flowing at a great speed is passed through a high magnetic field. Then due to changing flux , can we induce electric current in water? If electric ...
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Electron capture and overlapping wavefunction

I'm sorry if the following question sounds a little homework-ish, but I need help understanding how to even proceed with the following problem. In a hydrogenic atom of atomic number $Z$, the ...
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Hydrogen atom as a free particle moving on a three dimensional sphere?

I have heard on various occasions that the Hamiltonian/Lagrangian of the Hydrogen atom or that of a particle moving in $1/r$-potential can be transformed into that of a free particle moving on the ...
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About normal Zeeman effect and Hydrogen atom

I think that the normal Zeeman effect is experimentally observed only when we can neglect the spin i.e. if we consider a spin-singlet state with total spin $S=0$. Since in the hydrogen atom there is ...
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Problems with selection rules and polarisation

On this website http://bingweb.binghamton.edu/~suzuki/QuantumMechanicsFiles/2-4_Polarization_of_photon_emitted_from_an_excited_atom.pdf#page72 they want to explain the selection rules and the ...
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Expectation value of $r^{-3}$ for $\ell=0$

I'm trying to calculate the hyperfine structure energy for Hydrogen. From Jackson's Electrodynamics book eq. 5.73 we have: $$H_{HFS}=\dfrac{\mu_{0}}{4\pi}\left[\dfrac{-8\pi}{3}\mu_{N}.\mu_{e}\delta(r)+...
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Atom orbitals physics vs. chemistry

Why draw chemists always $|2p_x\rangle$, $|2p_y\rangle$ and $|2p_z\rangle$ orbitals instead of $|2,1,m\rangle$? Since $|2p_x\rangle$ or $|2p_y\rangle$ are a superposition of energy eigenstates they ...
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Is this really $SO(4)$ algebra?

The commutation relations involving the components of Runge-Lenz vector of the Hydrogen atom problem, ${\vec A}$ and the angular momentum ${\vec L}$ are given by $$ [L_i,L_j]=i\hslash\varepsilon_{ijk}...
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Probability of Finding an Electron Close to the Nucleus of a Hydrogen Atom

I am trying to calculate the probability of finding an electron in a small volume element close to the nucleus of a ground state hydrogen atom. I then need to calculate the same but replacing the ...
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Probability of finding the electron of a hydrogen atom within a cone of a given angle

I am working on a question where I must find "the probability of an electron being located within a cone of semi-angle 20 degrees w.r.t the z-axis" for the 2s and 2p states of a Hydrogen ...
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Spherical harmonics visualization as oscillations on sphere

I‘ve watched this video where they visualize the spherical harmonics as oscillations. Since the spherical harmonics determine the spatial shape of the orbitals of the hydrogen atom their time ...
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Current distribution in hydrogen atom

In the Bohr model of hydrogen, the electron orbits the proton at selected radii such that the angular momentum is quantized in integer multiples of $\hbar$. We can then associate a current with such a ...
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How can hydrogen have negative enthalpy?

According to the data found here, the enthalpy of hydrogen at very low temperatures can be negative. I fail to wrap my head around how this can be in line with $H = U + pV$, where $U$, $p$, and $V$ ...
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