Questions tagged [orbitals]

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What is the total degeneracy for the orbital motion for the case where the principal quantum number $n=4$? [closed]

I assumed the answer would be 9, as the angular number $l = n$ and the degeneracy is given by $2l+1$. However, my professor says my answer is wrong. I cannot spot the mistake.
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What is the difference between angular momentum of electron by Bohr and orbital angular momentum?

Orbital angular momentum of an electron is $\hbar \sqrt{\ell(\ell+1)}$ where $\ell$ is angular quantum number. Angular momentum of an electron by Bohr is given by $mvr$ or $\frac{nh}{2\pi}$ (where $v$ ...
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Are angles ($\theta$ and $\phi$) in spherical coordinates treated as operators in quantum mechanics?

Position is specifically considered as an operator in quantum mechanics. I want to know if $\theta$ and $\phi$ are explicitly considered as operators in quantum mechanics for solutions to 3D ...
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Can multiple electrons transition simultaneously?

In my atomic lecture notes we prove (at least for electric dipole transitions), that only one electron can 'jump' at once. The proof is as follows. The matrix element (assuming distinguishable ...
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4answers
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Why does the antibonding orbital has higher energy (than the bonding orbital) if the Coulomb repulsion is lesser?

Depending on its strength, the attractive double dirac delta potential shown below can support two bound states. They are called the bonding and the antibonding orbitals as shown in the figure below ...
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1answer
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How can real $d$ orbitals be computed from complex orbitals?

I recently completed MIT's 8.04 quantum mechanics course on edX and have been writing python code to compute hydrogen-like electron orbitals, basically just for fun. My program computes the ...
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1answer
42 views

When is the principal quantum number $n$ a good quantum number?

Since I don't know an associated operator to the principal quantum number $n$, I don't know when it is a good quantum number. By 'good quantum number' I mean a quantum number that is conserved over ...
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1answer
107 views

Why does exchange occur between electrons having same spin in “degenerate orbitals”?

I am a high schooler and I read in my chemistry book that half filled degenerate orbitals are stable because electrons with the same spin in those degenerate orbitals exchange their positions and this ...
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3answers
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Can protons have orbital if they gain more energy? [closed]

I read that because a proton is much more massive than electron but an electron has slightly more energy so it doesn't fall into the nucleus and orbital is due to the constructive interference of ...
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1answer
69 views

How to “restrict to a subgroup” to explain representation structure?

I am reading from Quantum Theory, Groups and Representations - Woit. In Chapter 21, on page 237 in discussion of the energy eigenstates of the Coulomb potential, the following figure is presented: ...
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6answers
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What actually makes a Bohr's radius stable?

I was told that Bohr introduced the concept of Stationary orbits in which electrons were stable in my school but I never got the reason behind this stability. So can someone explain why is Bohr's ...
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4answers
45 views

How charge is explained, in context of bonding?

In ionic compounds, like NaCl, Cl gains negative charge due to addition of electrons. In covalent compounds, like Cl2, sharing of electrons occurs. Here too, Cl atom has got an extra electron in its ...
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1answer
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How to calculate the sphere of influence of a planet?

I'm making an orbit simulator, and to make it simpler, I'm only simulating one celestial object(planet, moon, sun) acting on each object(sattelite). So that the sattelites and rockets can switch ...
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1answer
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Why do relativistic effects that cause contraction of the $s$ and $p$ orbitals impact the $s$ and $o$ orbitals further away from the nucleus?

I was reading quite a well-known paper titled 'Relativistic Effects in Structural Chemistry' by PEKKA PYYKKO. In the paper, he describes how the electrons occupying $s$ and $p$ orbitals move at ...
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Fermi Energy and single particle state

I have a question about Fermi energy and the single particle state. I have it a bit hard, on how to formulate my question, for the below text, because I don't fully understand the concepts so well. ...
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3answers
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Do actually sharing of electrons make bond?

My book says, "When we bring two atoms close, respective repulsion and attraction happens between the two's electrons and protons. Experimentally it is found that the attraction force is greater ...
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4answers
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Are all atoms spherically symmetric? If so, why are atoms with half-filled/filled sub-shells often quoted as 'especially' spherically symmetric?

In my atomic physics notes they say In general, filled sub-shells are spherically symmetric and set up, to a good approximation, a central field. However sources such as here say that, even for the ...
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Is it better to think of electron shells as depending on the value of $n$, or depending on the energy difference between sub-shells?

Often electron shells are defined as 'states with the same principal quantum number $n$' which would suggest that $3s$, $3p$, $3d$ sub-shells are all in the same shell. Conversely, it is also often ...
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4answers
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What significance can we assign to 'breathing' of $s$-shell orbitals?

In my atomic lecture notes it says We can visualise an $l=0$ $s$-state as a spherical cloud expanding and contracting - breathing, as the the electron moves in space. Similarly in the video enter ...
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Does Hartree-Fock method always converge to global energy minimum?

I'm not asking about whether the Hartree-Fock method will always converge, but, if it does, it seems like Wikipedia is saying that it will always converge to the global minimum energy of the TISE. ...
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1answer
43 views

What is the ratio of energies of Hydrogen in its first excited state and second excited state?

Since Energy of an electron in a particular orbit is given by $E= -13.6/n^2$ eV So, I equated the energies in $n=2$ and $n=3$ which are the 1st and 2nd excited states respectively. The answer was 9/4. ...
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2answers
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Planetary model of the atom

Wherever I look about the early planetary model of the atom, it says the electron must lose energy while revolving around the nucleus. And therefore fall into the nucleus. Thus, the atom is unstable. ...
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1answer
42 views

First-order perturbative correction of the ground state energy of the Helium atom in spherical coordinates [closed]

I want to calculate the first order perturbative corrected ground state energy $E^{(1)} = \langle H'\rangle$ with the perturbing Hamilton $H' = \frac{e^2}{|\vec r_1 - \vec r_2 |}$ using the spatial ...
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Orbital Angular Momnetum Explained as simply as possible

TL;DR Why is $\ell$ defined as the shape of an orbital? And what relation does angular momentum of the electron have with the shape of the orbital? Why is the $z$-component the magnetism (why does ...
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2answers
351 views

What exactly is the “energy” of an orbital?

In chemistry, I have come across the term " energy of orbital" several times but I am unable to understand it from a physics point of view. 1) is the energy of an orbital the electric ...
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What do atomic orbitals represent in quantum mechanics?

I am learning the basics of quantum mechanics and am familiar with the Schrödinger equation and its solution, but I was confused about what the familiar atomic orbital shapes represent? Do they ...
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81 views

Why does perturbation theory work for helium atoms?

I saw the following argument for calculating the energy levels of a helium atom. First, ignore the Coulomb interaction term between two electrons. For this simplified model, we have the same solutions ...
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35 views

How do we know that the probability of finding an electron on some specific pattern is high and on other patters are not?

I have watched a video about the positions of electrons inside of an atom and on some specific patterns probability of finding an electron is high for example in- s,n,d why?
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Why $d,f$ electrons are more localized than $s,p$ electrons?

I’m reading Assa Auerbach’s Interacting electrons and quantum magnetism. In 1.3, it says In contrast, transition metals and mixed valence rare earth compounds contribute $d$ and $f$ electrons to the ...
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Why ferroelectricity requires empty $d$-orbitals?

I understand that ferroelectricity needs off-centre cations, but why it requires the cation to be empty or fully filled d-orbits? Ref: Multiferroicity in atomic van der Waals heterostructures Nature ...
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How should we think about excited electron states?

Consider an atom of your choice. I wanted to have a way of organizing all of its electron excited states into some kind of mathematical structure, that was a sufficiently generic framework that the ...
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1answer
27 views

Do the 'fine transitions' of electrons in a molecule or atom involve jumps between sublevels within a 'level', e.g. from $s$ to $d$?

Hyperfine transitions, like the (astronomically) famous 21cm line, involve just spin flips, right? And the electron jumps we first learn about in school, that release visible or near-visible photons, ...
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Does 1 electron in each orbital guarantee maximum multiplicty?

An easy question(which most of the students of my class got correct) asks: Pairing of electrons in the orbitals belonging to the same subshell (p, d or f) does not take place until each orbitals ...
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2answers
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Ionisation energy of hydrogen: Why are kinetic and potential energy connected by $2 T = -V$ in the derivation? [duplicate]

The following is a translation from the German Wikipedia: The ionisation energy or binding energy B is the sum of potential energy V and kinetic energy T of the electron. $B = V+T$. Since the ...
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What is the continuum function in $R$-matrix basis?

In the book $R$-Matrix Theory of Atomic Collisions it is defined the $R$-matrix basis \begin{aligned} \psi_{k}^{\Gamma}\left(\mathbf{X}_{N+1}\right)=& \mathcal{A} \sum_{i=1}^{n} \sum_{j=1}^{n_{c}}...
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Why the energy of a state increases with increasing azimuthal quantum number $\ell$?

I am reading Griffiths' introduction to quantum mechanics. But I am confused about the relation between the state energy and azimuthal quantum number $\ell$, which is discussed in Chapter 5.2.2 where ...
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2answers
23 views

Does the intrinsic spin of electrons change when we excite electrons for hybridisation?

Have had been told that electrons can move from one energy level to another by transmitting or taking in energy, and do that profitably in hybridisation. I am interested in knowing how exactly do they ...
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291 views

How do electrons move from one energy level to another?

EDIT 1: Many people are recommending that the question Do electrons really perform instantaneous quantum leaps? is very similar to mine. However, that question is very specific to quantum leaps. I am ...
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Conceptual question about the transition dipole moment and its components

I'm looking at the following transition: $1s\to2p$. For this transition, my textbook says: We have to consider three degenerate $m$-components $m=0,\pm1$. We put the quantization axis in the $z$-...
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1answer
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How do delocalized electrons conduct electricity?

I have learned that for a meterial to be conductive, it must contain free charge carriers, in most cases electrons. Graphite does conduct electricity parellel to its graphene layers, which is due to ...
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How to couple the $^3p_2$ with $d_{5/2}$?

The first two electrons' LS term is $^3p_2$,how can this term coupling with another electron $d_{5/2}$?(this electron in another shell) for $^3p_2$,$L_{12}=1$,$S_{12}=1$,and only the orientation of $...
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Why does exchange energy of electrons lead to stability?

My text book says "electrons of the same spin in degenerate orbitals tend to exchange their positions and this leads to stability". Firstly, why should the electrons exchange their positions?...
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1answer
28 views

Why is there a third extremum radial coordinate for “rosace-like” trajectories in the Schwarzschild geometry?

I drop a test-particle in the Schwarzschild geometry, at an initial radial coordinate $r_0 > a \equiv 2 G M$ ($a$ is the Schwarzschild "radius"), with initial velocity $v_0 < 1$ ...
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1answer
92 views

Do electron (or other quantum particles) move in a 4th spatial dimension? [closed]

I was told that an electron can change its position from position A to position B without appearing anywhere between A and B. So i think that it might be possible for an electron to move in 4D or ...
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1answer
27 views

If atomic electron is in $f$-state, what values of the total angular momentum are allowed?

What I know is total angular momentum $\vec{j}= \vec{\ell}+\vec{s}$, here $\ell=3$ for $f$ state, $s=1/2$ for spin, so the total angular momentum should be $7/2$, is this correct? is there any other ...
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Bohr's second postulate

What was Niels Bohr's reasoning behind his second postulate (about integer angular momentum)? How did he come up with it without any knowledge of matter waves? I only have a high school level ...
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1answer
34 views

How do we know that $^1S$ is the ground state of the helium atom?

Let $\psi=a_1\phi(1s(2) \ ^1S)+a_2\phi(1s(1)2s(1) \ ^1S)+a_3\phi(2s(2) \ ^1S) +... $ be a state of the helium atom. Applying variationally calculus we can found the energy expectation value of this ...
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Do different eigenstates of total angular momentum have necessarily different energies?

Let $H$ be the Hamiltonian of a specific atom (not hydrogen) and $J$ the total angular momentum. Since $H$ and $J$ commute, they have common eigenstate. So we can label the atomic states by their ...
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2answers
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Is $1s2s$ state an upper bound of Helium excited states? [closed]

Generally people says that the state $1s(1)2s(1)$ $ ^1S$ is an excited state of Helium atom . Variation theorem guaranties that the expectation value of this state is greater than the ground state, ...
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1answer
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Locating a point in circular orbit on the Cartesian plane after some $t$ seconds

The second hand of an analog clock has angular velocity $\omega=\pi/30$ rad.s-1. The blue body in the image below mimics the hand's clockwise motion on the Cartesian plane with the center of ...

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