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Questions tagged [quantum-mechanics]

Quantum mechanics describes the microscopic properties of nature in a regime where classical mechanics no longer applies. It explains phenomena such as the wave-particle duality, quantization of energy and the uncertainty principle and is generally used in single body systems. Use the quantum-field-theory tag for the theory of many-body quantum-mechanical systems.

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Action of unitary operator on operator: two different definitions?

Defnition 1: Ref: Section 2.4.9, Quantum Mechanics: Concepts and Applications By Nouredine Zettili Ket $|\psi \rangle $ transform as $\underline{|\psi' \rangle = \hat{U} |\psi \rangle }$. Given ...
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When a diatomic ion dissociates, how does the electron density on each atom change?

Let's say, we have a fluorine molecule (F2), and we take 3 electrons from it, so now its bond starts to stretch, and at last the bond breaks and the two atoms are farther and farther away from one ...
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Uniqueness Theorem and the 1D Infinite Square Well

Consider the 1D infinite square well problem: $$\frac{d^2\psi (x)}{dx^2} = -k^2\psi (x)\tag{1}$$ along with the boundary conditions $\psi (0) = \psi (L) = 0$. This seems to be a well posed problem ...
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1answer
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Properties of Expectation Values Under Variable Substitution [on hold]

I am working on a homework problem from Griffiths QM (Problem 2.11, 3rd Edition). Specifically, I'm working on finding $\left<x^2\right>$ for the ground state and the first excited state of the ...
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Solving the problem using Many Body Perturbation Theory

I am trying to solve the following Hamiltonian using Many body perturbation Theory. $$H=\sum_{i=1}^{N}\Bigg[\frac{P_{i}^{2}}{2m} -\sum_{i,j}\frac{1}{|\vec{r}_{i}-\vec{R}_{j}|}\Bigg]$$. I split this ...
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Women who are science communicators for quantum physics? [on hold]

I've been asked about books or other sorts of materials (video, etc.) on quantum physics produced by women. If you can recommend any, thanks in advance!
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31 views

Normalising squeezed position eigenket?

I want to find the effect of squeezing operator $S(r) = \exp \big[r(\hat{a}^2 - \hat{a}^{{\dagger}^2})\big]$ on $|q\rangle$ i.e. $S(r)|q\rangle$. I proceed as follows: $$S(r)\hat{q}|q\rangle = S(r)...
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Rayleigh-Schrödinger series and pertubation theory

If we look at the time-independent perturbation theory and the non degenerate case where $H=H_0+\lambda H_1$ and $H_0$ has eigenvalues $E_i^{(0)}$ and normalized eigenfunctions $|\Psi_i^{(0)}>$. ...
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3answers
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Symmetry of the spin function and T0 and S states

$|T_0\rangle = \frac{1}{\sqrt{2}}(|\uparrow \downarrow\rangle + | \downarrow\uparrow\rangle )$ is a triplet state, whose spin function has to be symmetric. $|S \rangle = \frac{1}{\sqrt{2}}(|\...
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Where does the inconsistency between QM and general relativity stem from? [duplicate]

I understand special relativity works with QM (as seen by QFT) and I know that there is some discrepancy between GR and QM. Could someone elaborate on that? Where does the inconsistency stem from? ...
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How does light 'choose' between wave and particle behaviour?

Light exhibits wave behaviour in phenomenon such as interference but particle behaviour in the photoelectric effect. How does light 'choose' where to be a wave and where to be a particle?
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Zeeman effect - accidental degeneracy for high fields?

For a low level field, the energy splittings are given by $$\Delta E(L, S, J, m_J) = \mu_B m_j g_L B_z$$ where $g_L$ is a g-factor. This predicts a separation into $(2J+1)$ levels, so completely ...
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1answer
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Why an insufficient overlap cause vanishing exchange interaction?

Why should the exchange interaction vanish if the atoms do not have sufficient overlap in their overfunctions? For exchange interaction not to vanish, the only requirement seems to be that the ...
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0answers
22 views

Symmetry operators of a Bloch Hamiltonian

Consider a lattice with a 3 atom basis, e.g. the Lieb lattice, and some completely arbitrary on-site energies and hopping energies and phases between the different atoms. In momentum space we can ...
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1answer
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Derivation of gradient of the expectation of local energy

Background: In Variational Monte Carlo, given a Hamiltonian $H$ and a wave function $\psi_\alpha$ dependent on some parameter(s) $\alpha$, we have defined a quantity known as the local energy, $$E_L =...
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When can quantum states be distinguished with certainty?

Through a few examples, I'd like to learn under what circumstances can different quantum states be distinguished from each other. So for example, the standard quantum teleportation scheme starts out ...
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Eigen value of a unknown system [closed]

Why we are using operator for finding the eigen value of a unknown system
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1answer
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Decomposition of the maximally entangled states

We know that the set of symmetric bipartite pure states is spanned by $S=\{|\phi\rangle^{\otimes 2},|\phi\rangle \in \mathbb{C}^d\}$. I want to know if the maximally entangled state $|\psi\rangle = \...
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Symmetry propertis of the 3-J symbol from the symmetry properties of CG coefficient

I use the relation between 3-J symbol and the CG coefficient. But using the symmetry propertis of CG coeficient, I am not able to arrive at the symmetry properties of 3-J symbol normally used. Even ...
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2answers
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How does the collision between two atoms work?

Considering the quantum mechanical model for an atom, what exactly happens when two atoms (say, two Ca2+ ions in a Brownian motion) collide with each other? As I know, this collision is not like a ...
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2answers
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Why can many distinguishable particles be in the same quantum state?

In Boltzmann distribution, it says we have no restrictions on how many particles there are in a same quantum state. BUT the contradiction is: Distinguishable particles mean we think their wave ...
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1answer
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Does the wave function contain the information of the particle's mass?

It is said that wave function contains "all" the information about a particle but I don't see how I can get the mass from wave function. Further, it is able to judge what kind of particle it is by ...
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Ladder Operators - Quantum Mechanics [closed]

In my lectures notes, we define the raising operator as $$L_+ \Phi_{l,m_l} = \hbar \sqrt{l(l+1) - m_l (m_l + 1)} \hspace{1mm} \Phi_{l, m_l + 1}$$ Then we later look at spin-orbit coupling using ...
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1answer
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Scalar product of free field and conjugate momentum

Given $[\Phi (x), \Pi(y)] = \delta^{3}(x-y)$,$ $ $\Phi|\phi\rangle = \phi(x)|\phi\rangle$ and $\Pi|\pi\rangle = \pi(x)|\pi\rangle$, I am trying to prove $\langle\phi|\pi\rangle \sim e^{i\int d^{3x}\...
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what is the cause of the first atom moving in moving?

I'm a computer student and I know nothing about physics and physics.stackexchange, so if I did something wrong, tell me to correct it (for example tagging). and ...
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1answer
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Can we exlain Bell's Theorem without dwelving into Quantum Mechanics?

I have recently started learning Quantum Mechanics. We learnt some things on light polarization, so I referred to this site. Luckily I stumbled upon this answer which links to this site which sated my ...
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1answer
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Matrix representation of spin-1/2 operators in Sakurai

Hello and thanks for reading. I'm an undergrad working through the first chapter of Sakurai's text and was going through the principles of the spin-1/2 system. The author demonstrates closure and ...
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2answers
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What is the escape velocity of a neutron particle (not neutron star)

I'm not sure if this question makes sense (if not maybe you can explain why) But if the neutron has mass and have a size, then it should have a escape velocity in the "surface" right? I know the ...
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1answer
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Schrödinger equation variable substitution

Here is the equation sets: $$\frac{-h^2}{2m}\frac{d^2\psi}{dx^2}+\frac{h^2}{2m}\left(\frac{\rho(\rho-1)}{\sinh^2x}-\frac{\lambda(\lambda-1)}{\cosh^2x}\right)\psi=E\psi\tag{1}$$ there is a variable ...
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Integration by Parts - Calculus of variations [closed]

I've encountered an expression while doing a QM question where we make a transformation $\Psi \to \Psi + \delta\Psi$ to the Hamiltonian. $\Psi$ is normalised, as is $\Psi+\delta\Psi$: $$\int d^3r \...
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Does an antiparticle leave a different (from the normal particle) mark in a bubble chamber experiment

Experimentalists usually have experiments where they scatter particles superheated transparent liquid, thus checking for the particle's traces. These particles can be for example quarks, and ...
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What is frequency entanglement?

If frequency (or energy) is regarded as a quantum property, can one generate a pair of photons with frequency entanglement? What would be the uses of this type of entangelment in say, sensing? How ...
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2answers
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Orbital angular momentum quantum numbers - subtracted?

Reading Griffiths' Quantum Mechanics. We have the electronic confirmation of Carbon as $$(1s)^2 (2s)^2 (2p)^2$$ in the ground state. He says There are two electrons with orbital angular ...
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Encoding infinite information in a qubit vs. classical system

In this Quantum Computing article by Michael Nielsen he argues about some of the limitations imposed by quantum measurement. In particular how the amplitude $\alpha$ of a single qubit $\alpha |0> +...
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Three spin states of a spin-1 particle

For a spin-1 particle at rest, it has three spin states(+1, -1, 0, along the z axis). If we rotate the z axis to -z direction, the spin +1 state will become the spin -1 state. Can we transfer the spin ...
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What Lie group structure is used for infinite-dimensional Unitary Groups in Quantum Mechanics?

Given an infinite-dimensional Hilbert space $H$, the set $U(H)$ of all unitary operators on $H$ forms a group, known as the unitary group. Now several subgroups of this group play an important role ...
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Expectation value of $x,y,z$ for general $nlm$ state of hydrogen atom

How to calculate expectation value of $\langle x\rangle, \langle y\rangle,\langle z\rangle$ for the general $\psi_{nlm}$ state? $x$ has $\sin(\theta)\cos(\phi)$ angular part which can be expressed as $...
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Feynman's derivation of the Schroedinger equation by expanding path integrals to first order in $\epsilon$

As discussed in the answer to How can one derive Schrödinger equation?, one should be able to "derive" the Schrodinger equation from the path integral formulation of quantum mechanics. However, ...
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Quantum Numbers/state multiplicity - specifically concerned with the differences between $L$, $l$ and $S$, $s$

To put this into context, I will write out exactly what was on the lecture slide that confused me (I've also attached a picture, but I appreciate that text is preferred on here for search engine ...
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What is the symmetry behind this degeneracy?

I was working on a quantum mechanics problem, involving the perturbation of the 3D cubical potential well: Suppose we perturb the infinite cubical well \begin{equation} V(x,y,z)=\begin{cases} 0, \...
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Griffiths Quantum Mechanics - Identical Particles (Wavefunctions)

An example in Griffith's Intro. to Quantum Mechanics is: Suppose we have two non-interacting particles both of mass $m$ in a infinite square well. The one particle states are $$\phi_n (x) = \sqrt{\...
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Potential well and uncertainty principle

Is it possible to have an infinite potential well of such width, that for ground state $\sigma_x\sigma_p=\frac{\hbar}{2}$?
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Potential with Diracs Delta [closed]

Could a potential with Diracs Delta exist so that the energy for a ground state would be positive? I'll appreciate any answear since this question is giving me a headache for 2 hours.
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Schrödinger equation from Dirac–von Neumann axioms

Using Dirac–von Neumann axioms, it isn't too difficult to derive $$\frac{d}{dt}\left|\psi\right>=ikH\left|\psi\right>,$$ where $H$ is some hermitian operator. However, how would one show that $k=...
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How to derive the energy of a parabolic confining potential in a wire [closed]

I couldn't find any sources or help on other websites with regards to this question... Anyways, how to derive the expression for energy of parabolic confining potential in a wire as shown below? I ...
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1answer
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What would happen if the opposite complementary variables of two entangled particles were measured at the same time?

Many explanations about the uncertainty principle and its related EPR paradox state that it is impossible to measure opposite complementary variables on different entangled particles; for example, ...
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Is polarization complementary along its different axes?

Is polarization complementary along its different axes -- much like the spin of a particle is -- thus implying that the uncertainty principle holds for polarization measurements on these different ...
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How to use Dyall basis sets for radial wave functions?

I have found the dyall basis set coefficients on the homepage http://dirac.chem.sdu.dk/basisarchives/dyall/ Does anyone in the forum has experience for using for example the "5p" basis set? I only ...
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Derivation of Yurke-Stoler state

The evolution of a coherent state $\vert\alpha\rangle$ in a Kerr-nonlinear medium $H=\Omega (a^\dagger a)^2$ is given by $$ \vert \psi(t) \rangle = e^{-\vert\alpha\vert^2/2}\sum_{n=0}^{\infty} \dfrac{\...
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1answer
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How can quantum operators be expressed as a matrices?

I have just started quantum mechanics with Shankar. In my understanding, quantum operators are linear operators in infinite-dimensional Hilbert spaces. Shankar has repeatedly treated quantum ...