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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Quantum spin and violations of special relativity

This past semester I finished the second year of my physics B.Sc. program with a course that covered quantum mechanics (wave formulation) up to the Schrödinger hydrogen atom. The accompanying textbook ...
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$LS$ vs $JJ$ coupling hamiltonian

While studying $LS$ coupling, I came across the fact that it is only applicable for lighter atoms, and the perturbation term in the hamiltonian is given as: $H_2 = A_{ij} \vec{L}.\vec{S}$, where $A_{...
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Confusion regarding addition of spin and $LS$ coupling

I've recently studied the addition of angular momenta in quantum mechanics, and faced a massive confusion during the addition of spin, in a two-electron system. When adding the two spin-1/2 electrons, ...
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The logic behind measuring the whole entangled spins?

Does the same logic for single particle spin apply to the maximally entangled state $|\Psi\rangle=\frac{|ud\rangle-|du\rangle}{\sqrt{2}}$ as well when it comes to $\sigma_z^A\sigma_z^B$ observable and ...
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What system has a spectral series in the radio spectrum?

As an electron energy level has an emission level depending upon the excitation, is it possible for a system such as a Rydberg atom to have emissions in the low side of the frequency range?
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Evolution operator of a $N$-particle system

Consider a $N$-particle system with a nearest-neighbour interaction of the form $$H = - \hbar g(t) \sum_i A_i B_{i+1}$$ where $A_i, B_i$ are an operators acting on the $i$-th particle with $[A_i, B_j] ...
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Commutation of Hamiltonian [closed]

Commutator with exponential $[A, \exp(B)]$ I am trying to apply this to my problem. My problem is to find commutation for $[b^\dagger a+a^\dagger b , e^{\alpha a^\dagger−\alpha^\ast a}]$. Should i ...
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Why the amplitude in the third region is decreasing as compare to that of first region? Please explain its physical significance

Consider a finite potential barrier and the case where the energy of the particle is lower than that of finite potential. The figure is shown and we have three regions. As I solved the Schrödinger ...
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Commutation of Hamiltonian $a^{\dagger}b +b^{\dagger}a$ with Displacement operator $\alpha a^{\dagger} -\alpha^*a$? [closed]

I have a Hamiltonian $a^{\dagger}b +b^{\dagger}a$. Does it commute with $\alpha a^{\dagger} -\alpha^*a$. Here $a$ and $a^{\dagger}$ are bosonic creation and lowering operator. This is my calculation $$...
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Position operator in momentum space generator

Position operator in momentum space generator How to get the position operator in the momentum representation from knowing the momentum operator in the position representation? derived the position ...
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Variational principle quantum mechanics: trial wave function is not expressed by basis of eigenfunctions

Im learning about the variation method for solving quantum mechanics problems. The principle is that eigenfunctions minimize the expectation value of the Hamiltonian on that state (provided $\Psi$ is ...
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Is there an interpretation of a momentum-space propagator in terms of a momentum eigenstate?

I know that a position space propagator is just a matrix element between two position eigenstates: $\langle x', t' | x, t \rangle$. Or we could just write this as $\langle x' | x \rangle$ where the ...
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Moore's function

In the dynamical Casimir effect, the Casimir force is given in terms of Moore's function R which satisfies $$R(t+L(t))-R(t-L(t))=2$$ where $L(t)$ is the trajectory of a moving mirror (while another ...
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Transfer of energy from radiated to conducted

Can a CO$_2$ molecule in the atmosphere that has been heated by the earth's radiation, transfer that energy to one of the many O$_2$ or N$_2$ molecules nearby? If so, what is the mechanism?
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Invariant quantity from commutators

I am reading this paper on conformal quantum mechanics by De Alfaro, Fubini, and Furlan. There, they find the algebra of the generators of $(0+1)$-D conformal transformations (Eq. 2.23) $$ [H,D] = iH\;...
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Operators with continuous eigenbasis

I came across the following relation, regarding commutators : if $[\hat{a},\hat{b}] = k$, then we can write the following, $\hat{a} = k\frac{\partial }{\partial \hat{b}}$ and $\hat{b} = -k\frac{\...
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Deriving the angular momentum operator using passive transformations

So I'm following the solutions here. For exercise 12.2.2, I'm not quite sure why the commutator relations aren't $$[X,L_z]=i\hbar\frac{\partial L_z}{\partial p_x}-L_zi\hbar\frac{\partial}{\partial p_x}...
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When is the expectation value of momentum equal zero in $3$ dimensions?

For all one dimensional bound states (be it relativistic or non-relativistic), user Gonenc has explained clearly why $\langle p \rangle =0$. However, I want to know whether $\langle p \rangle=0$ is ...
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Coherent state and continuous basis of Harmonic oscillator

While deriving the coherent states of the harmonic oscillator, I used a heuristic argument : Since $\hat{a}$ and $\hat{a^\dagger}$ form a continuous basis, and $[\hat{a},\hat{a^\dagger}] = 1$, we can ...
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1answer
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When is expectation value of momentum $0$

When is the expectation value of momentum for a quantum mechanical wavefunction $0$ ? One of the possible cases, that I think of, is when the wavefunction is symmetric. In that case, the particle is ...
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Why do helium balloons deflate?

Helium balloons deflate–even when they are made from metal foil. How does this happen exactly? Is there any chance that quantum tunneling plays a role here? The thought is that the helium atoms inside ...
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Trajectory of an asteroid hit by a micro black hole? [closed]

I am not asking about how long it would take for the black hole to acquire the mass of the asteroid. I am specifically asking only about the trajectory of the asteroid (or the merged object), how that ...
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Difficulties in proving the area-law conjecture in higher dimensions

A very famous and important open conjecture in condensed matter physics is the area law of entanglement entropy, which claims that in a locally-interacting quantum many body system, if the ground ...
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A conjecture on energy distribution of product states in locally interacting systems

Let $\hat{H}$ be a locally-interacting quantum many body Hamiltonian, for example the nearest-neighbor interacting quantum Heisenberg model or Hubbard model, and let $|\psi \rangle$ be an arbitrary ...
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Question about Sakurai's $SO(4)$ symmetry section

In Sakurai's Quantum mechanics book, he says the hydrogen atom has $SO(4)$ symmetry by explicitly exhibiting operators $I_i,K_i$ that satisfy the commutation relation of the Lie algebra $so(4)$. ...
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Does the observer effect mean there is free will? [closed]

If it is true that the observer effect exists, then does it follow that there is a concept of "free will" for human beings?
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Invariance of commutator relations under change of basis

Consider the following: Let for operators $\hat A$ and $\hat B$ the following commutation relation holds: $$[\hat A,\hat B]=\hat C \tag{1}$$ and now we know that this relation holds, $$[\hat A',\hat B'...
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Which other physical quantities, other than the polarization of classical light be used as a qubit?

This article mentions that polarization of classical light can be used as a qubit. link Are there other physical quantities in the classical world which can be act as qubit.
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If all time simply is, i.e. “not flows”, why do we equate physical existence with the speed (time) of light (E=mc2)? [closed]

Could something like gravity (since no time) help us understand reality better? If location was not measured in speed-of-light-dependent terms, wouldn't "spooky action at a distance" ...
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Variation of a time-ordered exponential

Consider the time-ordered exponential (Wilson line): $$ U(t_{f},t_{i}) = \mathcal{T}\text{exp}\left(-i\int_{t_{i}}^{t_{f}}\mathcal{A}(t)dt\right)\tag{1} $$ Where $\mathcal{A}(t)$ is some matrix-valued ...
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Experimental electron configurations of copper and chromium and Aufbau principle

I was reading the Aufbau principle article on Wikipedia Aufbau principle after seeing so many student questions on Chem SE asking about writing electron configurations. Every general chemistry student ...
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Could wavefunction values be discrete?

According to standard quantum mechanics, Hilbert space is defined over the complex numbers, and amplitudes in a superposition can take on values with arbitrarily small magnitude. This probably does ...
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Does the many worlds interpretation of quantum mechanics mean in some worlds dice always land on the numbers you guessed?

I suspect this is a common misconception of the Many Worlds interpretation of quantum mechanics. I'd like to preface this by paraphrasing Hawking's book A Briefer History of Time: He refers to ...
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A flashlight shines on a wall. Emmission or Reflection?

A flashlight shines on a wall. As an observer, I can see the illuminated area from different positions in the room. How do the rays of light come about that get into my eye? For my understanding it ...
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Does the wave functions are in Hilbert Space? [closed]

From MWI worlds exist in same time and each new situation creates new world. So basically, my question is does this all infinite worlds points in Hilbert Space?
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Do all of the energy levels listed in Hydrogen Spectral Series assumed to be s ('sharp') sublevels?

When reading about the hydrogen spectral series (Lyman, Balmer, etc.) I noticed that nowhere are suborbital or sublevels (azimuthal) mentioned; only principal quantum numbers n .... Are there any p or ...
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Hamiltonian in terms of angular momentum [closed]

How to write the Hamiltonian \begin{align} H = ω(a^†a)+ω_0(b^†b+(b^†)^2(b^2)+g(a^†b + b^†a)) \end{align} in terms of angular momentum \begin{align} J_+ &= a^†b, \\ J_− &= ab^†, \\ J_z &= \...
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How to convert a ket vector into a bra vector? [closed]

Suppose I have been given a ket vector $$|i\rangle = \frac{1}{\sqrt{2}}|u\rangle + \frac{i}{\sqrt{2}}|d\rangle$$ and I want to find the corresponding bra vector. How can I do that? My attempt : My ...
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Evolution using interaction time

I was reading a paper called "Generation of two mode entangled coherent state". The paper is attached here. In that book I have a $N(|α\rangle±|-α\rangle)⊗|0\rangle$ in equation 9. How did ...
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What do these straight lines in the graph mean?

Does it mean thermal equilibrium? If the answer is yes, then how do we get to the molecular dissociation stage when we have already reached thermal equilibrium in vibrational mode? Because equilibrium ...
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If product of variance of two operators, $\Delta A \Delta B \geq 0$ then prove that operators $A$ and $B$ commute

I am trying to prove that if product of variance of two operators $A$ and $B$ is zero, then simultaneous accurate measurement of two operators is possible. My approach:- If I prove that if variance ...
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1answer
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Recovering the position-space probability density from QFT

Lately I've been trying to wrap my head around the relationship between quantum fields and the wave function of non-relativistic quantum mechanics. It is well-known that QFT, through its demotion of ...
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Question about spin states in Susskind's Quantum Mechanics

This is my first question on this site so please improve the question if needed. I am a beginner in Quantum Mechanics and so to study about it, I purchased the book "The Theoretical Minimum : ...
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Singularity and charts-problems met in Dirac quantization condition

Context: 45'23'' in a lecture given by Professor Wu, https://www.koushare.com/video/videodetail/4619. Consider a vector field $\vec{A}(\vec{x})$, with $\nabla\times\vec{A}(\vec{x})=\vec{B}(\vec{x})=g\...
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Can incompatible observables share an eigenvector?

I was recently introduced to the concept of compatible and incompatible observables and specifically the generalized uncertainty principle, which is written in my textbook as: $$ \sigma_A^2\sigma_B^2 \...
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How do you adapt the saddle-point integration where the amplitude function has a phase shift?

If anyone can help me with this, I'd be very appreciative. I have tried researching online, but I haven't been able to find many articles/textbooks which are helpful. I am trying to do an integral of ...
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1answer
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Can you identify Hong-Ou-Mandel (HOM) interference using the $E$-field operator?

The fastest way of showing the HOM interference effect is through the following procedure: $|1, 1\rangle = a^\dagger b^\dagger|0,0\rangle = \frac{1}{2}(c^\dagger+d^\dagger) (c^\dagger-d^\dagger)|0,0\...
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Electron configurations beyond hydrogen

In 1990, the Journal of Chemical Education (American Chemical Society) published a little bit controversial article titled "The nature of the chemical bond—1990: There are no such things as ...
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Underlying Hilbert space of Kitaev's exactly solvable models

In Kitaevs's paper (Anyons in an exactly solved model and beyond) section 2.1-2.2, he seems to be extending the Hilbert space of a multi-spin system using Majorana operators. More specifically, if ...

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