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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How is energy defined in QM?
I have been introduced to QM this spring semester. One thing that I couldn't understand is that how do they define the energy of an electron.
For example while solving "Particle in a box" ...
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3
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Entangled particles and the Andromeda paradox experiment
I know there are other questions linking the two subjects. I am not asking about an explanation, rather I am curious whether an experiment would be possible.
To explain the experiment let's start with ...
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Divergence of the Berry connection
Given the Berry connection
\begin{equation}
\boldsymbol{\mathcal{A}}(\mathbf{R}) = i \langle u(\mathbf{R}) | \nabla_\mathbf{R} | u(\mathbf{R}) \rangle,
\end{equation}
the Berry curvature can be ...
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Showing the Variance of an observable in a determinate state is always zero
I am working through Introduction to Quantum Mechanics by David J. Griffiths, and part 3.2.2 shows that the standard deviation of an obervable, $Q$, is always $0$ but I do not understand the steps ...
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Entropy during spontaneous symmetry breaking
Spontaneous symmetry breaking (SSB) occurs when the state of a system possesses less symmetry than the underlying laws governing it. This raises the question: What happens to the entropy of the system ...
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Square Integrability of spherical symmetric wave
In my class we were discussing some wave equation for a spherical symmetric wave $u(t,r)$ and my professor investigated the behaviour of the solutions asymptotics $r \rightarrow \infty$. The solution ...
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Usage in the Clebsch-Gordan (CG) coefficients' recursive relation
In Sakurai's Modern Quantum Mechanics, its 3.8.39 is
\begin{aligned}\sqrt{(j\mp m)(j\pm m+1)}\langle j_1j_2;m_1m_2|j_1j_2;j,m\pm1 \rangle \\=\sqrt{(j_1\mp m_1+1)(j_1\pm m_1)}\langle j_1j_2;m_1\mp1,...
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Effects of measurement on the entropy in quantum mechanics
How does measurement in quantum mechanics (QM) impact the system's entropy? The measurement process in QM is considered time-irreversible. Are there principles akin to the second law of thermodynamics ...
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Is it possible to model unidirectional evolution of an atom-photon system using the Schrodinger equation?
The physical system I'm imagining is pretty simple: suppose there's an atom in free space with three states $|g_0\rangle, |g_1\rangle, |e\rangle$ initialized to some superposition of $|g_0\rangle$ and ...
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Lifting Projective representations [migrated]
D. J. Simms in his book "Lie groups and quantum mechanics" (page 9) says that: any representation of $\sigma$ of $\tilde{G}$ (the simplgy connected covering group of the Lie group) in $U(H)$ ...
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Question about impurity scattering of multiband Hamiltonians in $T$-matrix approximation
Imagine, we have a 2-by-2 Hamiltonian:
$$
H = H_0+H_{imp}.
$$
The impurity Hamiltonian $H_{imp}$ is
$$H_{imp}=u_{imp}I_2\sum_{r_{imp}}\delta(\vec r - \vec r_{imp}).$$
$I_2$ is identity matrix of order ...
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How to prove that the spin operator commutes with the position operator? [duplicate]
In the lecture notes on Quantum Mechanics I'm reading, the author claims that the position operator $\hat{q}$, the square spin operator $\hat{s}^2$ and the spin operator component $\hat{s}_0$ (in a ...
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Quantum oracle for boolean function
As a math student, I am doing some quantum computing. In the course notes of Ronald de Wolf, he says that any Boolean function $$f:\{0,1\}^n\to\{0,1\}^m$$ can be made into a unitary operation that ...
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Tightly-focusing a single photon
I'm curious about the feasibility, both theoretically and experimentally, of tightly focusing (or spatially trapping) a single photon to guarantee its precise targeting. If it’s possible, isn’t it ...
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Unusual example of Compton Scattering (+Four-momentum approach, +nonrelativistic) [closed]
An electron of kinetic energy $k=100 keV$ (first note, doesn't this mean that its energy is much lower than $0.511 MeV$, and thus that it is a nonrelativistic electron we are dealing with?), collides ...
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Lindblad evolution as continuum limit of a discrete process coupling system and environment
I have trouble understanding the derivation of the Lindblad evolution in terms of the time evolution under a Hamiltonian $H$ in a system-environment Hilbert space $H_S\otimes H_E$, where we trace out ...
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Rotational quantum states for atoms
It is well-established that molecules possess rotational and vibrational quantum states, due to molecular symmetries, in addition to electronic states. In contrast, it is generally accepted that atoms ...
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Positronium radius more than expected
I am trying to derive the basic properties of positronium. Usually it is stated that we can just set reduced mass of hydrogen atom as $\dfrac{1}{2}m_e$ and obtain:
\begin{equation}
r_n=\dfrac{8\pi\...
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Limit for big system size of Fokker-Planck eigenfunctions
I am learning how to use diagonalization methods applied to Fokker-Planck equations with Gardiner's book and these notes. The idea is to find the probability density, $ P[X_t\in[x,x+dx]]=\rho_t \, dx$,...
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Does the density matrix still holds for infinite subsytems?
Consider a maximally mixed state composed of $n$ equal qubits. Its density matrix is then:$$\hat{\rho}^{(n)}=\hat{\rho}^{\otimes n}.$$ The eignevalues of such a matrix, $\lambda$ will all be equal (...
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Relation between classical probability and quantum probability formulae
Assuming superposition state
$$
\Psi = C_1 \psi_1 + C_2 \psi_2
$$
,one can write the expectation value $\langle A \rangle$ of a physical magnitude A as follows
$$
\langle A\rangle = P_1 \langle A\...
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If $H$ anniliates a state, must $Q$ and $Q^\dagger$ also annihilate the state?
Suppose we have a a Hamiltonian, $H$. And suppose also we have some operator $Q$ such that $\{Q, Q^{\dagger}\} = H$, and $Q^2 = 0$.
If we find a state $|\psi \rangle$ such that $Q|\psi \rangle = Q^{\...
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Why are $j=0,\frac{1}{2},1,\frac{3}{2},\ldots$ representations of $SU(2)$ algebra irreducible? [migrated]
In quantum mechanics, we construct all the representations of the $SU(2)$ algebra $$[J_j,J_k]=i\varepsilon_{jk\ell}J_\ell$$ in the $|j,m\rangle$ basis. The allowed values of $j$ turn out to be $$j=0,\...
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Trying to learn input-output formalism: 2-photon Fock state entering a cavity
I really don't understand the input-output formalism. I've gone through the 1985 paper, and I get completely lost when the following formula is written:
$$b_{out}(t)-b_{in}(t) = \sqrt{\gamma} c(t)$$
...
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108
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Uncertainty principle for incompatible observables whose probability distributions lack well-defined moments
The Heisenberg uncertainty principle states that the product of standard deviations (or variances) for incompatible observables has a non-zero lower bound (with a zero lower bound reserved for ...
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The relative phase of photons in two-photon absorption
In two-photon absorption (TPA), the relative polarization of the two photons about to be absorbed simultaneously by an atom is crucial in determining the TPA rate. However, there is a lack of ...
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3
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Nature of a superposition of states: is it true or only theoretical?
For quantum mechanics, a certain property of a subatomic particle, e.g. the spin of an electron, which can be either up or down, is a "superposition of states," and one of the two conditions,...
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Can we create a space in which density of all electromagnetic fields is 0?
so i was just thinking that may be the reason of that we cannot measure a quantum particle position and velocity may be because continuously it's being hit by electromagnetic radiation and since we ...
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Can we get quasiprobability distributions other than $P,Q,W$ from generalised characteristic functions?
It's a standard result that the three well-known quasiprobability distributions can all be expressed in terms of the "$s$-ordered characteristic functions" as
$$
W(\alpha) = \int\frac{d^2\...
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Why does quantum random walk show ballistic transport?
For the topic of quantum random walk, a simple example is considering a 1D lattice fermionic model, expressed as
$H = \sum_{i} c_i^\dagger c_{i+1} + h.c.$
Assuming there is one fermion located at the $...
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Can we prevent the Higgs field interaction with particles? [closed]
if not why?
is it possible in future?
Because if it is possible there may be chance of more advance invention
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Planck's quantum explanation of black body radiators
Oscillators in a black body (electrons) can only have energy equal to $E = nhf$ ie it is a linear relationship. so if an electron drops from energy level $n$ to a lower energy (jumping 1, 2,3 ... ...
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Expectation Value Involving $s$-Wave Solutions to Central Potential
I previously posted a question regarding the expectation value described below, but it was closed because the question was not developed enough. Since I was given the option to delete it, I deleted it;...
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How can a QFT field act on particle states in Fock space?
Recently I asked a question that was considered a duplicate. However I felt that the related question didn't answer my doubts. After a bit of pondering I have realized the core of my discomfort with ...
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What does it mean for classical mechanics to be based on the category of sets?
It is quite common[1][2] in the study of physics in the context of category theory to say that one of the fundamental difference between classical mechanics and quantum mechanics is that classical ...
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The limit of time evolution operator
Through reading Nenciu's rigorous proof on the Adiabatic Theorem and Gell-Mann-Low Theorem, I found:
Since the limit $t_0\to-\infty$ does not exist for $U(t,t_0,\epsilon)$, in order to make use of ...
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Single Photon Interference and the Hong-Ou-Mandel (HOM) Effect in an Interferometer
Reading on the Hong-Ou-Mandel (HOM) effect, I came to wonder how exactly we can be certain that interference that occurs in apparatuses such as a Michelson interferometer and Mach-Zehnder ...
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Why are the angular momentum raising and lowering operator coefficients real?
I had a homework problem where I had to find the coefficients for the angular momentum raising and lowering operators. I know the answer is supposed to be $\sqrt{l(l+1)-m(m\pm1)}$. I have figured out ...
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How to interpret QFT fields (in relation with QM)? [duplicate]
In QM we deal with the Schrödinger equation:1
$$i\frac{\partial}{\partial t}\psi = H \psi$$
the wave function $\psi(x)$ is the main object of interest: it can be interpreted as a scalar field, in the ...
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Why we cannot transfer information using a spin entangled singlet? [duplicate]
In QM and QE effects an entangled particle pair is called also a singlet with some properties of the two particles like spin, non-locally correlated. However, there is no transfer of information ...
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Deducing the ground state from a known first excited state
I am studying Schrodinger equation with a potential of hyberpolic functions.
$$ H \psi = - \psi''(x) + \Big[1-\frac{12}{1+b \cosh{(2x)}} + \frac{15\,(1-b^2)}{[1+b \cosh{(2x)}]^2}\Big]\psi(x) $$
The ...
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Experimental implications of unitarity [duplicate]
I have two questions with regards to unitarity:
To which extent it has been verified experimentally that quantum systems evolve in a unitary way when dealing with unbounded Hamiltonians?
Let us ...
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5
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What is the distinction between a ket and a state in quantum mechanics?
Sorry if this has been asked before in some manner, but I'm just a bit confused about the distinction between a state $\alpha$ and its ket $|\alpha\rangle$. I was recently told that a state $\alpha$ ...
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1
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Probability distribution for the momentum of a quantum harmonic oscillator
I was wondering if anyone could point me towards the analytical solution for the probability distribution for the momentum of a quantum harmonic oscillator in the canonical ensemble. I've come across ...
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A quantum machine that works unitarily and doesn't produce heat
For decades since my physics master's I daydreamed about machines that work on a quantum level unitarily. The reason I found it interesting was because I knew that unitaries map pure states to pure ...
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Intuitive intermolecular explanation required for why same force applied farther from a hinge causes larger acceleration than if applied closer?
Intuitive intermolecular explanation required for why same force applied farther from a hinge causes larger angular acceleration than if applied closer? No math or equation required. This question is ...
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Are the SHO's ladder operators induced from a Lie group action?
Consider a quantum system with a hamiltonian $\hat{H}$, which is invariant under the action of a lie group $G$, meaning we have a unitary representation of $G$, $\hat{U}(g)$, in Hilbert space, and $\...
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Question about quantum mechanics wave equation deduced by mechanics-optics analogy [closed]
I have a question about quantum mechanics wave equation deduced by the analogy between eikonal equation and fermat least principle (in optics) with, respectively, hamilton jacobi equation and ...
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1
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What will be wave function after application of operator?
In the mathematical treatment of quantum mechanics, we have a wave function ($ψ$) that helps us to know the different information (like position, velocity, energy, etc.). To measure such a quantity we ...
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A theoretical issue in the mathematical description of the Aharonov-Bohm experiment
Mathematically viewing, the Aharonov-Bohm experiment shows that the magnetic field creates a connection with a nonzero holonomy on a multiply-connected domain. This means that there isn't a state ...