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 do you account for all the photons and plethora of quantum particles in the box between the double slits and the back wall?
Does the particle being shot not interact with the all the particles that must be consuming the space in the box before the back wall? How do we know it’s the same photon traveling the distance to ...
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Invariant nature of mass and particle annihilation [closed]
Since mass is a Lorentz invariant, it can never change to preserve the vectorial nature of the four-momentum and the other four vectors. Thus the only interpretation of the energy-mass equation that I ...
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Why does the minimum energy of motion in a Coulomb field differ from the theoretical value?
I would like to find the minimum energy of Coulomb potential motion using matrix method.
$H=-\frac{1}{2}\Delta-\frac{1}{r}$
I have chosen Slater Type Orbitals as a basis functions $R(r)=Nr^{n-1}e^{-r}$...
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The probability density function $|Ψ|²$ [duplicate]
Max Born stated that $|Ψ|²$ is the probability density of a particle, given its wave function to be $Ψ$. But why is this? Where does this come from?
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Correlation function and Green's function
I have referred to several materials on Green's function but I found those notions pretty confusing. Now what I have tried to do is to calculate the inverse matrix of $G^+(E)=E-H+\mathrm{i}\eta$ where ...
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Relationship between $S$ matrix elements and $T$ matrix elements
In hadron physics, we define the $S$ matrix as $S=I+iT$, where $T$ is transition matrix. But on the question: Finding relation between matrix $S$ and matrix $M$ for wave propagation, it gave an ...
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References on the difference between Dirac's and Von Neumann's approach to Quantum Mechanics (rigged Hilbert Space vs Hilbert Space only)
I have not found any clear and comparative explanation between the Dirac and von Neumann versions of Quantum Mechanics (rigged Hilbert Space vs Hilbert Space only). I have found some short articles. ...
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Is the Hamiltonian some sort of connection/gauge field?
I'm not sure if this is a well-defined question, but I was just looking through some old notes and noticed that the Hamiltonian in usual QM has a similar transformation as gauge fields in QFT: under ...
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Analytic continuation of the many-body spectral density
For an observable $A$, define the real-time autocorrelation function
$$
C(t) = \langle A A(t) \rangle_{\beta} = \dfrac{1}{Z} \mathrm{Tr}\left[ e^{-\beta H} A e^{i H t} A e^{-i H t}\right],
$$
with $Z =...
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Expected Kinetic Energy in Exponentially decaying potential
This is a question that I don't know how to solve, because I keep getting negative kinetic energy.
Suppose there is a quantum mechanical system in d-dimensions that has the following Hamiltonian, ...
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Is the operator $P=-i\hbar\frac{d}{dx}$ self-adjoint given the Hilbert space of the problem of particle in a box?
The operator $P=-i\hbar\frac{d}{dx}$, is a symmetric operator in the domain $$D(P)=\left\{f(x) \big|f\in L_2[0, a], f(0)=f(a)=0\right\}$$ i.e. the domain is the subspace of square-integrable functions ...
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Experimental ways of studying the structure of atomic orbitals
As far as I understand, the notion of atomic orbital was introduced by Bohr in 1913 in terms of his semi-classical model of atom. Bohr assumed that each electron in the atom has a certain energy and ...
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Microstates in a quantum gas and the occupation number
The occupation number of a gas of fermions or bosons is typically defined as:
$$\bar{n} = \sum_n n\ p_n$$
where n is the number of particles in each state.
But $n$ is just a number of fermions (or ...
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Density of states of Fermi gas derivation
I'm going over this book. While deriving the gensity of states for a gas of fermions the author makes the following argument:
Remember that we are treating the
gas as having a set of states that can ...
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How to derive bound and unbound states for an absolute value potential?
How do you find for what range of energies the absolute value potential has bound and unbound states?
What I have understood from my previous Intro to Quantum lectures are that in order to derive the ...
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Can we call it "quantization" when we specify Hilbert space and operators to write a classical field theory into a quantum theory?
Can we call it quantization when we specify Hilbert space and operators to write a classical field theory into a quantum theory? Suppose there is a single spin 1/2 system with Hamiltonian $\hat{H}=\...
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Unclear passage, autoket and eigenvalues
I am not understanding a passage that our professor wrote: those are the lines.
$e_0 \cdot \hat\sigma$ is an operator, whose eigenvalues are $\pm 1$. He applied this to a ket $|e_0, \pm 1 \rangle$:
$$...
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Does any experimental data demonstrate radio frequency radiation or its absence in Stern Gerlach experiments?
I asked a question on this site about whether there is radio frequency radiation from particles passing through Stern Gerlach devices due to changes of spin state. One contributor predicted radio ...
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Difficulty solving conformal-bootstrap-like crossing equations using semidefinite-programming (SDP) via SDPB software
My question involves semidefinite programming (SDP) in the sense of attempting to find some vector $\alpha^{\mu}$ that satisfies the following conditions:
Normalisation: $\alpha^{\mu}n_{\mu} = 1$
...
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How the braking radiation fit into the photon picture of light?
The continuous part of the x ray spectrum is due the deceleration of electrons. I know that a decelerating charged particle emits a braking radiation according the EM theory. However, what's in the ...
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$X$ and $P$ representation
I wonder why the wave function of quantum particle can be represented either in the position space, $(x,t)\mapsto\Phi (x,t)$, or in the momentum space, $(p,t)\mapsto \Psi (p,t)$.
For a classic ...
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Question regarding interaction picture [closed]
$$\newcommand{ket}[1]{\left|#1\right>}$$The relation between the interaction picture and Schrödinger picture states and operators are
$$\ket{\psi(t)}_{I} = e^{iH_0t} \ket{\psi(t)}_S$$
$$\mathcal{O}...
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Double-slit experiment: How do we know the particle effect comes from the nature of light rather than its interaction with the detector?
In the double-slit experiment, we shine a light wave through two closely-spaced parallel slits at a screen, and observe an interference pattern on the screen. We then reduce the intensity of the light ...
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Formation of a hydrogen atom from delocalized electron and proton
From the initial state of an electron and a proton in a box. I would like to find a reasonable hamiltonian, or way to describe the interaction that leads to the formation of a Hydrogen atom.
Here is ...
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Can we study the angular momentum of a plane wave scalar particle?
In a plane wave we are in a $p_z$ eigenstate. I saw that $[L_z,p_z]=0$, but $[L^2,p_z] \ne 0$. Is it enough to say that the particle has a defined angular momentum along $z$?
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Calculation of phase error
In papers on quantum sensing, I often see the following formula:
$\delta \phi=\frac{\Delta M}{|\partial \langle M \rangle /\partial \phi|}$
(for example in this paper: Quantum-Enhanced Measurements:...
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It is possible to express the convolution between two quantum functions $\psi_{1}(x)$ and $\psi_{2}(x)$ in terms of a inner product?
The question of the title is due to the following methodology. Let us consider two arbitray quantum functions $\psi_{1}(x)$ and $\psi_{2}(x)$, such that the convolution between them is
$$\left\lbrace \...
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If the particle moves with group wave, what $\lambda$ in De Broglie equation should we use?
according to De Broglie equation
\begin{gather} p=\frac{h}{\lambda} \end{gather}
and knowing also that a particle moves with the group velocity not the phase velocity, indicates that has a range of $\...
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P-n junction diode and tunnelling current [duplicate]
In reverse bias p-n junction diode, why the tunnelling current cannot be explained by accounting the particular electrons which have enegy higher than the barrier's energy?
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Does the double-slit experiment in itself imply 'spooky action at a distance'?
Silly question, but when sending a single electron at a time through a double-slit and observing the interference pattern over time ... how does the single electron that popped up ('measured') at a ...
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"Stationary" vs. moving wave packet
I am working through a quantum mechanics problem involving the time evolution of a free particle (the particle is a proton) given that the initial state is a Gaussian wave packet of the form:
$$
\psi(...
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Reverse bias p-n junction diode and Kinetic energy
In reverse bias p-n junction diode, which energy is lower than the energy of the potential barrier: the average kinetic energy of the would-be-tunnelling electrons or the kinetic energy of the ...
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Eigenstates of time dependent Hamiltonians
I am trying to figure out how to make sense of a time dependent Hamiltonian. In the Schrödinger picture, the one dimensional Hamiltonian is written:
$$\hat{H} = -\frac{\hbar^2}{2m}\frac{\partial^2}{\...
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Are all eigenfunctions of the Hamiltonian Operator stationary waves?
I had a quick question regarding eigenfunctions, wave functions, and the TISE.
To put it frankly, are all eigenfunctions of the Hamiltonian operator stationary waves? And thus, are wave functions ...
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What is the interpretation of this integral equation $\psi(x,t_2)=\int G(x,y)\psi(y,t_1)dy$ of the generalized Huygens principle?
I have came recently across the following equation
$$\psi(x,t_2)=\int G(x,y)\psi(y,t_1)dy$$
I want to understand its interpretation. Here is what i understand, I see that this equation gives the form ...
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If electromagnetic waves do not need any medium, then why communication under water is hard or why light does not pass through wall?
If electromagnetic waves do not need any medium, then why communication under water is hard, or why light does not pass through wall?
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Is there a way to convert a CNOT into a single Hilbert space unitary?
Imagine the following CNOT gate:
Knowing that the secondary system state is fixed to $|0\rangle \langle0$|, I have the feeling that it can be converted into the following single Hilbert space unitary:...
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Is there a way to fix the ordering ambiguity of canonical quantization?
This question arose from my question on whether the vacuum energy is actually present for a free quantum scalar field
What is the right way to treat the vacuum energy?
Part of this discussion is that ...
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Probability inequality for Quantum Approximate Optimization Algorithm (QAOA)
In arXiv:2207.14734 the authors claim that it is "straightforward to show that" their equation 8 holds:
$$\mathrm{Pr}_{x\sim q}[x:f(x)\geq \mu] \geq \frac{1}{M}$$
where we have an objective ...
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Do gravitons undergo Shapiro delay?
Light undergoes Shapiro delay, and light travels along a null geodesic. Do gravitons also travel along a null geodesic? Do gravitons undergo Shapiro delay? If so, then how cold do they need to be in ...
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When/where (if at all) is radiation emitted in Stern-Gerlach sequential experiments?
Assuming an experimental setup like experiment 2 on this Wikipedia page:
https://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment#Experiment_2
100% of fermions entering the second apparatus are ...
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Erratum in Schlosshauer's "Decoherence" textbook (on the outcome of a SG experiment)?
In his textbook on Decoherence, Schlosshauer is trying to explicate the difference between a superposition and a classical ensemble. Considering a two-level system and using fairly standard notation, ...
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How to construct a POVM to discriminate between 3 non-orthogonal states?
In Nielsen and Chuang's Quantum Computation and Quantum Information textbook, there is an example (on page 92) of a POVM containing elements:
$E_1=\frac{\sqrt{2}}{1+\sqrt{2}}|1\rangle \langle1|$,
$E_2=...
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Physics prerequisites for quantum mechanics and general relativity [closed]
I’m a mathematician with a specialization involving differential equations and probability theory, and I’ve also spent a lot of time studying topics like differential geometry, algebraic topology, and ...
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What is the physical meaning of acting an operator on an arbitrary quantum state? [duplicate]
Let |+> and |-> are the two eigenstates of Pauli-X.
If an incoming arbitrary state is |psi> = c1|+> + c2|-> and we act on this with pauli-X,
then the outcoming state is a different ...
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Identical Particles Combinatorial approach Wave function (anti)symmetrization
We are given five idential spin-half particles in Linear Isotropic
Harmonic Oscillator potential. It is required to find the degeneracy
of the ground state of the system.
I understood the approach ...
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Giving arguments why indistinguishability is removed for high $T$ or low occupation nr. of an energy level (ideal quantum gas)
The average occupation nr. of an energy level in a bosonic/fermonic gas is:
$$\langle n_i \rangle=\frac{1}{e^{\beta(\epsilon_i-\mu)}\pm 1}.$$
If we make the assumption that we have a quantum gas of ...
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Rotations and angular momentum
Cohen tannoudji. Vol 1.pg 702
"Now, let us consider an infinitesimal rotation $\mathscr{R}_{\mathbf{e}_z}(\mathrm{~d} \alpha)$ about the $O z$ axis. Since the group law is conserved for ...
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Relating the Lindbladian to Kraus operator: why do we assume the specific Kraus: $K_0=I+L_0 dt$ and $K_{\alpha \neq 0}=\sqrt{dt} L_{\alpha}$
In open quantum systems, for a Markovian evolution, we can derive a Lindblad form for the evolution.
There is a way to relate this Lindblad form to the Kraus decomposition of the quantum map ...
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When hydrogen's energy level is changed I don't know where photon has gone
Before answering my question, I am student of KSA of KAIST (Korea Science Academy). Therefore I have little bit wrong conception or supposition.
If we want to change only one hydrogen's energy level, ...
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