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 to explain the minus (negative) sign before Kinetic energy operator in Hamiltonian operator?
The Hamiltonian operator is normally written in this form:
\begin{align}
\large
H_\mathrm{operator}
= \ &
\large
\frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2}
&+ \quad &
\large V(x)
\\...
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Dispersion Relation in Spin Chains in Terms of Quantum Speed Limits
Going off the dispersion relation derived by He and Guo
$$E_k = \sqrt{\left(\frac{J}{2}\right)^2 + h^2 + Jh\cos k}$$
Where $J$ is the nearest neighbour interaction strength and $h$ is the external ...
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Why do correlation amplitudes tend to zero as time increases?
I'm reading a section about correlation amplitudes in Sakurai. We are considering the correlation amplitude $C(t)$ of state $|\alpha \rangle$ which is in a superposition of energy eigenstates $\{|a'\...
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Reference request: Introductory books on Quantum Logic
I'm looking for introductory materials that presents the logic of quantum mechanics originally developed by von Neumann and Birkhoff suited to someone who took a full introductory undergraduate course ...
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What happens if we shoot a single photon to a mirror?
I understand mirrors absorb a small energy portion of the light and reflect most of it. What happens if we shoot a single photon to a mirror? Would it be reflected?
If the answer is Yes, then I would ...
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Does an electron have multiple positions BEFORE being observed?
I realize that an electron can only ever be detected at a single location in space when it is observed. That is, post-detection, an electron can only ever be at a single position in space. But prior ...
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Canonical momentum not observable vs energy is observable
I have seen explanations that canonical momentum for charged particles $p = mv + qA/c$ is not a measurable quantity/observable because it is not gauge invariant. However, there are many quantities ...
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Bose-Einstein Distribution function
Why is the following equation :
$\bar n_\epsilon=\frac 1 {e^{\beta (\epsilon_p - \mu) } \pm 1}$
called a distribution function?
In wikipedia the definition of a distribution function (a.k.a Cumulative ...
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Confusion regarding the average occupation number for a Boson/Fermion
Regarding the average occupation number for a Bose/Fermi gas we have:
$$\bar n_\epsilon=\frac 1 {e^{\beta(\epsilon_p - \mu)} \pm 1}$$.
Now the problem I am having has to do with the nomenclature of ...
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How does a particle passing an atom perturb the atom's QM system?
Consider the quantum mechanical system of an atom with Hamiltonian $\hat{H}_0$ and assume that we know the solutions of the eigenvalue problem $\hat{H}_0|n\rangle=E_n|n\rangle$ for $n=0,1,2,\ldots$. ...
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Odd function (probability density function) describing the measurable quantity of a quantum particle [closed]
This question is in reference to Quantum Physics of Atoms, Molecue, Solids, Nuclei and particles by Robert Eisberg and Robery Resnick.
The setup for an infinite square quantum well with the potential ...
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Meaning of different phases in two-state system
Assuming we have the state of some particle (maybe in this case an electron).
Is there an intuitive explanation what the difference between
$|\psi\rangle = a|0\rangle+b|1\rangle$ and $|\psi\rangle = a|...
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The Dirac Delta function for a fraction
I was studying quantum theory and came to know that the unrealistic function representing the position of a particle can be represented in the form of an infinitely peaked Gaussian or simply the Dirac ...
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Path integral formulation in Twistor theory?
Sir R. Penrose in his article (https://doi.org/10.1007/BF00668831) has shown that there are close similarities in various aspects of twistor theory and quantum mechanics. In twistor theory, one can ...
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On existence of orthonormal basis for each subsystem in Separable state
A separable state in $\mathcal{H}_{a}\otimes\mathcal{H}_{b}$ is given by
$$\rho_{s}=\sum_{\alpha,\beta}p(\alpha,\beta)|\alpha\rangle\!\langle\alpha|\otimes|\beta\rangle\!\langle\beta|.$$
Now, my ...
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Parity transformation in quantum mechanics
It was written in a book that if parity commutes with Hamiltonian and for some operator $\hat O$ if $P\hat O P^{-1} = -\hat O$ then $\langle\hat O\rangle = 0$.
I know how to show $\langle\hat O\rangle ...
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what does the photoelectric effect really demonstrate?
I'm an engineering graduate revisiting my text on semiconductor physics, and I've hit a snag.
My book claims that demonstrations of the photoelectric effect clearly show that "...light energy is ...
user306661
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On Dirac notation: orthogonal projector
Considering the following operator, in the position representation:
$$P:= \int _a ^ b dx | x \rangle \langle x|$$
It's an orthogonal projector.
Let's examine its action on the wave function $\psi$:
$(...
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Can a complex wave function just be seen as two real functions describing a particle and antiparticle state? [duplicate]
Let's assume electromagnetism. There are two charges. The wave function is complex but can be seen canonically as a vector in $\mathbb{R}^2$. Can we see one of the components as the electron and the ...
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Difference between uncertainty relations
I have come across two different uncertainty relations for position and momentum, namely
$$\Delta x\Delta p\geq\frac{\hbar}{2}$$ and $$\Delta x\Delta p\geq h.$$ These two are clearly different and ...
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Superposition of potentials
I'm trying to understand what superposition of potentials means.
For example, let be $$V_0(x) = \begin{cases} 0 &x \in [0,2a]\\ +\infty & \text{otherwise}\end{cases}$$ and $$V_1(x)=-\lambda\...
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What is the minimal time to move from the left to the right in a double potential well?
Consider a particle in a double potential well with Hamiltonian $\hat{H}$ and two basis states $|l\rangle, |r\rangle$ which correspond to the particle being maximally localized in the left resp. right ...
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On Dirac notation: inner product vs base representation
The Dirac notation $\langle a | b \rangle$ seems somewhat ambiguous.
On one hand, it can be seen as inner product of elements $a(x)$ and $b(x)$ of the Hilbert space $\scr H$, namely: $$\langle a | b \...
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Slater Determinant in Python or Mathematica [closed]
Is there any way, maybe a library or package, to calculate Slater determinant in Python or Mathematica? What I want basically is this: I would define a single particle wavefunction and specify the ...
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Is entropy a measure of how entangled a system is?
I recently learned about the Entropy of entanglement and now asking myself, what is the difference between the classical entropy and the entropy of entanglement? Is there even any or is entropy really ...
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Can a single QFT field configuration be written as a set of spinor/vector valued functions?
If a QFT state is a probability distribution over field configurations, then a single field configuration must correspond to an observable situation, right? As far as I can gather, each field ...
- 2,423
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Matrix mechanics tensor products in Fock space
In quantum mechanics, bra-ket notation is often used to represent the state vectors of the system. It is also possible to write these state vectors as "actual" vectors, for example if the ...
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Constructing orthonormal bases in sub-Hilbert spaces of $L^2(\mathbb R)$ defined for particular spatial regions?
I've explored the entanglement of modes by expanding the ground-state solution of a many-body problem as an infinite sum of Slater determinants of one-particle Hermite functions. The one-particle ...
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Is time the collapse of quantum superposition across the universe?
Just to preface, I admit I'm not a physicist, or even well-read on science-related things (though I do read articles from various aggregate sites and sometimes purchase New Scientist) but I am ...
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On Born's rule and Cauchy-Schwarz inequality
Citing Born's rule:
If an observable corresponding to a self-adjoint operator ${\textstyle A}$ with discrete spectrum is measured in a system with normalized wave function ${\textstyle |\psi \rangle }...
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How to use the translation operator in order to find the eigenstates in a perturbed QM system?
Given a quantum mechanical system with Hamiltonian $\hat{H_0}$, introduce a perturbation $\lambda \hat{H_1}$ with $\lambda$ sufficiently small. Define now the spacial translation operator to be $\hat{...
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How can neutral elementary particles have spin magnetic moment? [duplicate]
The spin magnetic moment of a particle is proportional to q/2m, where q and m are charge and mass of the particle respectively. So if an elementary particle is neutral (like neutrinos or photons), ...
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More Rigorous Laughlin pumping argument
I'm having a hard time understanding how the concept of spectral flow helps us compute the hall conductivity, and in particular, Laughlin's pumping argument. I think that this question summarizes the ...
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Inner product linearity on Dirac notation
I was starting to learn Dirac notation with MIT's notes on QM. The introduction states that Dirac notation starts from turning inner products from:
$$ \langle{u}, v\rangle $$
to, substituting the ...
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Find eigenvectors/eigenvalues for this density matrix [closed]
Consider the following density matrix defined on two coherent states
$$\rho = a\vert\alpha\rangle\langle\alpha\vert + b\vert\alpha\rangle\langle\beta\vert + + b^*\vert\beta\rangle\langle\alpha\vert + ...
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Is application of a Unitary operator equivalent to a phase change of a wave function?
From my understanding, the application of a unitary operator does not change the physics of the system. It is just a more convenient representation in a new eigenbasis.
Now, I have also read today ...
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If in the quantum world reality is just based on observation then why molecules (with electrons and protons) are real?
If in the quantum world reality is just based on observation then why atoms and molecules are real?
I mean, it is said that when a quantum particle is not observed it's neither spinning up nor down (...
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The Heisenberg model using the duality analysis
I would like to express the Heisenberg model using the duality analysis. It is shown here how to express the Ising model using Pauli matrices but I cannot get the relation $ \sigma _{i}^{z}= \prod_{...
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Derivation of $Z = \operatorname{Tr}e^{-\beta H + \mu N}$ [closed]
I've never studied quantum statistical mechanics myself, but I've read that the partition function of a quantum system in the canonical ensemble is given by:
$$Z = \operatorname{Tr}e^{-\beta H}$$
...
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Why is it selectively OK to rely on intuitive analogues when solving problems in Quantum Mechanics, such as the step potential problem?
I was looking at solved example (3.13) in the Schaum's Series book on QM by Yoav Peleg et al (2nd edn), where they solve for a step potential where a high energy particle is coming from the left and ...
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"Conservation of energy, or lack thereof," in quantum mechanics
In answering another question(1) on this site, I started to consider the conservation of energy in Quantum Mechanics. Doing some research, I came across this recent paper.(2) The abstract of the paper ...
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votes
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Forbidden translation generators due to a finite radius of convergence of $\psi(x)$
In Quantum Mechanics, a translation by a distance $a$ is given by $\hat T(a)$-
$$\hat T(a) \psi(x)=\psi(x-a)$$
which shifts the graph towards the right by a distance $a$.
When we want to find the ...
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How to choose Clebsch-Gordan coefficients?
I just started learning Clebsch-Gordan coefficients recently. I want to use the expression on Wikipedia (relation to Wigner $3j$ symbols):
$$
\langle j_{1},m_{1},j_{2},m_{2}|J,M\rangle=(-1)^{-j_{1}+j_{...
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What "happens" to the components of a composite quantum system?
Consider several separate systems described by their respective Hilbert spaces $\mathscr H_i$, a composite system $\mathscr{H=\bigotimes_i H_i}$ described by the tensor product of those Hilbert spaces,...
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Why is the probability that one state $|i\rangle$ ends up in the state $|f\rangle$ given by $|\langle i|f\rangle|^2$? [duplicate]
I've come across this relation numerous times, textbooks use it as if it is obvious. But I have never come across a proof or an intuitive explanation about why is it true.
It would be helpful if ...
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Where do the group and phase velocity terms come from in the wave function of the propagation of a dispersive wave packet?
I am reading Zettili's QM concepts and applications (second edition), and in section 1.8.3 where he discusses the motion of wave packets, we consider a wave packet where angular frequency $\omega$ is ...
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Finding the units of a variable in the wave function
I have the following figure which shows the wave function of an electron. The wave function is not realistic due to the discontinuities in slope, but consider its to approximate a possible smooth wave ...
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2
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What is the correspondence between operators and measurement in quantum mechanics?
Leonard Susskind, in his "Quantum Mechanics: The Theoretical Minimum" says the following
The correspondence between operators and measurement is fundamental in quantum mechanics. It is also ...
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If position is fundamentally undefined at quantum scales, how can strings be Planck length?
From how I understand Quantum Mechanics and the Uncertainty Principle, quantum objects position is fundamentally undefined. So how could an object like a string with a clearly defined geometry exist ...
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What happens to an electron if given quantized energy to jump to a full orbital?
Let's consider the element neon. Its ground-state electron configuration is: $1s^2 2s^2 2p^6$.
What would happen if enough energy was given for one electron in the $1s$ orbital to jump to the $2s$ ...
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