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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Explaining Quantum Entanglement to a layperson

I have developed an idea to explain quantum entanglement to a layperson. I would like to discuss it over here and question about issues with it. Suppose if person A has a box with a robotic hand ...
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Does the abstract wavefunction change in this following example?

Suppose, we have a basis $|u\rangle$, described by the function $u=g(x)$. We can normalize this basis, using our standard $|x\rangle$ basis using the following : $$\hat{I}=\int |x\rangle\langle x|dx=\...
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Free particle approaching a square 1D potential well

I am trying to find when the probability of being reflected by the potential well vanishes, where the approaching particle has energy $E$ and the square, 1D potential well has depth $V_0$ and width $...
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Form of the interaction representation time evolution operator

I am a bit confused about the interaction representation picture. Consider the time independent Hamiltonian $H = H_0 + V$. My question concerns the interaction representation time evolution operator: $...
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Displacement measurement in quantum optics

I was following a talk where it was stated that performing a displacement $D(\alpha)$ on an optical mode, followed by a single-photon detection without number resolution, corresponds in the case of no-...
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Are the Orbital Angular Momentum Operators Linear?

I'm a bit confused whether or not I can distribute out the operators in the definition of the raising and lowering operators. For example, given: $$L_{+}|l \space m\rangle = a |l\space m+1\rangle$$ $$...
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Vector operator formulation

I showed that the ladder operators: $ \hat{\overrightarrow{a}}=(a_x, a_y , a_z)$ and $\hat{\overrightarrow{a}}^{\dagger} = (a_x^{\dagger}, a_y^{\dagger} , a_z^{\dagger})$ can form a vector operator by ...
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Translation operator and position operator

I am a little bit confused about the translation and position operator and hope for some clarification. Let $\hat{x}$ be the position operator, which satisfies $\hat{x} \vert x \rangle = x \vert x \...
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Normalizable wavefunctions for bound states

In my quantum mechanics book I read the following sentence: If we want the wave function to be normalizable, one must impose boundary conditions: $$\lim_{x \to \pm\infty} \psi(x)=0.$$ My question is ...
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Isn’t this a paradox?

Imagine a plate beeing irradiated from all sides with a specific wavelength. One side of the plate is smoth and reflects most of the incoming radiation, the other side is covered in grooves with the ...
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What happens after the collision of an electron in the ground state of a hydrogen atom at rest and a high energy electron? [closed]

Consider an electron, 1, in the ground state of a hydrogen atom at rest. At a certain time t0 a high energy electron, 2, collides with the proton in the atom, and is captured, undergoing the weak ...
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One-dimensional Hamiltonian in SUSY Quantum Mechanics

I've seen in many places, in "Supersymmetry and Quantum Mechanics" of F. Cooper for example, that the hamiltonian in one dimension is defined: $$\hat H_1=\hat A^\dagger\hat A=-\frac{\hbar^2}{...
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Total energy of hydrogen-like atoms

Hello I am currently working on a problem regarding hydrogen-like atoms. It is given: $$ \hat H = \sum \limits_{i=1}^{n} \hat h(\mathbf{r_i}) $$ I need to show why the solutions of the Schrödinger-...
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How to perform Fourier transform of this Hamiltonian?

I am reading this article (arXiv:1505.01908 ) in which author is calculating linear response of a perturbation. The perturbation Hamiltonian is $H$ (Eq. 2 of article) given as $$ H = \frac{JS}{a^3}\...
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Apparent paradox in statistical mechanics

I can't understand why the likelihood of a particle to be in state $\epsilon_i$ in a canonical ensemble, does not depend upon the number of particles in that state. The probability that the a single ...
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Two vectors in quantum mechanics represent the same state

From what I understand, if $c$ is a complex number then $|\psi\rangle$ and $c|\psi\rangle$ represent the same physical state. Now, let $|\psi\rangle$ be an eigenvector of a Hermitian operator O then, $...
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Momentum in Classical mechanics and quantum mechanics

In the following question $\nabla, \hbar$ means gradient r(position vector) and reduced Plank's constant respectively.
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Inserting a position operator in the path integral in QFT

With the usual path integral description, we have the formula $$\langle q''t''|q't'\rangle =\int\mathcal{D}q \exp{(iS)}$$ where $S=\int_{t'}^{t''}L(q,\dot{q})$ is the action evaluated for $t\in (t',t''...
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Prove that $i\hbar\dot{\hat{\Omega}}=[\hat{\Omega},\hat{H}]$ , where $\hat{\Omega}(\hat{x},\hat{p})$ is Taylor expandable

How do I show that $i\hbar\dot{\hat{\Omega}}=[\hat{\Omega},\hat{H}]$ , where $\hat{\Omega}(\hat{x},\hat{p})$ is Taylor expandable, $\hat{H}(\hat{x},\hat{p})$ is the Hamiltonian of the system and $\hat{...
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Where did quantum numbers arise from?

I was told that there are 4 quantum numbers which help in locating the position of an electron inside an atom. And they are Principal quantum number Azimuthal quantum number Spin quantum number ...
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How do we choose the basis of an Hilbert Space?

When we define a basis for the Hilbert Space for a spin half particle I understand it being done using the principle of mutual exclusivity that is if $S_z = +\hbar/2$ then it cannot be $S_z = -\hbar/2$...
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What does the term "condense" mean in the physics literature

When reading the physics literature, we often see the term "condensate". Some examples: in the string net model (Wen, Levin), one will say the string "condensate". in QCD, people ...
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Is This Quantum Eraser Video from Fermilab Incorrect?

In a video from Don Lincoln of Fermilab here, the following claim is made about a quantum eraser-type setup (about 2 min 50 sec in): "If A & B are turned off, you see a wave pattern. That ...
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Heisenberg’s Formulation of Quantum Mechanics and Particle Trajectories [duplicate]

In the article, under the history section, Heisenberg recounts a discussion with Einstein regarding Heisenberg’s matrix mechanics: “ He pointed out to me that in my mathematical description the notion ...
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Average squared of angular momentum

Consider spin $S$ particles. Is there an easy way to prove that $\langle \vec{S} \rangle \cdot \langle \vec{S} \rangle \leq S^2$ for all states? I understand intuitively that the bound is realized ...
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On the invariance of $ \rho |\psi \rangle$? (For Wigner and his friend)

Motivation So let's say I shoot circular polarized light which is either right $| R \rangle$ or left $| L \rangle$ polarized. My friend notices that I choose left and right equally $50$% of the time. ...
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Measure Position of Electron Directly

What is the most directly way to measure the position of free electron in an expirement? I don't asking about theoretical suggestion but rather on practical expirement which people have done.
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Questions on quantum spin [closed]

I have been researching quantum physics by myself through books and videos and I can understand the property of particle “spin” but only to a limit. I understand it’s intrinsic angular momentum and ...
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Normalizer of $SU(2)\times SU(2)$ in $SU(4)$

What is the normalizer of $SU(2)\times SU(2)$ in $SU(4)$ or how would I find it? Reason for the question: with 2 qubits, if I was interested in conjugation of 2-qubit gates with generic $SU(2)$ ...
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Can weak measurement principles of Quantum mechanics be used in similar questions in classical estimation problems?

This is a surface level question and I don't want to go into detail. Imagine an algorithm which when used with a sensor output gives the statistical moments of a variable in nature (for example mean ...
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What would be the proper distribution to model the number of particles in a state in canonical ensemble

Suppose my system has $N$ particles, and I want to find a distribution for $n_i$, the number of particles in the $\epsilon_i$ energy state. What I do know is the boltzmann probability, which tells me ...
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Arbitrary helicity in the nonrelativistic quantum free particle

One of the Casimir operators of the Euclidean group $E(3)$ is $\vec{J}\cdot\vec{P}$. For a quantum free particle (described in abstract as) $|p,\lambda,\hat{\vec{p}}\rangle$, the given Casimir ...
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How Heisenberg defined the $P$ $Q$ matrices terms?

I learnt in some Wikipedia articles that the terms of the $P$ and $Q$ matrices designed by Heisenberg, were composed of Fourier coefficients. Could you provide some explanation on how these ...
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Measuring a qubit in the wrong basis

I'm a computer scientist and right now I'm trying to end the research work for my master thesis and a basic problem of quantum mechanics is blocking me. I'm trying to do a probability calculus of the ...
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Is there a (semiclassical) electric field operator?

So I come from a chemistry background, where the electronic structure of atoms and molecules is central. For practical purposes, we usually work with a charge density operator $$ \hat{\rho}(r) = q \...
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What are $δ$ and $ε$ in the list of Dirac's five 'fundamental constants', concerning his 'Large number hypothesis'?

From Jean-Philippe Uzan's Varying Constants, Gravitation and Cosmology: Dirac formed five dimensionless ratios among which1 δ ≡ H0ħ/mpc2 ∼ 2h × 10−42 and equation M1 and asked the question of which of ...
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Negative potential infinite square well

A 1D finite square well is generally defined either by \begin{equation} V(x)=\begin{cases} 0\qquad -a\leqslant x\leqslant a\\ V_0\qquad \text{otherwise} \end{cases} \tag{1} \end{equation} or \begin{...
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Deriving Hartree from fundamental atomic units leads to error?

I am trying to find how many newton meters are in a Hartree by using the following definition in terms of other physical constants: $$E_h = \frac{\hbar^2}{m_ea_0^2} $$ The values of the other physical ...
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Is there a potential energy for every wave function?

In my early ventures of quantum mechanics, the pattern seems to be: choose a simple potential energy and try to solve for the corresponding wave function. (We can stick to a particle in a box for my ...
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Changing the basis of a wavefunction [closed]

Usually wavefunctions are given in problems of quantum mechanics. I have the following question in which we want to find thee wavefunction: The state of a system is given by the wavefunction $\psi(x)=...
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Why are rotation operators and Pauli rotations defined so that $R_x(\pi)\neq X$?

When applying a Hadamard Gate Wikipedia defines it as $XR_y(\frac{\pi}{2}) = H$. The effect of a Pauli Gate $X$ is defined as a Rotation of $\pi$ radians about the x-Axis on the Bloch sphere. The ...
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Ehrenfest theorem: On which classical circle can we find the electrons in an homogenous magnetic field?

In the French wiki article about the Ehrenfest theorem I found these formulas. $${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\langle {\hat {x}}\rangle ={\frac {1}{m}}\langle {\hat {p}}\rangle }...
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Can the hydrogen Rydberg decay cascade be resolved into individual lines?

This is a technical question as a follow-up to the discussion in Would a high energy Hydrogen atom start emanating electromagnetic radiation? Suppose that I have a hydrogen atom that is excited, at $t=...
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How can a single electron in a crystal become excited, given Bloch's theorem?

Suppose I have an infinite crystal. A physically realistic, though non-stationary, state for the crystal is to have one electron excited, and the rest of the electrons not excited. However, Bloch's ...
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Translation matrix for multiple spins

I need to do a translation, but not a translation in the classical definition, I need to, in a system with a operator, for example, $$ \sigma^z_j=1\otimes \cdots \otimes 1\otimes \sigma^z\otimes 1 \...
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Expected value of the electric field operator in the superposition of two number states

Good morning everyone! I have recently started a course of introductory quantum optics, but unfortunately by quantum mechanics background is not very good - so I'm having some problems when it comes ...
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Ladder Operators and Angular momentum

The angular momentum operator $L_z$ can be expressed in terms of $L_z = xp_y - yp_x$ where $ p = \hat{p} = -i\sqrt{hwm/2}(a^{\dagger} - a)$ and $x = \hat{x}\sqrt{\hbar/2mw}(a^{\dagger} + a)$. We want ...
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In Stern-Gerlach experiment, where does wavefunction collapse?

I was reading Sakurai's Modern QM and it talks about Stern-Gerlach experiment in chapter 1. As silver atom passes through non-uniform magnetic field and enters detector downstream, a measurement is ...
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What is a good book to understand artificial atoms?

I'm working on a thesis about artificial atoms and population transfer, and all I could find for now are articles (very useful, but can be hard to fully understand). Can anyone recommend a good book ...

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