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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Quick question on Deriving Klein–Gordon equation from Dirac equation

On page 172 of Schwatz’s QFT book, he derives the Klein–Gordon equation from Dirac equation as following: $$(i \not{\partial} +m) (i \not{\partial} -m)\psi=(-\frac{1}{2} \partial_\mu \partial_\nu {\...
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Is back in time (classical ) information transfer possible (based on delayed choice entanglement swapping )?

Delayed choice entanglement swapping. Two pairs of entangled photons are produced, and one photon from each pair is sent to a party called Victor. Of the two remaining photons, one photon is sent to ...
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Find the normalized state and function such that the equation holds for arbitrary unitary matrix

I have been recently puzzled with a problem I do not know how to solve. Here is the setting and some of my thoughts on the problem. Given: Let us denote the set of all unitary $d \times d$ matrices ...
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Splitting a single particle wave function

The wikipedia article on the double slit experiment contains the following animation: https://upload.wikimedia.org/wikipedia/commons/transcoded/a/a0/Double_slit_experiment.webm/Double_slit_experiment....
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Implementation of neural network quantum states of the anti-ferromagnetic Heisenberg model

I'm studying this Science paper "Solving the quantum many-body problem with artificial neural networks" and looking into the implementation of the Anti-ferromagnetic Heisenberg model. The Hamiltonian ...
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Why does copper become more thermally conductive the colder it gets until 10 Kelvin?

I was wondering about superconductors and how much more conductive copper is when it is super-cooled. I mistakenly stumbled upon this page that describes copper's thermal conductivity with no external ...
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Schrödinger equation on operated states

I understand that we can apply the Schrödinger equation to any wavefunction. Now, my question is, can we apply it to states that are being operated upon? Because, when we apply an operator on a state, ...
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Quantum mechanical picture of “electrons orbiting the nucleus”

Given the wavefunction, $\psi$ is explained as the flow of probabilities or in other words probability density over a certain region of space. In the case of electrons, say in $s$ orbital, the ...
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Matrix representation of the operator $\hat{S}_x$ in the standard basis

I have recently been introduced to the idea of spectral decomposition of spin angular momentum operators in Quantum-Mechanics. Out of curiosity I was wondering if the the spin angular momentum ...
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Use the commutation relation between the position operator $\hat X$ and the momentum operator $\hat P_x$ to show the given equivalence relation

I am attempting to prove the following relation $\frac 1 2$$(\hat X^2 \hat P_x+\hat P_x \hat X^2)$ = $\hat X \hat P_x \hat X$ My attempt: $\hat X=x$ , $\hat P_x=-ih\frac d {dx}$ I commuted the ...
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Transforming Cartesian Position Operators into Spherical Coordinates

Context: (not asking for solution) I'm attempting to show $\langle n,l',m'|\hat z|n,l,m \rangle = 0$ for $m\neq m'$ using the explicit form of $Y_{l,m}(\theta,\phi)$. Question: I wasn't sure how ...
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Eigenfunction in momentum space

For the wavefunction $\Psi(x, 0)=A e^{-a x^{2}}$, its momentum representation is as follow: \begin{equation} \begin{aligned} \tilde{\psi}(p, 0) &=\frac{1}{\sqrt{2 \pi \hbar}} \int_{-\infty}^{+\...
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Standing Waves vs. Energy Eigenstates

The energy eigenstates for a particle in an infinite 1d square potential well and the modes for displacement of a standing wave of a string (length L) between two rigid posts have a similar form: $y(...
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What is the actual entropy of the Universe

I know that it depends on scale, but for example, for an experimenting coissing a coin, What was the maximum run, 20 , 30, that doesn´t influiate taking account that the velociy of the information is ...
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Calculate the commutator between the operator $S_z$ and the operator $S_x$ using the Dirac notation [closed]

Calculate the commutator between the operator $S_z$ and the operator $S_x$ using the Dirac notation. In standard matrix notation I proved the relation $[S_z,S_x]=ihS_y$ My attempt in Dirac Notation: ...
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Is fermion mass imaginary instead of real?

This seems to be an absurd question, but bare with me. In quantum field theory, the Dirac fermion mass Lagrangian term reads $$ m\bar\psi \psi = m(\bar\psi_L \psi_R + \bar\psi_R \psi_L) = m(\psi_L^\...
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Smallest object at rest (in the lab frame) vs Avogadro's number?

I am not asking about a boundary between classical and QM objects. There are many questions on this site about such a boundary. I am simply asking about an object being able to be at rest (in the lab ...
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A system of two identical non-interacting bosons, one in a stationary state with pos. parity, the other in a state with neg. parity

In a system of two identical non-interacting spinless bosons, one particle is in a stationary state $\psi_{1}(\textbf{r})$ with positive parity and another is in stationary state $\psi_{2}(\textbf{r})$...
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Virial theorem for atoms

How Virial theorem applies to atoms? For example we can use it to calculate the average kinetic energy for an electron in the hydrogen atom. But Virial theorem states that: $$\langle{K_{sys}}\rangle=\...
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Infinitesimal Translation

This is a part from modern quantum mechanics - J J Sakurai, about infinitesimal translation operators. I am not able to understand how the commutation relation between operators x and K came to be. ...
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Everett's interpretation of QM - does the world actually branch, physically?

Under Everett's interpretation I understand the key notion is that every quantum possibility is physically realized in its own universe, such that any one universe only contains any one physical ...
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Vector field under Symmetry Operation

Which vector fields are invariant under a symmetry operation of the type explained in this answer? https://physics.stackexchange.com/a/201571 My context is the following: I have a magnetic point ...
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Why can't I add the energies in this WKB approximation example to get the allowed energies for the given potential?

Use the WKB approximation to find the allowed energies ($E_n$) of an infinite square well with a "shelf", of height $V_0$ extending half-way across: $$V(x)=\begin{cases} V_0 &, \text{ if} \...
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In the Dirac equation, if the $\alpha$ is the mean velocity, why does it commute with $x,y,z,t$ if the velocity is related to the momentum?

In the Wikipedia talk page for the Dirac equation I found the following passage: The Dirac equation can be proved with the help of the correspondence principle. The energy and momentum of a ...
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Books about notions of subatomic physics [duplicate]

The Master 2 "Nuclei, Particles, Astroparticles and Cosmology" of the Université Paris-Saclay lists the following requirements as pre-requisites: Quantum Mechanics Notions of Sub-atomic Physics ...
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Is there a violation in the Onsager-Kelvin relation in Normal metal - superconductor junctions?

I'm currently reading the paper https://journals.aps.org/prb/pdf/10.1103/PhysRevB.48.15198 It is about charge $I$ and heat $I_H$ currents in NS junctions due to temperature $(\Delta T)$ and voltage $(...
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When equations are written the describe quantum experiments, does the equation necessarily include the set up conditions of the subatomic particles?

What I am wondering is e.g. when an experiment is done on the LHC, this involves colliding particles traveling in opposite directions, at a certain speed and one might assume angular momentum. The ...
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Help using the definition of Hermitian operator in $\int\psi^*(\hat F-\left<F\right>)^2\psi dr$

In my lecture the professor said that the mean value of a physical quantity- since it must be real- must satisfy the following condition: $$\begin{align} \left<F\right>=\left<F\right>^* \...
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Simple question about projection operators

In John Preskill's lecture notes, I've encountered a brief discussion about observables. He writes some operator $A$ in a Hilbert space as $$A=\sum_n{a_nP_n},$$ where $P$ is the projection operator ...
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Solutions to David Tong's exercises?

I am a mathematician taking a course in relativistic quantum mechanics and I am really struggling with a lot of it. I am trying to do a few more questions to get used to the material, but it would ...
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Question regarding the derivation of equation for anomalous density

I was reading this paper and have a question regarding the derivation of equation $3.7$ for $\tilde m$ from equation $3.4c$. In particular, I was having trouble seeing why \begin{align} (h^\text{sp}(\...
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Problem in deriving Pauli equation

I’m trying to derive the equation from the Hamiltonian $$H=\frac{1}{m}[\vec{\sigma} \cdot (\vec p-q\vec A)]^2+ q \phi$$ and $$[\vec{\sigma} \cdot (\vec p-q\vec A)]^2= (\vec p-q\vec A)^2-iq \vec{\...
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Counting spin degeneracies

Let's say that given an ensemble of $N$ spin-1 particles, I want to find the number of degeneracies. For this example, let's just assume that only the spin of the particle gives us degeneracies. For ...
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Does the many worlds interpretation require a 100% spin up branch?

In his Q&A after his Brief History of Quantum Mechanics talk Sean Caroll mentioned that when writing his book he made 50 quantum spin measurements for an example. He admitted that in some universe ...
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Peculiar dip pattern in Frank Hertz I-V curve

What's the real reason behind the much lower second dip than the other dips in Frank Hertz I-V curve performed with Argon gas?
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Most probable trajectory of a constantly measured particle?

Background Let us assume I have a particle which is in the position basis. I was wondering what was the most probable trajectory taken by such a particle when constantly measured. Let the particle ...
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Is it possible to observe a quantum probability distribution?

Is it possible to observe the probability distribution of a quantum particle in real time? So not to observe A state, which would collapse the wavefunction, but observe the whole wave and its ...
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Is the vacuum state different after a Bogoliubov transformation of phase space operators?

I am looking at the problem in the context of quantum optics. Consider this logic: In the Schrodinger picture, the state evolves in time. The time evolution of a state is given by a unitary that is ...
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Input-output theory: interpretation of the final expression

I am trying to understand this paper- https://journals.aps.org/pra/abstract/10.1103/PhysRevA.30.1386 I will try to give my understanding of the paper first. We start with the quantum Langevin ...
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Inner product of $x$ and $p$ [duplicate]

I’m taking this for granted and using it to show that the wave function in momentum space is the Fourier transform of $x$ space. But suddenly I don’t why this stands. $$\langle p |x \rangle =e^{-ipx}/...
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Problem in derivation or momentum as generator of translation

A derivation of momentum as generator of translation In this page, part 2 “Momentum as generator of translations”, I don’t understand this step: $$T(x)=\lim_{N \rightarrow \infty}(T(x/N))^N =\lim_{...
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Expression of Spin operator in function of the creation and annihilation operators [closed]

How can I prove the following relations: $S^z=S-a^{\dagger}a$ $S^+=\sqrt{2S}\sqrt{1-\frac{a^{\dagger}a}{2S}}\cdot a$ $S^-=\sqrt{2S}a^{\dagger}\sqrt{1-\frac{a^{\dagger}a}{2S}}$ where $a$ and $a^{\...
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Why does the Kubo formula not give us Curie's law?

Suppose our system is a spin-$\frac{1}{2}$ particle kept in a magnetic field along the Z direction. $$ H = -BS_z $$ Suppose we want to calculate the magnetic response of the system to a small time ...
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Finding the lowest eigenvalue using VQE-like circuits

I want to find the lowest eigenvalue for the following hamiltonian using VQE_like circuits (in python), but I don't want to use quantum libraries. \begin{eqnarray} H &=& \frac{-3}{2} I \...
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Is nature itself is random? [closed]

Quantum mechanics teaches us that with Every physical system you can associate a state vector $|{\Psi(t)}\rangle$ and For every physical obeserable there is an operator ,eigen values of these ...
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What is meant by “spontaneous creation” in this paper?

I have some questions in regard to the paper "Spontaneous creation of the universe from nothing". If I am not mistaken it is akin to Alexander Vilenkin's proposed cosmological model that has the ...
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Have researchers managed to “reverse time”? If so, what does that mean for physics?

According to press releases, researchers have reversed time in a quantum computer and violated the second law of thermodynamics. What does that mean for physics? Will it allow time travel? Further ...
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Electronic component of the Hamiltonian operator and uncertainty principle

This question has to do with the concept of uncertainty principle. The Hamiltonian operator has the electronic component that takes the inverse of the distance between any two electrons. My question ...
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Is electronic coupling between a donor and an acceptor molecule a contant?

We take electronic coupling between two molecules to be a constant. Is there a way that this electronic coupling can be time dependent? also is there a way that the dipole moment of a molecule can be ...
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Unitary evolution of a state [closed]

If a quantum system is under a potential, doesn't it mean it is interacting with the surrounding. If so, why do we say that the state of system will evolve under unitary transformation? (as by ...

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