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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37 views

Creation operator on coherent state and issue with commutation relations

If our bosonic annihilation and creation operators are $[a,a^\dagger] = 1$, then for any complex number $\varphi$ we can define the (unnormalized) coherent state $$ | \varphi \rangle \equiv e^{\varphi ...
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Results concerning computation of differential cross section of one (two) particle scattering in non-relativistic QM

I have developed a method of numerically computing the differential cross section $f(\textbf{p'} \leftarrow \textbf{p})$ for an arbitrary time independent potential $V(\textbf{r})$ in 3D. Since two ...
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38 views

Is there a commonly accepted definition of a quantum phase definition for a finite lattice/set of particles?

As noted by Sachev, and in a previous question, https://www.physicsoverflow.org/41602/, there cannot be quantum phase transitions for finite systems (with bounded local Hilbert space dimension). The ...
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67 views

Unitary transformation of bosonic operators

Consider two sets of bosonic operators $\{b^{\dagger}_k, b_k\}$ and $\{B^{\dagger}_i, B_i\}$ satisfying $[b_k,b^{\dagger}_{k^{\prime}}]=\delta_{kk^{\prime}}$ and $[B_i,B^{\dagger}_j]=\delta_{ij}$. The ...
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28 views

How to find the operator $A$ in the second order correction to the ground state energy of Hydrogen (Stark effect)

Consider that the hydrogen atom is placed in a constant electric field $F$ along the negative $z$-direction. Hamiltonian $$H=H_0+H_1$$where $$H_0=-\frac{1}{2}\bigtriangledown^2-\frac{1}{r}\qquad,\...
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1answer
57 views

Why do we call $x$ a constant while mentioning eigenfunctions of the position operator?

For a position eigenstate $\psi_x$, we say $$\hat{x}\psi_x=x\psi_x$$ Since it's an eigenfunction, we say that the result of using the operator is "a constant times the function itself". But $x$ isn't ...
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44 views

3D harmonic oscillator magic numbers

I know that, $$V(r) = (1/2) m \omega^2 r^2 ,$$ $$\omega \approx 40(Z+N)^{-1/3}\ \rm{MeV} $$ and $$E = (n+3/2) \hbar \omega.$$ How do you find the magic numbers of protons and neutrons which ...
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33 views

Has anyone tried to incorporate other forces in GR-like theory rather than incorporate gravity in QM? [duplicate]

Has anyone tried to imagine other three forces in a way similar to gravity, i.e., warping some higher dimensions of spacetime to create a curvature. If there is such a theory, where does it fail?
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208 views

Differentiation of a ket vector with respect to a spatial dimension

Consider a state $|\psi\rangle$. While discussing the Schroedinger equation, we say $$\hat{H}|\Psi(t)\rangle=i\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle$$ We also define the hamiltonian operator ...
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15 views

Simple way to modify the diagonal elements of the hamiltonian after adding a strong interaction

Let's say we have a hamiltonian of a non-interactive system of two particles. We have correctly worked out the matrix form of the hamiltonian. Now, if we add a very strong attractive interaction ...
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67 views

Single-particle Green's function

Define a single-particle Green's function as \begin{equation} i\hbar G(xt;x't') = \langle x| e^{-iH(t-t')/\hbar} | x'\rangle. \end{equation} By inserting the completeness relation, we have \begin{...
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Determining the state of a system

My textbook says: "To determine the state of a system at a given instant, it suffices to perform on the system a set of measurements corresponding to a complete set of commuting observables (CSCO)" ...
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24 views

Relation between Wigner flows and entropy

After reading this question What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum Mechanics? and some papers of Steuernagel’s group,...
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1answer
61 views

Solutions that are part of the Hilbert space

Why do we omit solutions that do not converge at $\pm\infty$ from the physical Hilbert space, what is the argument for us being allowed to do so?
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73 views

Why any expectation value can be computed by this path integral, and not just the time-ordered ones?

This is quite a basic question about the path integral. In Polchinki's String Theory book, Chapter 2, he says: Expectation values are defined by the path integral $$\langle \mathscr{F}[X]\...
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1answer
80 views

Does No-cloning theorem hold in time domain?

So for an unknown quantum state $|A\rangle$, it's impossible to make a copy of $|B\rangle$ such that $|A\rangle=|B\rangle$. However, I want to know that, suppose the unknown $|A\rangle$ is time ...
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16 views

Wave-like and particle-like nature [duplicate]

My notes say Radiation emitted from any source moves discontinuously in form of small packets of energy. Each packet is called quantum. Does this mean that when light travels, it is actually a ...
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3answers
56 views

Expected value of Momentum in a square infinite well [closed]

Say I have a particle with mass $m$, in a potential infinite well centered at $x=0$ with length $d$ which wave function at $t= 0$ is represented by: $$\Psi(x)=\begin{cases} \frac{1}{\sqrt{2}}\left[\...
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1answer
79 views

Question about the “wave function” on Relativistic Quantum Mechanics (RQM) and Quantum Field Theory (QFT)

I'm enrolled on a short and conceptual couse on RQM and QFT and the professor made a distinction about the Klein-Gordon (K-G) equation on RQM and the K-G equation on QFT. Roughly speaking, he said ...
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4answers
200 views

Why can't quantum field theory be complex instead of imaginary?

In the following question 1, the author claims that a QFT is defined as: $$Z[J] \propto \int e^{iS[\phi]+J.\phi} D[\phi]$$ Then uses this definition to explore the possibility of formulating a QFT ...
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2answers
61 views

Does the static gravitational field move with the source instantly?

I have read this question: How fast does gravity propagate? where hawkeye says: So what does that mean? It means that the "speed of gravity" is the speed of light … technically. Changes in ...
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Heisenberg uncertainty principle in daily life

I need some examples of the Heisenberg uncertainty principle on a basic level, or if possible in daily life. Or maybe a simple explanation for validity of the principle in easier words. I cannot get ...
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19 views

Is the Quantum Zeno Effect real? [duplicate]

When a decaying particle is observed using the Quantum Zeno Effect, can the decay of this particle be delayed or prevented?
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1answer
43 views

Euler-Lagrange Equations for Molecular Dynamics

In the Car-Parrinello (CP) method for molecular dynamics simulation, the Euler-Lagrange equations are given as $$ \begin{aligned} \frac { d } { d t } \frac { \partial \mathcal { L } _ { \mathrm { ...
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27 views

Different Quantum Measurements

What is the difference between an interaction-free measurement, a weak measurement and a counterfactual measurement in quantum mechanics?
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21 views

Bohr’s Quantisation Condition [duplicate]

I am a grade 12 student from India and my physics textbook does not delve deep in the bohrs quantisation condition but has given us a paragraph to figure out what it is: “Consider Motion of an ...
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1answer
39 views

Has many-body tunneling at the level of nuclei been studied?

In a recent paper, the authors stress the difference between single-body tunneling and many-body tunneling (at the atomic level): "In contrast to the well-studied incoherent single-particle tunnelling,...
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2answers
130 views

Trouble understanding Sakurai's calculation of $\exp\left(\frac{iS_Z\phi}{\hbar}\right) \;S_x \; \exp\left(\frac{-iS_Z\phi}{\hbar}\right)$

I'm having some trouble with a derivation in Sakurai's Modern Quantum Mechanics (specifically Derivation 1 on §3.2, p. 159), where he computes $$ \exp\left(\frac{iS_Z\phi}{\hbar}\right) \;S_x \; \exp\...
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0answers
54 views

Tunneling elements in the Hubbard model

Consider the tunneling Hamiltonian in the Hubbard model for a 1D lattice of quantum dots. $$\begin{align}\hat{H}_t=t\displaystyle\sum_{i,j,\sigma}c_{i,\sigma}^{\dagger}c_{j\sigma}+c^{\dagger}_{j,\...
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1answer
41 views

Inverse covariance matrix for a Gaussian state

I was reading an article about Gaussian Boson Sampling (https://arxiv.org/pdf/1801.07488.pdf) and following some calculation appear an inverse covariance matrix when he defines the following matrix A. ...
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1answer
27 views

Why and in what cases we use $E$ instead of $\Delta E$ in energy-time uncertainity relation?

sometimes in problems of energy or time uncertainity we use the value of energy of the particle/system to calculate the delta t. Now what are those cases specific examples, where we can apply this?
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109 views

Do virtual particles follow worldlines (spacelike, timelike, lightlike)?

When we talk about static force fields, we use virtual particles to describe the interaction of such fields with other particles. These virtual particles are not real particles, they are a ...
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What is the easiest system to take the matrix representation of a Hamiltonian?

To understand how the unitary operator, preserve the inner products, I wanted to explore the unitary operator as a matrix. Now the equation for the unitary operator (time evolution operator) has a ...
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1answer
52 views

Number operator - annihilation operator commutation

Is there a rigorous way to prove that $$ (N+1)^{-1/2} a = a N^{-1/2} $$ where $a$ is a bosonic annihilation operator and $N=a^\dagger a$ is the corresponding number operator?
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1answer
91 views

What potentials have infinitely many bound states? [on hold]

Some potentials have only finitely many bound states (the finite square and delta function are two good examples) Others have infinitely many bound states (for example the infinite square well and $1/...
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17 views

Does electron affinity decrease with charging?

Lets consider two materials A having higher electron affinity than B while both are not too far off. If they are both insulators and A accumulate charges due to triboelectric effect. Than after a ...
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2answers
77 views

Electric field of a point source in QM

We know that radiation from a point source vanishes at large distances. So when an atom emits a photon, the expectation value of the electric field must vanish at large distances. How we can explain ...
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1answer
62 views

Gaussian wave packet with a step potential

In principle of quantum mechanics by Shankaar on page 170, while doing transmission and reflection index for a step potential for a Gaussian wave packet moving to the right. We come to this ...
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74 views

Why does light come as quanta of the harmonic oscillator?

I've recently been learning the basics of Quantum optics and it seems to be a fundamental concept that light is best described in the framework of the Quantum Harmonic Oscillator. This lead to a ...
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2answers
61 views

Path of an electron around a nucleus [duplicate]

I have just started reading about quantum mechanics and i stumbled upon this silly question - Quantum mechanical model of an atom says that path of an electron around the nucleus is uncertain. Then ...
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2answers
69 views

Why doesn't the amplitude of a wave-function fall off to zero immediately at a potential barrier?

When a wave function in QM potential well problems interact with a potential barrier with height more than the energy of the wave, the amplitude of the wave doesn't immediately falls off to zero, ...
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3answers
106 views

What is the result that that differs by many orders of magnitude between QM and GR? [duplicate]

It is well known that QM and GR are deemed incompatible due to a discrepancy in some calculations which I have read can differ by large magnitudes. What are these calculations to which people are ...
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2answers
50 views

Observing a system in an energy eigenstate when the eigenstate is not normalized

In the following notes from an MIT OCW course, Zweibach claims that energy eigenstates are not necessarily normalized. https://ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/lecture-...
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1answer
47 views

Unitary Transformation of an Interfering Beam Splitter

I was reading this research paper Quantum interference enables constant time quantum information processing and was confused by one particular expression involving the Hamiltonian of a beam splitter. ...
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1answer
36 views

Calculating $\langle \psi_j^0 | \delta\psi_i\rangle$ in perturbation theory [closed]

Within first order (or linear order) quantum perturbation theory, the Schrödinger equation (for a state $i$) can be written: $$\delta H |\psi_i^0 \rangle + H^0 |\delta\psi_i\rangle=\delta\...
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1answer
55 views

Derivation of the Guiding Wave equation

I've been searching around the internet for a derivation of the guiding wave equation, but I can't find a derivation anywhere. I know that Bohmian Mechanics is not a mainstream idea but I was hoping ...
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1answer
40 views

Modifying the Hamiltonian when there is a presence of the Coulomb interaction

Referring to the Hamiltonian of a system of free electrons, $$ H_0= \sum_{\sigma} \int d^3rd^3r' \psi_{\sigma}^{\dagger}(\mathbf{r})\left(- \frac{\hbar^2}{2m}\nabla^2\right)\delta(\mathbf{r}-\mathbf{...
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If time is always passing, at what increment/speed is it passing in? [duplicate]

If time is always unfolding around us it must be occurring in some increment or speed, if answerable, what speed or increment would time be moving in? If not answerable, would there be a way to figure ...
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3answers
77 views

Negative energy in bound states of a particle in a finite potential well

Consider you have a particle in a finite potential well as depicted in the photo attached. Now we have three regions: $$V(x) = \begin{cases} 0, & \text{for } x<-a & (1)\\ -V_0, & \...
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1answer
80 views

Do potentials make sense in relativistic quantum theory?

In Peskin & Schröder QFT, just before equation 7.93, he writes in passing, Next let us examine how $\Pi _2 (q^2)$ modifies the electromagnetic interaction, as determined by Eq. (7.77). In the ...