Quantum mechanics Definition and 994 Threads

We can calculate ##\langle zHz \rangle## by noting that ##[z,[H,z]] = 2zHz - z^2H - Hz^2##. The author says this is the quick way: it takes "four lines". Embarrassingly, it took me way longer than that. For instance, in the calculation of ##[H,z]##, he says that $$[p_x^2 + p_y^2 + p_z^2, z] =...

This is part of a larger, more difficult question. But this is the part I was a bit stuck with. ##\langle r \rangle = \frac{3}{2}a## and apparently ##\langle x \rangle = 0##. I could arrive at this by plugging the numbers in but I would like to know how to come to this result without doing any...

This figure is from "Introduction to Quantum Mechanics" by Griffiths (3rd edition). It is available to download. It is from page 142. I am hoping the usual people on this site will give me a hand understanding what is going on in the figure. After the equation (4.50) it says "It is customary...

I just finished my three year bachelor degree in physics with an overall average grade of 7.4 on all courses. In september I will be starting my 2 year masters degree at University. I am reading through modern quantum mechanics third edition of J.J Sakurai and Napolitano. However, I do find I...

The last problem I posted on QM made it into advanced homework help, that is why I am putting it here. I am sorry for any hassle imposed on the moderators by myself. Part (a) is quite easy. We get $$\sigma_1 = 2\lambda, \mathbf{v}_1 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \sigma_2 =...

While studying for my exam I came across the addition of two angular momenta, in the simple case of ##J=L+S## where ##L## is the angular momentum operator and ##S## is the spin (in this case a fermion with ##s=1/2##). I have some doubts on the derivation of the basis and the eigenvalues. From...

From results in my book (which I think are fairly standard across quantum mechanics) the answer to a) is ##\psi_1##. I will ask about c) later. It might come to me when I understand b). I can state with confidence that if ##B## is measured then we are either going to get ##b_1## or ##b_2##...

The following sentence appears in my book (Introduction to Quantum Mechanics by Griffiths (3rd Edition)) : "## \langle x | p \rangle ## is the momentum eigenstate (with eigenvalue p) in the position basis." Page 122. I am not sure why this is the case. I have done some digging... on a previous...

Hello everyone, I've been thinking about the standard physical picture of a neutron star reaching the TOV limit, and I've run into a conceptual question that I can't quite shake. I'd appreciate your perspective on it. The textbook explanation is a beautiful balance of our two great theories...

Consider a square well of length ##2a## centred at the origin. With a constant potential in the well ##-V_0##. That is, the potential function is an even function. Consider the solution to the problem outside the finite square well. First the left hand side: $$\psi(x) = Ae^{ikx} + Be^{-ikx}$$...

In my book the following potential is given: $$V(x) = -\frac{h^2 a^2}{m}\text{sech}^2(ax)$$ then the task is: Check that this potential has the ground state ##\psi_0(x) = A\text{sech}(ax)##. I first thought to put it in the time-independent Schrodinger equation and show that it is a solution...

In my textbook I have the following: "The double well is a very primitive one-dimensional model for the potential experienced by an electron in a diatomic molecule (the two wells represent the attractive force of the nuclei). Does the electron tend to draw the nuclei together, or push them...

Answers to questions like this assume that the quantum state in a Hilbert space is only a function of time, that is ##\partial_i\vert\psi(t)\rangle\neq0## only when the variable ##i## is ##t##. Is this a postulate of standard quantum mechanics, that in Schrödinger's equation the state in...

Wikipedia says that the equation in the title is defined to be true. But is it true always? Working with the right-hand side, $$\langle \mathbf r\vert\hat A\vert\Psi\rangle=\int\langle \mathbf r\vert\hat A\vert\mathbf r'\rangle\langle\mathbf r'\vert\Psi\rangle\ d\mathbf r'$$ If we assume...

Questions and the figure is given. This is a task from previous years, fortunately there are no answers. I have tried to solve, starting from part a, but I think I do not understand how to set boundary conditions. I ended up with: 2*C*e^(i*a*k2)=B*e^-i*a*k1*(1-(i*k1/i*k2) When I proceed with...

I'm not entirely sure how to approach part 1. I tried the following: Call the span of the two states S. We can decompose H = S + S_perp. We consider some projector P: H -> V where V is some subspace of H. I made a new operator P_s P P_s where P_s is the orthogonal projector onto S. However, I...

Disclaimer: I’m new to QM, so things I say might be wrong, please forgive me for possible inaccurate info. So as for what I know, quantum gravity is the study of merging GR and QM on gravity. Here’s a few questions I’m interested about on the quantisation of gravity: String theory 1. What is a...

Hi. I have got question about the damper term in lorentz oscillator model. I understand that "spring" term comes from interaction between nucleus and electron, driving term form external electric field acting on molecule. But I don't understand damping term. So my question is where does it come...

I came across an interesting quote from Freeman Dyson: (start of quote): The Collapse Of The Wave-Function Four and seven years ago, Erwin Schrödinger invented wave functions to describe the behaviour of atoms and other small objects. According to the rules of quantum mechanics, the motions of...

Consider a general ket ##|\psi\rangle##, expressed in the ## |\alpha_i\rangle## basis: $$ |\psi\rangle = \sum \langle \alpha_i | \psi \rangle |\alpha_i\rangle, $$ where## \langle \alpha_i | \alpha_j \rangle = \delta_{ij} ##, with ## \delta_{ij} ## being the Kronecker delta function, and ##...

Let me explain the situation: I have an experimental setup consisting mainly of a Michelson interferometer, so I want to use it to measure the coherence length for two different situations, when the beam is focused on a mirror and when it is focused on a sample. Therefore, before analyzing the...

How did you find PF?: Google search Hello, I am a semi-retired management consultant specializing in procurement and supply chain. Most of my work involves collection and analysis of a company's procurement data and identification of value creation opportunities from initiatives such as...

Consider 1 and 2 point sources. Then corpuscular view gives a single peak in those cases, whereas considering the undulation of a quantity along the path gives fringes in both cases. My question is : is it true that the case single point source gives a single peak and two would give fringes...

Suppose we have in a box a particle that is travelling left and right at some speed, bouncing off the walls of the box. The wall on the right is then removed such that the particle would be free to escape the box. Does the wave function of the particle get "updated" instantly the moment the...

It is common to say that ##t## and ##-i\hbar\partial_t## are not operators in quantum mechanics. But I haven't seen a satisfying justification. As an example of the precision of our discourse, someone has said that ##-i\hbar\partial_t## satisfies the definition of Hermicity, but it is not an...

In *An Introduction to Thermal Physics* by Schroeder, while deriving the multiplicity of an ideal gas makes the following statements (image below): Even in quantum mechanics, the number of allowed wavefunctions is infinite. But the number of independent wavefunctions (in a technical sense...

Hi. I'm studying a degree in maths/physics with the open university through the "combined STEM" degree programme. I wanted to know, do i require quantum mechanics modules if I wanted to focus on cosmology in the future? My current module options are: Mathematical Methods and Fluid Mechanics...

I have one tremendous doubt about it. On ##t=0## the state of the oscillator is ##| \Psi (t) \rangle = | 1 \rangle ##. The perturbation is ##V(x)=\alpha x^3 = \alpha (\frac{\hbar}{2m\omega})^{3/2} (a+a^{\dagger})^3 = \gamma (a^3+3Na+3Na^{\dagger} + 3a + (a^{\dagger})^3)##. The only possible...

There is an hydrogen atom on a electric field along ##z## ##E_z= E_{0z}## . Consider only the states for ##n=2##. Solving the Saecular matrix for find the correction to first order for the energy and the correction to zero order for the states, we have: ##| \Psi_{211} \rangle##, ##|...

Section 3.3 titled 'Solutions of the time-dependent Schrödinger equation' states in its 1st line that the time-dependent solution is not an eigenvalue equation: The same section ends with a comment on eigenstates: How do you reconcile this: are solutions to the time-dependent equation...

If I understood it correctly, at enormous timescales into the future, it is theoretically expected that eventually stable massive structures (like white/black dwarfs) will suffer quantum tunneling events that would make small pieces of them slowly turn into black holes that would rapidly decay...

This textbook claims ##m_j## is not a "good" quantum number because the total angular momentum (of an electron of a hydrogen atom placed in a strong uniform magnetic field) is not conserved. I don't understand why ##m_j## is not a "good" quantum number. Since ##J=L+S##, ##J_z=L_z+S_z##. Since...

I suppose that when carried out a Poisson spot with photons one at a time should have to be observed. I tried to find such experiment but i got 0 result. it seems that nobody cared about that. But i think such experiment would be very important as it will show that the wavefunction can have...

Below is the derivation of E1so, the first-order correction to the Hamiltonian due to spin-orbit coupling of the election in hydrogen atom. My question is whether it's valid to use [6.64] (see below). ##<\frac{1}{r^3}>## I believe is ##<\psi_{nlm}|\frac{1}{r^3}|\psi_{nlm}>##, but ##\psi_{nlm}##...

Good day to all. I'm a new member who is fascinated by and interested in physics, quantum mechanics, astronomy, and cosmology, though I never took classes during my schooling. Admittedly, I have basic math skills and worked hard to attain mediocrity in math classes (a living example of the law...

TL;DR Summary: Do not get confused with the title, standard thunderstorms are not generally classified as EMP storms because they don't have the power, the scale and nature of the electromagnetic energy involved are greatly different compared to a nuclear EMP detonation, that's exactly the...

Can energy be stored in a single particle without it being lost over time? I mean, photons would be an exampld in principle, but they get redshifted as the universe expands and become less energetic as time goes by We could store that energy in form of kinetic energy for individual...

Hello, The idea I had was to time evolve the state ##U(\hat{\textbf{b}}, \omega t)| \phi(t) \rangle##, but I'm confused on how to operate with ##H## on such state. I Iwould be glad if anyone could point some way. Thanks!

When computing the projection of time-evoluted state ## |x_j> ## on ## |x_{j+1}> ## it uses the 'completeness' of momentum basis ## \int \frac{dp}{2\pi} |p><p| ##. Next it explicitly states the form of Hamiltonian ## \hat{H} = \frac{\hat{p}^2}{2m}+\hat{V}(\hat{x_j},t_j) ##. Thereafter i believe...

Just a question: how would the wavefunction "collapse" in a time-reversed universe? Let's take Alice. If she's taking a backward time travel to -say- 2021 and finds herself in 2021, wouldn't that be a (prohibited) quantum cloning of an already measured quantum state? Say, the |Alice 2021⟩ ket is...

Are there good textbooks which explain electricity in a chemical context better. i.e. for use in measurements (cyclic voltammetry and others), the physics (suitable for a chemist) of electricity in solutions and how solutions can be modelled in circuit diagrams. I have some knowledge of...

A very massive charged black hole could reach a near-extremal state in the right conditions supressing the rate of emission of Hawking radiation (https://physics.stackexchange.com/questions/490524/evaporation-of-large-charged-black-holes) Meanwhile, the radiation emitted by a black hole can be...

I am not able to use Latex for some reason. It is very glitchy and if I do one backspace then it fills my whole screen with multiple copies of the same equation. Thus I am pasting a screenshot of handwritten equations instead. Apologies for any inconvenience. In Introduction to Quantum...

I have a hard time to grok QM. I wonder if it is my fault. Probably QM and all the interpretations are incapable to explain the world as we observe it (either in Newtonian mechanics, or in Special Relativity), not counting gravitation, also not allowing "objective collapse" (which would be a...

I know that completeness (roughly) implies that (almost) all functions can be decomposed into a sum of eigenfunctions of a Hermitian operator. ##\psi=\sum_n \alpha_n \psi_n##. Clearly, there have to be some restrictions on the function itself, and the operator as well. But what is that? My...

First I calculated ##(\vec{n} \cdot L) \psi(r) = -i\hbar(n_{x}(3y-z)+n_{y}(z-3x)+n_{z}(x-y))f(r)## and then tried to solve for ##n_{i}## such that I get (x+y+3z)f(r), and then divide ##n_{i}## by the magnitude of ##\vec{n}## to get the unit vector and m, but when I try doing this, I get the...

https://www.nature.com/articles/s41598-019-43323-2. Please note Figure 4 p.4 of the article, and the following description: " In Fig. 4(h), the right slit is completely blocked, the interference fringes have disappeared completely and only the diffraction image of the open left slit remains."...

so I thought that when a system was measured there could be an interaction between the measurement device or environment and the system but overall energy was conserved, but I came across these 2 articles which seem to imply this is not the case...

I will include a photo. Essentially, we need to prove that dp/dt + the gradient of J(x,t) = 0. We are getting that the two are instead equal and opposite, and have checked with multiple people.