Quantum mechanics Definition and 994 Threads

In a) I get that T should be largest where V_0 is least wide, because when V_0 is infinitely wide the particle would be fully reflected. But I don't get how height in b) and energy levels height in c) correlates to T and R. Is it because of their k? I get the opposite answer from the correct...

I have solved c), but don’t know how to solve the integral in d. It looks like an integral to get c_n (photo below), but I still can’t figure out what to make of c) in the integral of d). I also thought maybe you can rewrite c) into an initial wave function (photo below) with A,x,a but don’t...

I have a problem with this Hamiltonian: two identical particles of mass ##m## and spin half are constrained to move on the surface of a sphere of radius ##R##. Their Hamiltonian is ##H=\frac{1}{2}mR^2(L_1^2+L_2^2+\frac{1}{2}L_1L_2+\frac{1}{2}S_1S_2)##. By introducing the two operators...

I have approached this question step by step as shown in the image attached. I request someone to please guide if I have approached the (incomplete) solution correctly and also guide towards the complete solution, by helping me to rectify any mistakes I may have made. I'm still unsure how to...

I am considering tunnel effect with a potential barrier of a certain height that is ##\neq 0## only for ##0 \le x \le a## . I write the Hamiltonian eigenfunctions outside the barrier as:## \psi_E(x)=\begin{cases} e^{ikx}+Ae^{-ikx} \quad \quad x \le0 \\ Ce^{ikx} \quad \quad x\ge a \\...

If I have two identical particles of ##1/2## spin, for Pauli's exclusion principle all physical states must be antysimmetrical under the exchange of the two particles, so ##\hat{\Pi}|\alpha\rangle=-|\alpha\rangle##. Now, let's say for example this state ##\alpha## is an Hamiltonian eigenfunction...

In a central potential problem we have for the Hamiltonian the expression: ##H=\frac{p^2}{2m}+V(r)## and we use to solve problems like this noting that the Hamiltonian is separable, by separable I mean that we can express the Hamiltonian as the sum of multiple parts each one commuting with the...

Just earlier today i was practicing solving some ODEs with the power series method and when i did it to the infinite square well i noticed that my final answer for ##\psi(x)## wouldn't give me the quantised energies. My solution was $$\psi(x) = \sum^{\infty}_{n=0} k^{2n}(\cos(x) + \sin(x))$$...

Very basic question here, about statistical independence in quantum mechanical experiments. The quote from PD below is what prompted the question. When we talk about "some kind of pre-existing correlation" are talking about a simple correlation in the sense of the correlation of sunglasses and...

Summary:: I understand the consensus on PF about studying for knowledge and not merely for "cracking Semester Exams" but I urge you all to go through below thread before attaching to that feeling in my case. Hi. So I have my Exams on Intro QM approaching very soon, which will be a combination...

Below I have attached an image of my possible solution. I have replaced all the relevant limits. For some reason, I am getting the final value for (i) part as ψ(x)= with an additional √2pi in the denominator. Have I made any errors or is it fine if I take it within the constant A.. In...

I need to know if I have solved the following problem well: A spin-less particle of mass m is confined to move on the surface of a cylinder of infinite height with a harmonic potential on the z-axis and Hamiltonian ##H=\frac{p_z^2}{2m}+\frac{L_z^2}{2mR^2}+\frac{1}{2}m\omega^2z^2## and I need to...

Can I conceive a countable position basis in Quantum Mechanics? How can I talk about the position basis in the separable Hilbert space?

I have read about several approcahes to bypass some classical restrictions to quantum facts such as the electron being in a torus-like shape to avoid ,the greater than speed of light, rotation paradox . Could you recommend websites , sources or books that give good classical analogy to quantum...

To solve a particle on a sphere problem in quantum mechanics we get the below equation :##\left[\frac{1}{\sin \theta} \frac{d}{d \theta}\left(\sin \theta \frac{d}{d \theta}\right)-\frac{m^{2}}{\sin ^{2} \theta}\right] \Theta(\theta)=-A \Theta(\theta) ## To solve this differential equation, we...

I've tried figuring out commutation relations between ##L_+## and various other operators and ##L^2## could've been A, but ##L_z, L^2## commute. Can someone help me out in figuring how to actually proceed from here?

Let ##|l,m\rangle## be a simultaneous eigenstate of operators ##L^2## and ##L_z## and we want to calculate ##\langle l,m|cos(\theta)|l,m'\rangle## where ##\theta## is the angle ##[0,\pi]##. It is true that in general ##\langle l,m|cos(\theta)|l,m'\rangle=0## ##(1)## for the same ##l## even if...

I am very interested in quantum mechanics/physics and i keep seeing the Heisenberg uncertainty principle and its making me think about other forms of viewing particles. We traditionally use Photons to view something (our eyes), or other forms of radiation/particles, but i know that merely...

I join as a 69 year old retired electrical engineer who is interested in physics. I have particular interest in particle physics and quantum mechanics. I don't expect to provide answers on this forum, but I do intend to ask questions.

We know that both momentum and position can not be known precisely simultaneously. The more precisely momentum is known means position is more uncertain. In fact, as I understand quantum mechanics, position probability never extends to 0% anywhere in the universe (except at infinity) for any...

In non relativistic quantum mechanics, the expectation value of an operator ##\hat{O}## in state ##\psi## is defined as $$<\psi |\hat{O}|\psi>=\int\psi^* \hat{O} \psi dx$$. Since the scalar product in relativistic quantum has been altered into $$|\psi|^2=i\int\left(\psi^*\frac{\partial...

Time indepedendent Schrödinger equation for a system (atom or molecule) consisting of N electrons can be written as (with applying Born - Oppenheimer approximation): $$ [(\sum_{i=1}^N - \frac {h^2} {2m} \nabla _i ^2) + \sum_{i=1}^N V(r_i) + \sum_{i < j}^N U(r_i,r_j)] \Psi = E \Psi $$ Terms in...

Is there a view in quantummechanics, of quantummechanics, without time as a concept?

I heard something today about the "informational interpretation" of quantum mechanics and a phrase used was "it from bit." Is there actually such a thing? What does it mean, and how is it distinguished from other interpretations like MWI or Copenhagen?

Hello, I am trying to figure out the right way to approach this. First of all, other than the different Bohr radius value, does the change to a negative pion make any other difference to calculating the probability? Also what would be the correct way to apply the "small volume"? What I'm...

* The general formula for the magnetic moment of a charge configuration is defined as ##\vec{\mu} = \frac{1}{2} \int \vec{r} \times \vec{J} \,d^3r##* For an electron it's said that the correct equation relating it's spin and magnetic moment is is ##\vec{\mu} =g\frac{q}{2m}\vec{S}## * It's...

Hi Fellas! My first post after a long hiatus from forums. Feeling nostalgia (this is the place where it all began, my fuel for quantum fascination so to speak). I am revisiting the mathematical formulation of quantum mechanics with the dimensional (MLT) perspective. I want to understand what...

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Summary:: I am in the highest level Quantum class at my university- technically considered a grad class. I am an undergrad and need advice on just how to learn it. What study tips? Good Youtubers? Physical simulations? Anything that helped you in quantum mechanics. Hello! I am an undergrad...

Is Modern Quantum Mechanics, 3rd Edition, by J. J. Sakurai and Jim Napolitano a good book to learn quantum mechanics from?

According to the uncertainty principle, when we measure a micro-object with a measuring device, we cannot predict what value the device will show. But if we knew exactly the wave function of this device, together with the wave function of the micro-object, could we exactly predict the result of...

Hi! I want to self study some of quantum mechanics so i need introductory textbook. I've taken courses on linear algebra that covers all "Linear algebra done right" by Sheldon Axler, multivariable calculus, two courses on general physics and the basics of differentials equations. I really like...

Can you please suggest a good introductory statistical and quantum mechanics book which can be self studied. My math background : I've done multivariate calculus, vector calculus, linear algebra ,some complex analysis all at the usual undergraduate level. The books I've self studied thus far...

Please be kind to help.I would be grateful

I recently started studying some quantum mechanics, so far I have been using online resources(like MIT OCW 8.04/8.05, and Tongs notes I think I have reached a stage where I understand the Schrodinger eqn and can solve it for various potentials(including for the H-atom) but I don't know anything...

"B0 is a static magnetic field (produced by a superconducting magnet) that initially causes the protons in the body to align with the field and precess at the larmor frequency along the z axis . From a mathematical perspective this precession around the B0 axis occurs due to the time evolution...

A lot of people say that Quantum Field theory (QFT) an Quantum Mechanics (QM) are equivalent. Yet, I've found others who dispute these claims. Among the counter-arguments (which I admittedly do not have the expertise to pick apart and check their validity in full) are the following: 1) While QFT...

I just finished a new paper, A. Neumaier, Quantum mechanics via quantum tomography, arXiv:2110.05294. (later renamed to) A. Neumaier, Quantum tomography explains quantum mechanics, arXiv:2110.05294. Abstract: Starting from first principles inspired by quantum tomography rather than Born's...

Hello all, So I've been working through the solutions to some simple introductory problems for the Schrodinger Equation like the infinite square well, and I'm trying to make sense of how to think about the phase component. For simplicity's sake, let's start off by assuming we've measured an...

Take a simple case: A system is prepared in state ##\rho_i## at time ##t_0##, and a projective measurement is performed at time ##t_2## with an outcome ##b##. We can retrodict a projective measurement outcome ##a## at time ##t_1## where ##t_0<t_1<t_2##$$p(a|b) =...

for undergraduate students how to explain this?

An axiomatization of classical mechanics such as the one by McKinsey et al. (1) does not contain any reference to humans or experiments, and does not contain the magic (irony!) words of quantum mechanics, i.e. observables and measurements. (1) McKINSEY, J. C. C., et al. “Axiomatic Foundations...

Now from the relevant equations, $$U(t) = \exp(-i \omega \sigma_1 t)$$ which is easy to compute provided the Hamiltonian is diagonalized. Writing ##\sigma_1## in its eigenbasis, we get $$\sigma_1 = \begin{pmatrix} 1 & 0\\ 0 & -1\\ \end{pmatrix} $$ and hence the unitary ##U(t)## becomes...

Summary:: The problem solutions contain a lot of unjustified steps, making them pointless. I am trying to use Griffiths Introduction to Quantum Mechanics. He states that the wave function ##\psi## approaches 0 as x approaches infinity to make normalization work. I can accept that. But then I...

I had two questions in the field of physics: We know that in quantum mechanics there is an electron in a certain distance from the distance to the nucleus as a cloud or a cover. But is motion for the cloud defined by the electron moving around the nucleus? And the main question is, can the...

Here is what I tried. Suppose ##f(\phi)## and ##\lambda## is the eigenfunction and eigenvalue of the given operator. That is, $$\sin\frac{d f}{d\phi} = \lambda f$$ Differentiating once, $$f'' \cos f' = \lambda f' = f'' \sqrt{1-\sin^2f'}$$ $$f''\sqrt{1-\lambda^2 f^2} = \lambda f'$$ I have no...

fidelity for pure state with respect to t=0 is 1. My teacher told me this. But I am not getting this. This is my detailed question the initial state(t=0)##|\psi\rangle=|\alpha\rangle|0\rangle## the final state (t) ##|\chi\rangle= |i\alpha\sin(t)\rangle|cos(t)\alpha\rangle## Fidelity between the...

Hello! If I place a particle with more energy levels (of the order of kT) in a well defined state, in a thermal bath at temperature T, how will the blackbody radiation affect the internal state of the particle i.e. will the distribution be classical or QM? Basically, if I prepare that particle...

I was reading in the Book: Introduction to Quantum Mechanics by David J. Griffiths. In chapter Time-independent Perturbation Theory, Section: Spin -Orbit Coupling. I understood that the spin–orbit coupling in Hydrogen atom arises from the interaction between the electron’s spin magnetic moment...

Robert Lawrence Kuhn: It seems that special relativity suggests time is like gravity and electromagnetism, not built into the absolute fabric of reality like logic and causation. David J Gross: Yes, time is dynamical. The phenomena are dynamical and are labeled by what we call time. Including...