Quantum mechahnics Definition and 190 Threads

Attempt: I'm sure I know how to do this the long way using the definition of stationary states(##\psi_n(x)=\sqrt{\frac {2} {a}} ~~ sin(\frac {n\pi x} {a})## and ##\int_0^{{a/2}} {\frac {2} {a}}(1/5)\left[~ \left(2sin(\frac {\pi x} {a})+i~ sin(\frac {3\pi x} {a})\right)\left( 2sin(\frac {\pi x}...

I need help with part d of this problem. I believe I completed the rest correctly, but am including them for context (a)Show that the hermitian conjugate of the hermitian conjugate of any operator ##\hat A## is itself, i.e. ##(\hat A^\dagger)^\dagger## (b)Consider an arbitrary operator ##\hat...

Hi everyone, was just wondering what people think is a good undergraduate QM book is as opposed to Griffiths. I've read through it, and I have looked and many people say it is good for people who've never been exposed to QM before, but when it comes to solving problems I struggle a lot, and...

⟨Lx⟩=⟨l,m|Lx|l,m⟩=−iℏ⟨l,m|[Ly,Lz]|l,m⟩

I worked out the expectation values of the components of a 1/2 spin particle. However, I'm confused about Griffiths notation for the x and y components. For the x component I got, ## \left< S_x \right> = \frac \hbar 2 (b^*a+a^*b)## which is correct, but Griffiths equates this to ##...

If I calculate ## <\psi^0|\epsilon|\psi^0>## and ## <\psi^0|-\epsilon|\psi^0>## separately and then add, the correction seems to be 0 since ##\epsilon## is a constant perturbation term. SO how should I approach this? And how the Δ is relevant in this calculation?

The wavefunction is Ψ(x,t) ----> Ψ(λx,t) What are the effects on <T> (av Kinetic energy) and V (potential energy) in terms of λ? From ## \frac {h^2}{2m} \frac {\partial^2\psi(x,t)}{\partial x^2} + V(x,t)\psi(x,t)=E\psi(x,t) ## if we replace x by ## \lambda x ## then it becomes ## \frac...

Summary: In the past, physicists talked of the phenomenon of "wave function collapse" very freely, whereas now there seems to be some reservation about it. Why? In reading older popular physics literature, physicists used to talk about "wave function collapse" freely and more often...

I am an undergraduate doing research on QC/QI. My current topic to learn is continuous-time quantum walks, but first I must learn the random quantum walk. That being said, I was wondering if someone could simply explain what a random quantum walk is and then explain how they could be useful with...

I've been reading Griffths' intro to elementary particles and I encountered this symbol that looks similar to "M" called amplitude, which can be calculated by analyzing the Feynman diagram of an interaction. What exactly is it? When I hear amplitude I imagine waves, but not sure what this one's...

If it is the asymptotic behavior of the Airy's function what it's used instead of the function itself: Does it mean that the wkb method is only valid for potentials where the regions where ##E<V## and ##E>V## are "wide"?

Firstly, I don't know in which Picture this equation holds (if I hadn't missed some words in the previous text...). I think it may be the Heisenberg Picture. But if it is, the rest target is to prove $$\frac{i}{\hbar}[H_R+H_{FR},(a^\dagger) ^ma^nO_A]=\langle\frac{d}{dt}((a^\dagger)...

Summary: As hawking radiation is based on quantum fluctuations, can they cancel out each other due to equal probabilities of a particle remaining in or drifting away? I recently learned how hawking radiation actually works. It is based on quantum fluctuations which happen randomly in space...

Hello! I am stuck at the following question: Show that the wave function is an eigenfunction of the Hamiltonian if the two electrons do not interact, where the Hamiltonian is given as; the wave function and given as; and the energy and Born radius are given as: and I used this for ∇...

Suppose that the spacetime is discrete, with only certain positions being possible for any particle. In this case, the probability distributions of particles have nonzero values at the points on which the wavefunction is defined. Do we need randomness in the transitions of particles in such a...

Quantum Electrodynamics (QED) has some observable effects such as the lamb shift, which is mainly caused by the vacuum polarization and the electron self-energy. These effects contribute to the "smearing" of the electron in an unpredictable manner, other than the uncertainty we already have...

The Uehling potential due to vacuum polarization by virtual electron-positron pairs is said to be the dominant contribution — 205.0073 meV — to the Lamb shift between the 2P1/22P1/2 and 2S1/22S1/2 states of muonic hydrogen. In the Wikipedia page (https://en.wikipedia.org/wiki/Lamb_shift), it is...

Look at the paper in the link below: https://link.springer.com/content/pdf/10.1007%2Fs10701-016-0026-7.pdf It introduces a pilot-wave model on a discrete spacetime lattice. However, the pilot-wave model is not deterministic; the motion of quantum particles is described by a |Ψ|^2-distributed...

Hello I am bemused by a sign convention for Holes, My questions are as follow: For an electron inside the 2D Circular Quantum Well. We can write our Hamiltonian as H = 1/2m * ( p - q A)^2 + 1/2 m w^2 r^2 (Should we use minus in the momentum term? I think for Holes, it is) If we expand this...

I want to know the differences between Newtonian Gravity and Quantum Gravity

Homework Statement Given the expression s_{\pm}|s,m> = \hbar \sqrt{s(s+1)-m(m\pm 1)}|s,m \pm 1> obtain the matrix representations of s+/- for spin 1/2 in the usual basis of eigenstates of sz Homework Equations s_{\pm}|s,m> = \hbar \sqrt{s(s+1)-m(m\pm 1)}|s,m \pm 1> S_{+} = \hbar...

Homework Statement Given: Ψ and Φ are orthonormal find (Ψ + Φ)^2 Homework Equations None The Attempt at a Solution Since they are orthonormal functions then can i do this? (Ψ + Φ) = (Ψ + Φ)(Ψ* + Φ*)?

Looking into the infinitesimal view of rotations from Lie, I noticed that the vector cross product can be written in terms of the generators of the rotation group SO(3). For example: $$\vec{\mathbf{A}} \times \vec{\mathbf{B}} = (A^T \cdot J_x \cdot B) \>\> \hat{i} + (A^T \cdot J_y \cdot B)...

Homework Statement Using the Schrödinger equation find the parameter \alpha of the Harmonic Oscillator solution \Psi(x)=A x e^{-\alpha x^2} Homework Equations -\frac{\hbar^2}{2m}\,\frac{\partial^2 \Psi(x)}{\partial x^2} + \frac{m \omega^2 x^2}{2}\Psi(x)=E\Psi(x) E=\hbar\omega(n+\frac{1}{2})...

Homework Statement My doubts are on c) Homework Equations $$< H > = \int \Psi^* \hat H \Psi dx = \frac{2}{a} \int_{0}^{a} sin (x\frac{\pi}{a}) \hat H sin (x\frac{\pi}{a}) dx$$ The Attempt at a Solution I understand that mathematically the following equation yields (which is the right...

Hi! When calculating ##(\hat{a} \hat{a}^{\dagger})^2## i get ##\hat{a} \hat{a} \hat{a}^{\dagger} \hat{a}^{\dagger}## which is perfectly fine. But how do I end up with the ultimate simplified expression $$\hat{ a}^{\dagger} \hat{a} \hat{a}^{\dagger} \hat{a} + \hat{a}^{\dagger} \hat_{a} +...

Homework Statement Consider an electron in a hydrogen atom in the presence of a constant magnetic field ##B##, which we take to be parallel to the ##z##-axis. Without the magnetic field and ignoring the spin-orbit coupling, the eigenfunctions are labelled by ##\vert n, l, m, m_s \rangle##...

Before you report this, yes I do know there was already another post like this one, but I don't feel like it fully answered the question. Note that I really don't know anything about quantum anything, but I'm trying to do some reading up on "randomness" and the consensus seems to be that this...

Homework Statement So in my problem, there's a given of 3 non interacting fermions in a harmonic well potential. I already got the wavefunction but i have problems in obtaining the ground state energy and its 1st excited state energy for 3 fermions (assuming they are non interacting and...

Homework Statement Let ##U_t = e^{-iHt/\hbar}## be the evolution operator associated with the Hamiltonian ##H##, and let ##P=\vert\phi\rangle\langle \phi\vert## be the projector on some normalized state vector ##\vert \phi\rangle##. Show that $$\underbrace{PU_{t/n}P\dots PU_{t/n}}_{n\text{...

Homework Statement Prove that the Clebsch-Grodan coefficients (in the notation ##\langle j_1j_2m_1m_2|j_1j_2jm\rangle##) for the decomposition of the tensor product of spin ##l## and spin ##1/2## to spin ##l+1/2## are $$\left\langle l,\frac{1}{2},m\mp \frac{1}{2}, \pm \frac{1}{2} \Bigg\vert l...

Homework Statement Let ##\vec{e}\in\mathbb{R}^3## be any unit vector. A spin ##1/2## particle is in state ##|\chi \rangle## for which $$\langle\vec{\sigma}\rangle =\vec{e},$$ where ##\vec{\sigma}## are the Pauli-Matrices. Find the state ##|\chi\rangle## Homework Equations :[/B] are all given...

Using that ##\hat{a} =a = \sqrt{\frac{mw}{2 \hbar}} \hat{x} +\frac{i}{\sqrt{2mw \hbar}} \hat{p}## and ## a \dagger = \sqrt{\frac{mw}{2 \hbar}} \hat{x} -\frac{i}{\sqrt{2mw \hbar}} \hat{p}## We can solve for x in term of the lowering and raising operator. Now, recently I read a derivation of...

Homework Statement A generic state represented by the wave function ##\psi (\vec(x)## can be expanded in the eigenstates with defined angular momentum. Write such an expansion for a plane wave traveling along the z direction with momentum ##p = \hbar k## in terms of unknown coefficients ##c ( k...

Let's consider two observables, H (hamiltonian) and P (momentum). These operators are compatible since [H,P] = 0. Let's look at the easy to prove rule: 1: "If the observables F and G are compatible, that is, if there exists a simultaneous set of eigenfunctions of the operators F and G, then...

Hello 1. Homework Statement The dipole moment of an ammonia molecule is ##d_0=5*10^{-30} C.m##.If we apply a static electric field of ##\mathcal { E }=1*10^{6 }V*m^{-1}## to an ammonia molecule initially in the state ## |ψG⟩## where the nitrogen molecule is considered to be on the left,we make...

Hello! Thanks for your time reading my questions. When I was studying quantum statistical mechanics, I get so confused about the relations between Pauli's exclusion principle and the Fermi-Dirac distributions. 1. The Pauli's exclusion principle says that: Two fermions can't occupy the same...

Homework Statement Consider the operator ##F_a(\hat{X}) =e^{ia \hat{p} / \hbar} \cdot F(\hat{X}) e^{-ia \hat{p} / \hbar}## where a is real. Show that ##\frac{d}{d_a} F_a(\hat{X}) \cdot \psi = F'(x) \psi## evaluated at a=0. And what is the interpretation of the operator e^{i \hat{p_a} /...

Homework Statement Homework Equations ##\hat{P}= -ih d/dx## The Attempt at a Solution To actually obtain ##\psi_{p_0}## I guess one can apply the momentum operator on the spatial wavefunction. If we consider a free particle (V=0) we can easily get obtain ##\psi = e^{\pm i kx}##, where ##k=...

Homework Statement Consider solar neutrinos of energy 1 MeV (EDIT: 10 MeV not 1 MeV) which are formed at the center of the sun in the ##\nu_2## eigenstate. What fraction of it do you expect to arrive at Earth as ##\nu_\mu## and what fraction as ##\nu_\tau##? Assume that it evolves adiabaticaly...

I just recently graduated with a mathematics degree. Lately, I've been very fascinated with quantum mechains and string theory, but when I try to do research I am a little overwhelmed by all the varying topics of advanced mathematics I have to know. Can anyone suggest mathematical topics to...

I have trouble with finding the eigenstates of a spherical pendulum (length $l$, mass $m$) under the small angle approximation. My intuition is that the final result should be some sort of combinations of a harmonic oscillator in $\theta$ and a free particle in $\phi$, but it's not obvious to...

I was recently reading about annihilation and creation operators in particle physics using the model of an harmonic oscillator, and then quantizing it. This is fine. I can understand it. But how does this quantization of the energy of the harmonic oscillator manifest physically? Is it that only...

Homework Statement I want to prove that ##\frac{\partial \langle x \rangle}{\partial t} = \frac{\langle p_x \rangle}{m}##. Homework Equations $$i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + V \Psi$$ The Attempt at a Solution [/B] So...

Hi guys, something has been bugging me for a while now and I thought I’d just ask it here in the hope someone can explain it to me. Ever since Elon Musk brought it up, I’ve been thinking about the simulation theory (I know it’s not his original idea, it’s just the event that brought it to my...

Homework Statement "Consider a system with three states, ##|1\rangle , |2\rangle ,|3\rangle ## with energies ##\hbar \omega_1 , \hbar \omega_2 , \hbar \omega_3 ##. the states are then separated by ##\hbar \omega_3 -\hbar \omega_1 = \hbar \omega_{13}## and ## \hbar \omega_3-\hbar \omega_2= \hbar...

Homework Statement The spectrum shows the series 3p - nd, n = 4 - 7 in Na as well as the resonance line 3s - 3p, with the experimental vacuum wavelengths in Å.Calculate the quantum defect for the nd ##^2D## n = 4-7 terms. Estimate, as accurately as possible, the wavelength for 3p - 8d. The...

Homework Statement We are investigating hydrogen in a plasma with the temperature 4500 ºC. Calculate the probability per atom and second for stimulated emission from 2p to 1s if the lifetime of 2p is 1.6 ns Homework Equations ##A=\frac{1}{\Sigma \tau}## $$A_{2,1} = \frac{8*\pi *h *...

Okey so I think this question or a similar one was here recently but I can't find it so creating a new. Homework Statement The figure below shows the hyperfine structure in the transition 6s ##^2S_{1/2} - 8p ^2P_{3/2}## in 115In (I = 9/2). The measurement is made using a narrow-band tunable...

Homework Statement The Hamiltonian is given below: ##H=\frac {\mathbf p^2}{2m} -\frac {\partial^2_z}{2m} +V(z) +\gamma V'(z)(\hat{\mathbf z} \times \mathbf p)\cdot \vec{\sigma}## Where this term ## \gamma V'(z)(\hat{\mathbf z} \times \mathbf p)\cdot \vec{\sigma}## represents the spin orbit...