Hamiltonian Definition and 833 Threads

Homework Statement Assuming psi is an eigenstate of the Hamiltonian (TISE) and that E=0, determine the potential V(x) appearing in the Hamiltonian. Homework Equations Time Independant Schrodinger Equation - 1 Dimensional (x) I am given the wavefunction psi = N/(1+x^2) I have found the...

Hello, I'm a second year physics student. We are going to use "hand and finch analytical mechanics", however the reviews I saw about this book are bad. I've already taken calculus for mathematicians, linear algebra, classical mechanics, special relativity, and electromagnetism. The topics it...

Suppose I am given some 1D Hamiltonian: H = ħ2/2m d2/dx2 + V(x) (1) Which I want to solve on the interval [0,L]. I think most of you are familiar with the standard approach of discretizing the interval [0,L] in N pieces and using the finite difference formulas for V and the...

This book should introduce me to Lagrangian and Hamiltonian Mechanics and slowly teach me how to do problems. I know about Goldstein's Classical Mechanics, but don't know how do I approach the book.

In literature I have read it is said that the Hamiltonian ##H## is the generator of time translations. Why is this the case? Where does this statement derive from? Does it follow from the observation that, for a given function ##F(q,p)##, $$\frac{dF}{dt}=\lbrace F,H\rbrace +\frac{\partial...

Homework Statement I have the matrix form of the Hamiltonian: H = ( 1 2-i 2+i 3) If in the t=0, system is in the state a = (1 0)T, what is Ψ(x,t)? Homework Equations Eigenvalue equation The Attempt at a Solution So, I have diagonalized given matrix and got...

Recently I have been asked to solve the problem of an electron in a Zeeman-field that couples the spin of the electron to the magnetic field. I am not sure how to correctly set up the problem. I think, however, that what I have done on the picture is correct. The usual p^2/2m + V term in the...

Suppose the initial radial position and radial velocity of the bead are ##r_0>0## and ##0## respectively. Then ##E## is negative. Is there any significance to the negative value of ##E##? Note that ##E## is defined by (5.52) and given by (5.144) below.

Dear all, The Hamiltonian for a particle in a magnetic field can be written as $$\hat{H} = \frac{1}{2}g\mu_B\textbf{B}\cdot\boldsymbol\sigma$$ where ##\boldsymbol\sigma## are the Pauli matrices. This Hamiltonian is written in the basis of the eigenstates of ##\sigma_z##, but how is it...

Homework Statement A spin-1/2 electron in a magnetic field can be regarded as a qubit with Hamiltonian $$\hat{H} = \frac{1}{2}g\mu_B\textbf{B}\cdot\boldsymbol\sigma$$. This matrix can be written in the form of a qubit matrix $$ \begin{pmatrix} \frac{1}{2}\epsilon & t\\ t^* &...

Dear all, The Hamiltonian for a spin-orbit coupling is given by: \mathcal{H}_1 = -\frac{\hbar^2\nabla^2}{2m}+\frac{\alpha}{2i}(\boldsymbol \sigma \cdot \nabla + \nabla \cdot \boldsymbol \sigma) Where \boldsymbol \sigma = (\sigma_x, \sigma_y, \sigma_z) are the Pauli-matrices. I have to...

Only thing I know about them is that they are alternate mechanical systems to bypass the Newtonian concept of a "force". How do they achieve this? Why haven't they replaced Newtonian mechanics, if they somehow "invalidate" it or make it less accurate, by the Occam's razor principle? Thanks in...

Homework Statement Consider a two-state system with a Hamiltonian defined as \begin{bmatrix} E_1 &0 \\ 0 & E_2 \end{bmatrix} Another observable, ##A##, is given (in the same basis) by \begin{bmatrix} 0 &a \\ a & 0 \end{bmatrix} where ##a\in\mathbb{R}^+##. The initial state of the system...

Hello, I'm stuck with this exercise, so I hope anyone can help me. It is to prove, that the density of states of an unknown, quantum mechanical Hamiltonian ##\mathcal{H}##, which is defined by $$\Omega(E)=\mathrm{Tr}\left[\delta(E1\!\!1-\boldsymbol{H})\right]$$ is also representable as...

I have a basic understanding of the reason why we look for derivative or integration in Physics, based on the water flow example, where integration is the process of accumulating the varying water flow rate "2x" , while we reverse to the water flow rate by differentiating " x squared " the...

Hello everyone Homework Statement I have been given the testfunction \phi(\alpha, r)=\sqrt{(\frac{\alpha^3}{\pi})}exp(-\alpha r) , and the potential V(r,\theta, \phi)=V(r)=-\frac{e^2}{r}exp(\frac{-r}{a}) Given that I have to write down the hamiltonian (in spherical coordinates I assume), and...

Hi, I have been trying to get my head around the effect of a time reversed hamiltonian ##H^B(t)=H(-t)=T^{-1}H^F T ## on a state ket ##|\psi>##, where ##H^F=H## is the regular hamiltonian for the system (energy associated with forward time translation) and ##H^B=H(-t)## is the time reversed...

I am using Jose & Saletan's "Classical Dynamics", where they introduce a rather contrived Hamiltonian in the problem set: H(q_1,p_1,q_2,p_2) = q_1p_1-q_2p_2 - aq_1^2 + bq_2^2 where a and b are constants. This Hamiltonian has several constants-of-motion, including f = q1q2, as can be easily...

Homework Statement The Hamiltonian H for a certain physical quantum mechanical system has three eigenvectors {|v1>, |v2>, |v3>} satisfying: H|vj> = (2-j)a|vj> Write down the matrix representing H in the representation {|v1>, |v2>, |v3>} . Homework EquationsThe Attempt at a Solution I though...

Homework Statement We have a mas m attached to a vertical spring of length (l+x) where l is the natural length. Homework Equations Find the Lagrangian and the hamiltonian of the system if it moves like a pendulum The Attempt at a Solution we know that the lagrangian of a system is defined as...

Homework Statement For the system: \frac{dx}{dt}=x\cos{xy} \: \: \frac{dy}{dt}=-y\cos{xy} (a) is Hamiltonian with the function: H(x,y)=\sin{xy} (b) Sketch the level sets of H, and (c) sketch the phase portrait of the system. Include a description of all equilibrium points and any saddle...

Good day everyone, The question is as following: Consider an electron gas with Hamiltonian: \mathcal{H} = -\frac{\hbar^2 \nabla^2}{2m} + \alpha (\boldsymbol{\sigma} \cdot \nabla) where α parameterizes a model spin-orbit interaction. Compute the eigenvalues and eigenvectors of wave vector k...

Given 1A.1 and 1A.2, I have been trying to apply the Schrödinger equation to reproduce 1A.3 and 1A.4 but have been struggling a bit. I was under the assumption that by applying ##\hat{W} \rvert {\psi} \rangle= i\hbar \frac {d}{dt} \rvert{\psi} \rangle## and then taking ##\langle{k'} \lvert...

System is composed of two qubits and the bath is one bath qubit. The interaction Hamiltonian is: $$\sigma_1^x\otimes B_1 + \sigma_2^x\otimes B_2$$ where $$B_i$$ is a 2 by 2 matrix. I try to interpret and understand this, is it the same as: $$(\sigma_1^x\otimes B_1)\otimes I_2 +...

A system consisting of two spins is described by the Hamiltonian (b>0) H = aσ1 ⋅ σ2 + b(σ1z - σ2z) where a and b are constants. (a) Is the total spin S = ½ (σ1 + σ2) conserved? Which components of S, if any, are conserved? (b) Find the eigenvalues of H and the corresponding...

Homework Statement Im trying to understand the Legendre transform from Lagrange to Hamiltonian but I don't get it. This pdf was good but when compared to wolfram alphas example they're slightly different even when accounting for variables. I think one of them is wrong. I trust wolfram over the...

I'm working on some classical mechanics and just got a question stated: Is the Hamiltonian for this system conserved? Is it the total energy? In my problem it was indeed the total energy and it was conserved but it got me thinking, isn't the Hamiltonian always the total energy of a system...

The Klein-Gordon field ##\phi(\vec{x})## and its conjugate momentum ##\pi(\vec{x})## is given, in the Schrodinger picture, by ##\phi(\vec{x})=\int \frac{d^{3}p}{(2\pi)^{3}}...

Homework Statement The Lagrangian density for a massive vector field ##C_{\mu}## is given by ##\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m^{2}C_{\mu}C^{\mu}## where ##F_{\mu\nu}=\partial_{\mu}C_{\nu}-\partial_{\nu}C_{\mu}##. Derive the equations of motion and show that when ##m...

What are Hamiltonian/Lagrangian Mechanics and how are they different from Newtonian? What are the benefits to studying them and at what year do they generally teach you this at a university? What are the maths required for learning them?

Homework Statement We are given a paramagnetic system of N distinguishable particles with 1/2 spin where we use N variables s_k each binary with possible values of ±1 where the total energy of the system is known as: \epsilon(s) = -\mu H \sum_{k=1}^{N} s_k where \mu is the magnetic moment...

H=p^2/2m+c What's c? It's of course a shift in energy, but can be thought also as a smoother and smoother real-space local potential that becomes a constant all over the space. On the other hand, why couldn't one think about it as a constant potential in reciprocal space? It's a shift in energy...

In the Hamilonian for an H2+, the kinetic energy of the electron (KE of nucleus ignored due to born-oppenheimer approximation) has a negative sign in front of it. I understand the signs for the potential energy operators but not for the KE apart from the strictly mathematical point of view. Can...

Suppose we have an electron in a hydrogen atom that satisfies the time-independent Schrodinger equation: $$-\frac{\hbar ^{2}}{2m}\nabla ^{2}\psi - \frac{e^{2}}{4\pi \epsilon_{0}r}\psi = E\psi$$ How can it be that the Hamiltonian is spherically-symmetric when the energy eigenstate isn't? I was...

Homework Statement This problem is from Zetelli 3.21 http://imgur.com/wYTNVwz http://imgur.com/wYTNVwz Homework Equations Just the standard probability via product between the eigenfunction and the wavefunction The Attempt at a Solution I've found the eigenvectors for the Hamiltonian...

I came across a previous exam question which stated: Do all physical states, ψ, abide to Hψ = Eψ. I thought about it for a while, but I'm not really sure.

Hello, This was part of my midterm exam that i couldn't solve. Any help is extremely appreciated. Problem: The K.E. of a rotating top is given as L^2/2I where L is its angular momentum and I is its moment of inertia. Consider a charged top placed at a constant magnetic field. Assume that the...

I'm a little confused about the hamiltonian. Once you have the hamiltonian how can you find conserved quantities. I understand that if it has no explicit dependence on time then the hamiltonian itself is conserved, but how would you get specific conservation laws from this? Many thanks

For each of the four fundamental forces (or fields), must one always specify the Lagrangian and Hamiltonian? What else must one specify for other fields (like the Higgs Fields)?

Hey, here's a quick question: What is the Hamiltonian operator corresponding to a decaying Carbon-14 atom. Any insight is quite appreciated!

Homework Statement [/B] Particle is moving in 2D harmonic potential with Hamiltonian: H_0 = \frac{1}{2m} (p_x^2+p_y^2)+ \frac{1}{2}m \omega^2 (x^2+4y^2) a) Find eigenvalues, eigenfunctions and degeneracy of ground, first and second excited state. b) How does \Delta H = \lambda x^2y split...

Homework Statement [/B] Particle in one dimensional box, with potential ##V(x) = 0 , 0 \leq x \leq L## and infinity outside. ##\psi (x,t) = \frac{1}{\sqrt{8}} (\sqrt{5} \psi_1 (x,t) + i \sqrt{3} \psi_3 (x,t))## Calculate the expectation value of the Hamilton operator ##\hat{H}## . Compare it...

I'm reading about stationary states in QM and the following line, when discussing the time-independent, one-dimensional, non-relativist Schrodinger eqn, normalization or the lack thereof, and the Hamiltonian, this is mentioned: "In the spectrum of a Hamiltonian, localized energy eigenstates are...

suppose that the momentum operator \hat p is acting on a momentum eigenstate | p \rangle such that we have the eigenvalue equation \hat p | p \rangle = p| p \rangle Now let's project \langle x | on the equation above and use the completeness relation \int | x\rangle \langle x | dx =\hat I we...

Hello everybody, As I mentioned in the title, it is about molecular symmetry and its Hamiltonian. My question is simple: For any molecule that belong to a precise point symmetry group. Is the Hamiltonian of this molecule commute with all the symmetry element of its point symmetry group...

Hi everyone, I need help for preparing a Hamiltonian matrix. What will be the elements of the hamiltonian matrix of the following Schrodinger equation (for two electrons in a 1D infinite well): -\frac{ħ^{2}}{2m}(\frac{d^{2}ψ(x_1,x_2)}{dx_1^{2}}+\frac{d^{2}ψ(x_1,x_2)}{dx_2^{2}}) +...

Homework Statement I am trying to get the hamiltonain operator equality for a rigid rotor. But I don't get it. Please see the red text in the bottom for my direct problem. The rest is just the derivation I used from classical mechanics. Homework Equations By using algebra we obtain: By...

Homework Statement Define n=(x + iy)/(2)½L and ñ=(x - iy)/(2)½L. Also, ∂n = L(∂x - i ∂y)/(2)½ and ∂ñ = L(∂x + i ∂y)/(2)½. with ∂n=∂/∂n, ∂x=∂/∂x, ∂y=∂/∂y, and L being the magnetic length. a=(1/2)ñ+∂n and a†=(1/2)n -∂ñ a and a† are the lowering and raising operators of quantum mechanics. Show...

Homework Statement Homework Equations $$E_n^{(2)}=\sum_{k\neq n}\frac{|H_{kn}'|^2}{E_n^{(0)}-e_k^{(0)}}$$ The Attempt at a Solution Not sure where to start here. The question doesn't give any information about the unperturbed Hamiltonian. Some guidance on the direction would be great...

Homework Statement Point transformation in a system with 2 degrees of freedom is: $$Q_1=q_1^2\\Q_2=q_q+q_2$$ a) find the most general $P_1$ and $P_2$ such that overall transformation is canonical b) Show that for some $P_1$ and $P_2$ the hamiltonain...