Hamiltonian Definition and 833 Threads

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...

Someone says we can choose the new eigenstate: exp(-iλx/hbar)*ψ,and let the momentum operator p acts upon this new state. At the same time, so does p^2. Something miraculous will happen afterwards. My question is: how to image this point? Thank you very much.

In the boxed equation, how would you get the right hand side from the left hand side? We know that ##H(1,2) = H(2,1)##, but we first have to apply ##H(1,2)## to ##\psi(1,2)##, and then we would apply ##\hat{P}_{12}##; the result would not be ##H(2,1) \psi(2,1)##. ##\hat{P}_{12}## is the exchange...

In a quantum mechanical exercise, I found the following Hamiltonian: Consider a particle of spin 1 constrained to move on the surface of a sphere of radius R with Hamiltonian ##H=\frac{\omega}{\hbar}L^2##. I knew that the Hamiltonian of a particle bound to move on the surface of a sphere was...

To calculate the Hamiltonian of a charged particle immersed in an electromagnetic field, one calculates the Lagrangian with Euler's equation obtaining ##L=\frac{1}{2}mv^2-e\phi+e\vec{v}\cdot\vec{A}## where ##\phi## is the scalar potential and ##\vec{A}## the vector potential, and then we go to...

We have commutation relation ##[J_j,J_k]=i \epsilon_{jkl}J_l## satisfied for ##2x2##, ##3x3##, ##4x4## matrices. Are in all dimensions these matrices generate ##SO(3)## group? I am confused because I think that maybe for ##4x4## matrices they will generate ##SO(4)## group. For instance for...

I've just started Quantum mechanics by McIntyre and have understood the following about operators which the author wrote till chapter 2: Each observable has an operator Operators act on kets to produce another kets. Only eigenvalues of an operator are possible values of a measurement. Now...

Is that possible to derive the Navier-Stokes equations with Lagrangian and Hamiltonian methods? If yes, how? and if it is not possible, why?

Hi, I was attempting a question about Hamiltonian systems from dynamic systems and wanted to ask a question that arose from it. Homework Question: Given the system below: \dot x_1 = x_2 \dot x_2 = x_1 - x_1 ^4 (a) Prove that the system is a Hamiltonian function and find the potential...

This isn't technically a homework problem, but I'm trying to check my understanding of the geometric phase by explicitly calculating the Berry connection for a simple 2x2 Hamiltonian that is not a textbook example of a spin-1/2 particle in a three dimensional magnetic field solved via a Bloch...

What I did was first noting that ##\hat{\vec{S}}_1\cdot\hat{\vec{S}}_2=\frac{1}{2}(\hat{\vec{S}}^2-\hat{\vec{S}}_1^2-\hat{\vec{S}}_2^2)##, but these operators don't commute with ##\hat{S}_{1_z}## and ##\hat{S}_{2_z}##, this non the decoupled basis ##\ket{s_1,s_2;m_1,m_2}## nor the coupled one...

Some introduction books on Lagrangian and Hamiltonian mechanics use classical mechanics as the theoretical framework, and when it come to special relativity it goes back to the basics and force language again. I would like to ask for some recommendations on good books that introduces Lagrangian...

Hello ! I require some guidance on this prove :I normally derive the Hamiltonian for a SHO in Hilbert space with a term of 1/2 hbar omega included. However, I am unsure of how one derives this from Hilbert space to Fock space. I have attached my attempt at it as an image below. Any input will be...

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...

I'm looking for complete and detailed references on constrained Hamiltonian systems and Dirac brackets. While my main interest is electrodynamics, I would prefer a complete exposition of the theory from the ground up. So far, my knowledge about the topic comes from books in QFT, like Weinberg...

For example if I consider H = (a^†)b+a(b^†), how will it act on even coherent state i.e. |α⟩+|-α⟩?. I know that |α⟩ don't act on (a^†) because |α⟩ is a eigenstate of lowering operator.

> Consider two particle with spin 1/2 interacting via the hamiltonian $H = \frac{A}{\hbar^2}S_{1}.S_{2}$, Where A is a constant. What aare the eigenstates, eigenvalues and its multicplity? $H = \frac{A}{\hbar^2}S_{1}.S_{2} = A\frac{(SS-S_{1}S_{1}-S_{2}S_{2})}{2\hbar^2 } =...

Hello, I have this band structure plot for five band Hamiltonian model. I want to know which bands are valence and which one is conduction bands. Also if they have any special name I like to know that. Thank you.

The Noether's theorem for finite Hamiltonian systems says that: My question is: If I know a symmetry how can I write the first integral?

For instance,how to systematically derive the equns 2.2 & 2.5 given a Hamiltonian on the article below?; arxiv.org/pdf/0904.2771.pdf .

In Classical Mechanics by Kibble and Berkshire, in chapter 12.4 which focuses on symmetries and conservation laws (starting on page 291 here), the authors introduce the concept of a generator function G, where the transformation generated by G is given by (equation 12.29 on page 292 in the text)...

I find a exercise in Leonard Susskind's book Classical Mechanics the Hamiltonian of a charged particle in a magnetic field(ignore the electric field) is $$H=\sum_{i} \left\{ \frac{1}{2m} \left[ p_{i}-\frac{e}{c}A_{i}(x) \right]\left[ p_{i}-\frac{e}{c}A_{i}(x) \right]...

Hello! I am reading some stuff about the effective hamiltonian for a diatomic molecule and I have some questions about relating the parameters of these hamiltonian to experiment and theory. From what I understand, one starts (usually, although not always) with the electronic energy levels, by...

Last week I was discussing with some colleagues how to handle time-dependent Hamiltonians. Concerning this, I would like to ask two questions. Here I go. First question As far as I know, for a time-dependent Hamiltonian ##H(t)## I can find the instantaneous eigenstates from the following...

I am getting that we have to operate the given Hamiltonian on the given state |α>. But what is confusing me is that since this H contains position and momentum operators which just involve variable x and partial derivative, how do I operate this H on the given α, since it seems like α is...

Hello, I'm trying to understand the counter-rotating terms of the Rabi Hamiltonian : ##a^\dagger \sigma_+## and ##a \sigma_-##. I find these terms rather strange, in the sense that naively I would interpret them as describing an electron that gets excited by emitting a photon (and vice...

I was reading a paper on Radical-Pair mechanism (2 atoms with 1 valence electron each) and the author used the hyperfine hamiltonian $$H_{B}=-B(s_{D_z}+s_{A_z})+As_{D_x}I_x+As_{D_y}I_y+as_{D_z}I_z$$ and found the following eigenvalues: a/4 (doubly degenerate) , a/4±B , (-a-2B±2√(A^2+B^2)) ...

We show by working backwards $$\hbar w \Big(a^{\dagger}a+\frac{1}{2}\Big)=\hbar w \Big(\frac{mw}{2\hbar}(\hat{x}+\frac{i}{mw}\hat{p})(\hat{x}-\frac{i}{mw}\hat{p})+\frac{1}{2}\Big)$$...

Given the hamiltonian: \hat{H} = \hbar \omega_{0} \hat{a}^{+}\hat{a} + \chi (\hat{a}^{+}\hat{a})^2, where ##\hat{a}^{+}##, ##\hat{a}## are creation and annihilation operators. Find evolution of the state ##|\psi(t) \rangle##, knowing that initial state ##|\psi(0)\rangle = |\alpha\rangle##...

Dear everybody, Let me ask a question regarding the unitary time evolution of a given Hamiltonian. Let's start by considering a Hamiltonian of the form ##H(t) = H_0 + V(t)##. Then, I move to the interaction picture where the Schrödinger equation is written as $$ i\hbar \frac{d}{dt}...

If Hamiltonian commutes with a parity operator ##Px=-x## are then all eigenstates even or odd? Is it true always or only in one-dimensional case?

He told me I "need to show that the Hamiltonian matrix elements you get by using those states have nonzero elements only on the diagonal." I understand what and how a diagonal matrix works, but what I don't understand is what those states are. Are they states I put in my "quantum mechanical...

Hello! I need some help with this problem. I've solved most of it, but I need some help with the Hamiltonian. I will run through the problem as I've solved it, but it's the Hamiltonian at the end that gives me trouble. To find the Lagrangian, start by finding the x- and y-positions of the...

The hamiltonian ´for a free falling body is $$H = \dfrac{p^2}{2m} + mgy$$ and since we are using cartesian coordinates that do not depend on time and the potential only depends on the position, we know that ##H=E##. For this hamiltonian, using the Hamilton's equations and initial conditions...

What I have tried is a completing square in the Hamiltonian so that $$\hat{H} = \frac{\hat{p}^2}{2} + \frac{(\hat{q}+\alpha(t))^2}{2} - \frac{(\alpha(t))^2}{2}$$ I treat ##t## is just a parameter and then I can construct the eigenfunctions and the energy eigenvalues by just referring to a...

If we can identify ##|c_n|^2## as the probability of having an energy ##E_n##, then that equation is just the bog standard one for expectation. But the book has not proved this yet, so I assumed it wants a derivation from the start. I tried $$ \begin{align*} \Psi(x,t) = \sum_n c_n...

I know that in the case of central potential V(r) the hamiltonian of the system always commutes with l^2 operator. But what happends in this case?

The issue here is that I don't know how to operate the final equations in order to get the phase diagram. I suppose some things are held constant so I can get a known curve such as an ellipse. I attach the solved part, I don't know how to go on.

Homework Statement:: ... Relevant Equations:: . What is the minimum mathematic requirement to the Lagrangian and hamiltonian mechanics? Maybe calc 3 and linear algebra?

Hello everybody, new here. Sorry in advance if I didn't follow a specific guideline to ask this. Anyways, I've got as a homework assignment two cannonical transformations (q,p)-->(Q,P). I have to obtain the hamiltonian of a harmonic oscillator, and then the new coordinates and the hamiltonian...

Once I know the Hamiltonian, I know to take the determinant ##\left| \vec H-\lambda \vec I \right| = 0 ## and solve for ##\lambda## which are the eigenvalues/eigenenergies. My problem is, I'm unsure how to formulate the Hamiltonian. Is my potential ##U(r)## my scalar field ##\phi##? I've seen...

Here's the problem and the solution provided online by the author (the problem numbers are different but it's the same question). I think I'm okay up until the last step where he declares the Hamiltonian is (1 1 1 -1). Where did he get those components?

Given the following $$L(\theta,\dot{\theta},\phi,\dot{\phi}) = \frac12ml^2((\dot{\theta})^2 + (sin(\theta)^2)\dot{\phi}^2) + k\theta^4$$ This is my attempt: I am not understanding if the conserved quantities (like angular momentum about the z-axis) impacts my formulation of the Hamiltonian or...

Let a mass m charged with q, attached to a spring with constant factor k = mω ^2 in an electric field E(t) = E0(t/τ) x since t=0. (Equilibrium position is x0 and the deformation obeys ξ = x - x0) What would the hamiltonian and motion equations be in t ≥ 0, in terms of m and ω?? Despise magnetic...

I am attaching an image from David J. Griffith's "Introduction to Quantum Mechanics; Second Edition" page 205. In the scenario described (the Hamiltonian treats the two particles identically) it follows that $$PH = H, HP = H$$ and so $$HP=PH.$$ My question is: what are the necessary and...

Is Hamiltonian mechanics a mathematical generalization of Newtonian mechanics or is it explaining some fundamental relationship that has a meaning that extends into our nature ? I guess my question is what would led William Rowan Hamilton to come up with his type of mechanics or anything...

Hello! This is probably a silly question (I am sure I am missing something basic), but I am not sure I understand how a Hamiltonian can be a scalar and allow transitions between states with different angular momentum at the same time. Electromagnetic induced transitions are usually represented...

Hi, I have a question which asks me to use the generalised Ehrenfest Theorem to find expressions for ##\frac {d<Sx>} {dt}## and ##\frac {d<Sy>} {dt}## - I have worked out <Sx> and <Sy> earlier in the question. Since the generalised Ehrenfest Theorem takes the form...

In an Introduction to Quantum Mechanics by Griffiths (pg. 180), he claims that "P and H are compatible observables, and hence we can find a complete set of functions that are simultaneous eigenstates of both. That is to say, we can find solutions to the Schrodinger equation that are either...