Hamiltonian Definition and 833 Threads

Consider the complete graph with 5 vertices, denoted by K5. E.) Does K5 contain Hamiltonian circuits? If yes, draw them. I know that a Hamiltonian circuit is a graph cycle through a graph that visits each node exactly once. However, the trivial graph on a single node is considered to possesses...

OK. An example I have has me stumped temporarily. I'm tired. General spin matrix can be written as Sn(hat) = hbar/2 [cosθ e-i∅sinθ] ...... [[ei∅sinθ cosθ] giving 2 eigenvectors (note these are column matrices) I up arrow > = [cos (θ/2)] .....[ei∅sin(θ/2)] Idown arrow> =...

(inspired partially by this blog post: http://johncarlosbaez.wordpress.com/2012/01/19/classical-mechanics-versus-thermodynamics-part-1/ To my understanding, the thermodynamic configuration space has a nice symplectic structure. For example, using the language of classical mechanics, starting...

How do I write the non-approximated Schrodinger equation Hamiltonian for a mixture containing 25% by partial pressure of H2 gas and 75% by partial pressure of He gas, at 100 KPa pressure and 298 K?

Homework Statement Consider the following Hamiltonian H=\frac{p^2}{2m}e^{\frac{-q}{a}} a: constant m: mass of the particle q corresponds to the coordinate, and p its momentum. note: q' stands for the derivative of q. a) Prove that for p(t) > 0 this system seems to describe a particle...

In most of textbooks, the canonical quantization procedure is used to quantize the hamiltonian with a simple form, the quadratic form. I just wonder how should we deal with more complex form hamiltonian, such like the ones including interaction terms?

Hi! I am trying to understand the statistical mechanics derivation of the ideal gas law shown at: http://en.wikipedia.org/wiki/Ideal_gas_law inder "Derivations". First of all, the statement "Then the time average momentum of the particle is: \langle \mathbf{q} \cdot \mathbf{F} \rangle=...

Homework Statement At what time does the particle reach infinity given that H(p,x)=(1/2)p^2 -(1/2)x^4. And initial values are x(0)=1 and p(0)=1 Homework EquationsThe hamiltonian equations i believe are given by the partial derivatives let d mean partial derivative so x'=dH/dp and p'=-dH/dx...

The regular spin orbit Hamiltonian is H_{SO} = \frac{q\hbar}{4 m^2 c^2}\sigma\cdot(\textbf{E}\times \textbf{p}) If I consider a 2D system where E = E(x,y) and p is treated as an operator, i.e. \hat{p} = \hat{i}p_x + \hat{j}p_y then, clearly E and p do not commute, so this doesn't look like...

Homework Statement Assume a Hilbert space with the basis vectors \left| 1 \right\rangle, \left| 2 \right\rangle and \left| 3 \right\rangle, and a Hamiltonian, which is described by the chosen basis as: H=\hbar J\left( \begin{matrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\...

Could hamiltonian of linear harmonic oscilator be written in the form? ##\hat{H}=\sum^{\infty}_{n=0}(n+\frac{1}{2})\hbar\omega |n\rangle \langle n| ##

help with electron evolution governed by Hamiltonian ,,, Homework Statement an electron evolution governed by Hamiltonian H=(p^2) /2m +(1/(4Piε))* (e^2)/(r1-r2) give an energy approximation and what's the physical interpretation of the such a Hamiltonian Homework Equations The...

I know in ordinary QM, the spectrum of the Hamiltonian \{ E_{n}\} gives you just about everything you need for the system in question (roughly speaking). So what happens to this spectrum in QFT where |\psi\rangle is now a multiparticle wavefunction in some Fock space? I've been trying to...

Hi I am looking for the MOST GENERAL statement that determines, when the Hamilton function and the energy are equal to each other in classical mechanics.

Homework Statement I have an assignment for my thesis to make Hamiltonian for Schrodinger equation. I won't go into physics part of it, because that is well understood. I need to somehow generate a specific matrix for Hamiltonian (H). Please see the attached file of what I need to get...

Homework Statement Let G be a graph. 1. Let n be a natural number. Use induction to show for all n >= 2 Kn has a Hamiltonian path. 2. Explain how you could use the proof from #1 to show that for all n (natural number) n > 2 Kn has a Hamiltonian cycle. Homework Equations The...

Homework Statement A quantum particle, i.e. a particle obeying Schrodinger equation and moving in one dimension experiences a potential ˆV (x). In a stationary state of this system show that ⟨x∂/∂x(ˆV(x)⟩ = ⟨ˆp2/2m⟩ Hint: Consider the time dependence of ⟨ˆxˆp⟩. Homework Equations...

Homework Statement A particle of mass m and charge q is subject to a uniform electrostatic eld ~E . (a) Write down the Hamiltonian of the particle in this system (Hint: consider the potential energy of an electric dipole); (b) Find the Heisenberg equation of motion for the...

Hi PF members, I am stuck with a problem about larmor precession. I cannot find the eigenstates of the hamiltonian given as H = \frac{\hbar}{2}\begin{pmatrix} \omega_{0} & \omega_{1}\delta(t-t') \\ \omega_{1}\delta(t-t') & \omega_{0} \end{pmatrix} Can anyone help me? Since it has time...

Hello, If we look at a system of two two-level atoms interacting with light, most papers start with a Hamiltonian H_{int}=(\sigma_{1}^{+}+\sigma_{2}^{+})a_{\textbf{k},\lambda} + h.c. That is, we absorb a photon and lost one excitation in the atoms or vice versa. Why do we never...

Hi guys, this is my first time posting, I'm studying physics at uni, in my third year and things are getting a bit tough, so basically my question is in relation to solving problem 1, (i included a picture...) I missed the class and don't really know what I'm doing. Any help would be appreciated.

Suppose I'm considering particles of mass \mu_i, 1 \leq i \leq 3, located at positions r_i. Suppose I ignore the potential between \mu_1 and \mu_2. Then the Hamiltonian I'd write down would be H = -\frac{1}{2\mu_1}\Delta_1 -\frac{1}{2\mu_2}\Delta_2 - \frac{1}{2\mu_3}\Delta_3 + V_1(r_3 -...

Hi can anyone explain how to derive an expression for the Dirac Hamiltonian, I thought the procedure was to use \mathcal{H}= i\psi^{\dagger}\Pi -\mathcal{L}, but in these papers the have derived two different forms of the Dirac equation H=\int d^{3}x...

Hey there, the question I'm working on is written below:- Let |a'> and |a''> be eigenstates of a Hermitian operator A with eigenvalues a' and a'' respectively. (a'≠a'') The Hamiltonian operator is given by: H = |a'>∂<a''| + |a''>∂<a'| where ∂ is just a real number. Write down the eigenstates...

I understand how to use Hamiltonian mechanics, but I never understood why you construct the Hamilitonian by first constructing the Langrangian, and then performing a Legendre transform on it. Why can't you just construct the Hamiltonian directly? Does it have to do with the generalized...

Hi. I hope this is in the right spot - I am not a physics major so not sure if it qualifies as classical, quantum, or other type of physics). I am asking the following to check the calculations of my graduate math thesis I am simulating a one dimensional chain of masses and linear springs...

Homework Statement The hamiltonian of a simple anti-ferromagnetic dimer is given by H=JS(1)\bulletS(2)-μB(Sz(1)+Sz(2)) find the eigenvalues and eigenvectors of H. Homework Equations The Attempt at a Solution The professor gave the hint that the eigenstates are of...

Please, help me with this problem! Two distinguishable particles of spin 1/2 interact with Hamiltonian H=A*S1,z*S2,x with A a positive constant. S1,z and S2,x are the operators related to the z-component of the spin of the first particle and to the x-component of the spin of the second...

I often see the EM Hamiltonian written as $$H=\frac1{2m}\left(\vec p-\frac ec\vec A\right)^2+e\phi,$$ but this confuses me because it doesn't seem to have the right units. Shouldn't it just be $$H=\frac1{2m}\left(\vec p-e\vec A\right)^2+e\phi,$$ since the vector potential has units of momentum...

Homework Statement The Hamiltonian for two particles with angular momentum j_1 and j_2 is given by: \hat{H} = \epsilon [ \hat{\bf{j}}_1 \times \hat{\bf{j}}_2 ]^2, where \epsilon is a constant. Show that the Hamiltonian is a Hermitian scalar and find the energy spectrum.Homework Equations...

Homework Statement Find the eigenvalues of the following and the eigenvelctor which corresponds to the smallest eigenvalue Homework Equations I know how to find the eigenvalues and eigenvectors of a 2x2 matric but this one I'm not so sure so any help would be appreciated The...

Suppose G is a HC (Hamiltonian-connected) graph on n >= 4 vertices. Show that connectivity of G is 3. I tried starting by saying that there would be at least 4C2=6 unique hamiltonian paths. But then I'm not sure where to go from here. Any hints would be appreciated.

In the QM Hamiltonian, I keep seeing h-bar/2m instead of p/2m for the kinetic energy term. H-bar is not equivalent to momentum. What am I missing here?

I'm a little confused about why the Lagrangian is Lorentz invariant and the Hamiltonian is not. I keep reading that the Lagrangian is "obviously" Lorentz invariant because it's a scalar, but isn't the Hamiltonian a scalar also? I've been thinking this issue must be somewhat more complex...

When I write down the Hamiltonian for the hydrogen atom why do we not include a radiation term or a radiation reaction term? If I had an electron moving in a B field it seems like I would need to have these terms included.

I'm watching a lecture on the Hamiltonian and can't figure out something. Here it is. Take a generic function G, and differentiate it with respect to p and q. What you get is the partial of G with respect to p TIMES the derivative of p (or p-dot), plus the derivative of G with respect to q...

Homework Statement The JCM has the Hamiltonian: \hat{H} = \hbar \omega \left(\hat{a}\hat{a}^{*} + 1/2 \right) + \frac{\hbar\omega_{0}\hat{\sigma}_{z}}{2} + \hbar g (\hat{\sigma}_{+}\hat{a} + \hat{\sigma}_{-}\hat{a}^{*} Find the eigenstates and energy eigenvalues in this non-resonant case...

given the Schroedinger equation with an exponential potential -D^{2}y(x)+ae^{bx}y(x)-E_{n}y(x)= 0 with the boudnary conditons y(0)=0=y(\infty) is this solvable ?? what would be the energies and eigenfunction ? thanks.

To cut to the chase, I have to solve for the evolution of a two-state system where the system's state at time t satisfies the equation \mathrm{i}\hbar\left( \begin{array}{cc} \dot{c}_1(t)\\ \dot{c}_2(t) \end{array} \right)=\left( \begin{array}{cc} 0 & \gamma...

Homework Statement If I have a Hamiltonian matrix, \mathcal{H}, that only depends on a kinetic energy operator, do the energy eigenvalues have to be non-negative? I have an \mathcal{H} like this, and some of its eigenvalues are negative, so I was wondering if they have any physical...

Hi, I just started self studying solid state and I'm having trouble figuring out what the hamiltonian for a square lattice would be when considering the Heisenberg interaction. I reformulated the dot product into 1/2( Si+Si+δ+ +Si+δ+S-- ) + SizSi+δz and use Siz = S-ai+ai Si+ =...

Hi, I was calculating the Edwards-Anderson Hamiltonian of a Hopf link. A hopf link is like attachment 1. I have drawn the Seifert surface of that link. The surface is shown in attachment 2. It also contains the Boltzmann weight. So, this is an Ising model. I am confused as there are more than...

My book writes a 5-step recipe for detemining the hamiltonian, which I have attached. However I see a problem with arriving at the last result. Doesn't it only follow if the matrix M is a symmetric matrix - i.e. the transpose of it is equal to itself.

Homework Statement Consider a charged particle of charge e traveling in the electromagnetic potentials \mathbf{A}(\mathbf{r},t) = -\mathbf{\nabla}\lambda(\mathbf{r},t)\\ \phi(\mathbf{r},t) = \frac{1}{c} \frac{\partial \lambda(\mathbf{r},t)}{\partial t} where \lambda(\mathbf{r},t) is...

Given the hamiltonian in this form: H=\hbar\omega(b^{+}b+.5) b\Psi_{n}=\sqrt{n}\Psi_{n-1} b^{+}\Psi_{n}=\sqrt{n+1}\Psi_{n+1} Attempt: H\Psi_{n}=\hbar\omega(b^{+}b+.5)\Psi_{n} I get to H\Psi_{n}=\hbar\omega\sqrt{n}(b^{+}\Psi_{n-1}+.5\Psi_{n-1}) But now I'm stuck. Where can I...

As the title suggests, I am interested in symmetries of QM systems. Assume we have a stationary nonrelativistic quantum mechanical system H\psi = E\psi where we have a unique ground state. I am interested in the conditions under which the stationary states of the system inherit the...

Title says it all, confused as to how I'm supposed to define the phase space of a system, in my lecture notes I have the phase space as {(q, p) ϵ ℝ2} for a 1 dimensional free particle but then for a harmonic oscillator its defined as {(q, p)}, why is the free particles phase space all squared...

Homework Statement Find the energy eigenvalue. Homework Equations H = (p^2)/2m + 1/2m(w^2)(x^2) + λ(x^2) Hψ=Eψ The Attempt at a Solution So this is what I got so far: ((-h/2m)(∂^2/∂x^2)+(m(w^2)/2 - λ)(x^2))ψ=Eψ I'm not sure if I should solve this using a differential...

I need to elaborate the equation ,and need to know what is the physical significance and how matrices will manipulate in the equation $$ \hat{H} = (\hat{\tau_3}+i\hat{\tau_2})\frac{\hat{p}^2}{2m_0}+ \hat{\tau_3}m_0 c^2 = \frac{\hat{p}^2}{2m_0} \left| \begin{array}{ccc} 1 & 1 \\ -1 &...

Given H=p^2/2 - 1/(2q^2) How to show that there is a constant of motion for this one dimensional system D=pq/2 - Ht ? I tried doing it in my usual way i.e. p'=-∂H/∂q and q'=∂H/∂p and then finding the constants of motion but that doesn't match with what I have to show. Please guide me as...