Hamiltonian Definition and 901 Threads

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...

Homework Statement Show that for a particle in a central potential; V=f(|r|) H is conserved. Homework Equations THe hamiltonian is H=\sum(piq'i)-L It is conserved if dH/dt=0 Euler-Lagrange equation d/dt(dL/dq')=dL/dq Noether's Theorem For a continuous transformation, T such...

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...

Hello everyone, this is my first thread. Hope to be helpful here, as well as to find some help! :D Homework Statement Given a Hamiltonian H(q,p)(known) and given a transformation of coordinates (q,p)\rightarrow (Q,P): a) Show that it is a canonic transformation b) Solve the...

Homework Statement Let be q(t)=e^{-t}\alpha and p(t)=e^{-t}\beta Can this be a Hamiltonian Evolution Homework Equations The Hamilton equations for \dot{p} and \dot{q}. The Attempt at a Solution Can be a Hamiltonian evolution if verifies Hamilton 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...

Homework Statement This problem has to do with a canonical transformation and Hamiltonian formalism. Below is my attempt at solving it, but I am not too sure about it. Please help! Let \theta be some parameter. And X_1=x_1\cos \theta-y_2\sin\theta\\ Y_1=y_1\cos \theta+x_2\sin\theta\\...

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...

In Rovelli's book, in chapter 7 it talks about the Hamiltonian operator for LQG. In manipulating the form for the Hamiltonian operator Rovelli makes the following expansions U(A,\gamma_{x,u})=1+\varepsilon u^a A_a(x)+\mathcal{O}(\varepsilon^2) where by fixing a point x and a tangent...

hi friends. i don't know how can i write a fortran code for expressing spins in Heisenberg model which have 3 dimension spin operator, sx,sy,sz? thanks for your help

Homework Statement The total energy may be given by the hamiltonian in terms of the coordinates and linear momenta in Cartesian coordinates (that is, the kinetic energy term is split into the familiar pi2/2m. When transformed to spherical coordinates, however, two terms are angular momentum...

Hi there, If the evolution operator is given as follows U(t) = \exp[-i (f(p, t) + g(x))/\hbar] where p is momentum, t is time. Can I conclude that the Hamiltonian is H(t) = f(p, t) + g(x) if no, why?

I'm currently reading Sakurai's 'Modern Quantum Mechanics' (Revised Edition) and at page 76 he introduces a spin half hamiltonian H = - (\frac{e}{mc}) \vec S \cdot \vec B. But what is c doing in this hamiltonian? Clasically the energy of a magnetic moment in a magnetic field is E = -...

Homework Statement "Show that if the Hamiltonian depends on time and [H(t_1),H(t_2)]=0, the time development operator is given by U(t)=\mathrm{exp}\left[-\frac{i}{\hbar}\int_0^t H(t')dt'\right]." Homework Equations i\hbar\frac{d}{dt}U=HU U(dt)=I-\frac{i}{\hbar}H(t)dt The Attempt at a...

Theorem: Prove that there exist $n$ edge disjoint Hamiltonian cycles in the complete graph $K_{2n+1}$. ---------------------------------------------------------------------------------- I have found two constructive proofs of this over the internet. But I would like to prove it...

Hi, I have a question about two different expressions of Jaynes-Cummings Hamiltonian H=\Delta_c a^{\dagger}a+\Delta_a \sigma_{+} \sigma_{-} + g (a^{\dagger}\sigma_{-} +a\sigma_{+} ) and H=\Delta_c a^{\dagger}a+\Delta_a \sigma_{+} \sigma_{-} +i g (a^{\dagger}\sigma_{-} -a\sigma_{+}...

Hi I have a question regarding the pump-term in the Hamiltonian on page 9 (equation 2.9b) of this thesis: http://mediatum2.ub.tum.de/download/652711/652711.pdf. This term is not very intuitive to me. a is followed by a CCW phase, whereas a^\dagger is followed by a CW phase. How does this...

I don't understand this idea. For example we have cubic crystal which has a lot of unit cells. We define spin variable of center of cell like S_c. And spin variable of nearest neighbour cells with S_{c+r}. So the cell hamiltonian is...

Why is minus sine in definition of hamiltonian H=-\sum_{i,j}J_{i,j}(S_{i}^+S_{j}^-+S_i^zS_j^z) Why not? H=\sum_{i,j}J_{i,j}(S_{i}^+S_{j}^-+S_i^zS_j^z)

I have a Hamiltonian, consisting only of angular momentum components Lx,Ly,Lz. I need to go from it to some coordinate representation. But I don't have derivatives Lx' etc. in H. So, when I'll go to coordinates and momenta I'll have Hamiltonian equations like p_i=0, which doesn't have sense...

How can you tell if the Klein-Gordan Hamiltonian, H=\int d^3 x \frac{1}{2}(\partial_t \phi \partial_t \phi+\nabla^2\phi+m^2\phi^2) is time-independent? Don't you have to plug in the expression for the field to show this? But isn't the only way you know how the field evolves with time is...

I'm currently using David J. Griffiths 'Introduction to Quantum Mechanics' to teach myself quantum mechanics and I had a quick question about the way he factors the Hamiltonian into the raising and lowering operators for the potential V(x)=(1/2)kx² On page 42 he writes the Hamiltonian as...

In this paper called "Stepping out of Homogeneity in Loop quantum Cosmology" - http://arxiv.org/pdf/0805.4585.pdf. On page 4 they say "where the sum is over the couples of distinct faces at each tetrahedron, U_{ff'} = U_f U_{f_1} U_{f_2} \dots U^{-1}_{f'} where l_{ff'} = \{ f , f_1; f_2; \dots...

Generally, what sort of problems are handled better by Hamiltonian mechanics than by Lagrangian mechanics? Can anyone give a specific example?

Hi there. I need help to work this out. A particle with mass m is studied over a rotating reference frame, which rotates along the OZ axis with angular velocity \dot\phi=\omega, directed along OZ. It is possible to prove that the potential (due to inertial forces) can be written as: V=\omega...

Homework Statement A particle moves in a one dimensional potential : V(x) = 1/2(mω2x Show that the function ψ(x) = a0exp(-βx2) is an eigenfunction for the Hamiltonian for a suitable value of β and calculate the value of energy E1 Homework Equations The Attempt at a Solution...