Hamiltonian Definition and 901 Threads

\hat{H}_{BCS}=\sum_{\vec{p},\sigma}\epsilon(\vec{p})\hat{a}^+_{\vec{p},\sigma}\hat{a}_{\vec{p},\sigma}+\sum_{\vec{p},\vec{p}'}V(\vec{p},\vec{p}')\hat{a}^+_{\vec{p}\uparrow}\hat{a}^+_{-\vec{p}\downarrow}\hat{a}_{-\vec{p}'\downarrow}\hat{a}_{\vec{p}'\uparrow} What is the meaning of the terms...

When one says that a system is in an eigenstate of the Hamiltonian, what exactly does this mean? I mean, if the Hamiltonian is the total energy of the system, then if it is in an eigenstate of the Hamiltonian, is this saying that its energy is a multiple of its total energy? Obviously this...

Hi I am a mathematics junior and I am doing a research project on hamiltonian systems and liouville integrability (don't ask why...). I am using the book by Vilasi, a graduate level book, but I am finding it quite difficult and badly written; for instance he uses functional analysis and...

Homework Statement Find the eigenvalues and eigenfunctions of H\hat{} for a 1D harmonic oscillator system with V(x) = infinity for x<0, V(x) = 1/2kx^2 for x > or equal to 0. Homework Equations The Attempt at a Solution I think the hamiltonian is equal to the potential + kinetic...

Hi, I need some help in writing the Hamiltonian function for the following dynamical systems. 1) u''+u=A (1+2*u+3*u^2) 2) u''+u=A/((1-u)^2); In both cases A is a constant and u is a function of t. Any help would be greatly appreciated. Thank you. Manish

Hi In almost every reference I have found the phonon part of the frohlich electron phonon interaction hamiltonian is given by (b_{q}+b^{\dag}_{-q}) notice the +, where b_{q} is a phonon creation operator and b^{\dag}_{-q})is the destruction operator of a phonon. however in a paper on...

Homework Statement The operator Q satisfies the two equations Q^{\dagger}Q^{\dagger}=0 , QQ^{\dagger}+Q^{\dagger}Q=1 The hamiltonian for a system is H= \alpha*QQ^{\dagger}, Show that H is self-adjoint b) find an expression for H^2 , the square of H , in terms of H. c)Find the...

Hey all, (As I mentioned in my previous post) I am trying to derive the Hamiltonian for a aeroelastic system, where the dynamical equations of motion (determined by Newtonian Mechanics) are known. My process has been to 1. "guess" a form of the Lagrangian, check that it recreates the...

Hi all, I am in a bit of a dilly of a pickle of a rhubarb of a jam with determining the Hamiltonian of a specific system. For background information it is an 2-DoF aero-elastic system where I am (temporarily) neglecting the aerodynamic lift and moment terms. Being an intrinsically...

Homework Statement Derive the Hamiltonian function H(A,B) such that A'=HB and B'=-HA. Plot the contours of this function in the range -0.005 =< H =< 0.01. Identify the approximate position an type of each of the three critical points which occur. We can look for solutions of the form x =...

Homework Statement use the hamiltonian equation H=H_x+H_y+H_z to show that wave functions of the form \varphi(r)=\phii(x)\phij(y)\phik(z) where the functions phi_i(x) are the energy eigenfunctions for a 1-d SHM , satisfy H*phi=E*phi , and find the followed values of E for the 3-d...

in the bose-hubbard model, we need to enumerate all the possible basis usually, the basis vectors are taken as the fock states The problem is that, how to arrange the basis and how to establish the matrix of the hamiltonian as soon as possible It is apparent the the matrix will be very...

So, I'm doing some undergraduate research in quantum spin systems, looking at the ground states of the Heisenberg Hamiltonian, H=\sum{J_{ij}\textbf{S}_{i}\textbf{S}_{j}}. But I think I have a critical misunderstanding of some fundamental quantum mechanics concepts. (I'm a math major, only had...

I have been studying the theoretical framework of quantum mechanics in an attempt to have a working understanding of the subject, if not a comprehensive one, and I have hit upon the following stumbling block. Now, given that the orthogonality of states is preserved with time, it is easily...

I am not posting this in the homework section because it is not really a homework problem. Its from the schaum outline and I am stumped in this: http://img379.imageshack.us/img379/688/67356569.jpg I have NO idea about 3.2.4 and 3.2.5. Its black magic! How the hell does that substitution work...

Homework Statement Use the generalized Ehrenfest theorem to show that any free particle with the one-dimensional Hamiltonian operator H= p^2/2m obeys d^2<x^2> / dt^2 = (2/m)<p^2>, Homework Equations The commutation relation xp - px = ih(bar) The Attempt at a Solution...

Hi everyone I'm trying to express each term of the Hamiltonian H = \int d^{3}x \frac{1}{2}\left[\Pi^2 + (\nabla \Phi)^2 + m^2\Phi^2\right][/tex] in terms of the ladder operators a(p) and [itex]a^{\dagger}(p). This is what I get for the first term \int d^{3}x...

I've just encountered the terms Hamiltonian and Lagrangian. I've read that the Hamiltonian is the total energy H = T + U, while the Lagrangian L = T - U, where T is kinetic energy, and U potential energy. In the case of Newtonian gravitational potential energy, U = -G\frac{Mm}{r}. So am I...

Homework Statement The Hamiltonian for a spin-half particle is H = 2a/ħ (Sx + Sy) where a is a positive constant and Sx , Sy are the x and y components of the spin. Initially (at time t=0) the particle is in the state |ψ> = (1/√2) (|↑>+|↓>) where up and down arrows denote...

Homework Statement Derive the hyperfine Hamiltonian starting from \hat{H}_H_F = -\hat{\mu}_N \cdot \hat{B_L} . Where \hat{\mu}_N is the magnetic moment of the nucleus and \hat{B_L} is the magnetic field created by the pion’s motion around the nucleon. Write down the Hamiltonian in the...

I have the Hamiltonian for an S=5/2 particle given by: H= a.Sz + b.Sz^2 +c.Sx where Sz and Sx are the spins in z and x directions respectively. The resulting matrix is tridiagonal symmetric but I can't find the eigenvalues..Any idea how to diagonalise it. N.B: a is a variable and must be...

I found the uncertainty between delta x (position) and delta H (Hamiltonian) to be greater or equal to (h_bar*<p>)/ 2m. Does this mean for stationary states, where <p>=0, the uncertainty can be zero? ie we can precisely measure the position and energy?

Homework Statement I'm following an article by Pitaevskii from 1956, and there's one mathematical transition which I don't understand. The Hamiltonian is given first in coordinate space, and then the density operator is transformed to its Fourier components. The continuity equation is also...

Homework Statement One dimensional harmonic oscillator has the Hamiltonian H(hat)=p(hat)/2m +0.5mw^2x(hat)^2 Show that the eigenvalue spectrum of H(hat) is En=(n+0.5)h(bar)w n=0,1,2... I've managed to show this Suppose the real constant C is added to the Hamiltonian H(hat) to give the...

Homework Statement I am trying to solve a problem of 1D electron system. Given a,a^\dagger,b,b^\dagger annihilation and creation operator which satisfy the fermion commutation relations diagonalize the following hamiltonian...

Switching to a Matrix Hamiltonian -- Conceptual Issues It's probably very clear and well-established for those who rigorously studied Quantum Mechanics but I don't think what I am going to ask is easily 'google'-able but if it is so - please send me to the correct source before spending time...

So if you have a 3D Shrodinger Equation problem, what allows you so assume that the wave function solution is going to be a product of 3 wave functions where each wave function is for a different independent variable? And also is it true that in general in these cases the eigen-energies are...

The problem is A particle of mass m and electric charges q can move only in one dimension and is subject to a harmonic force and a homogeneous electrostatic field. The Hamiltonian operator for the system is H= p2/2m +mw2/2*x2 - qεx a. solve the energy eigenvalue problem b. if the...

Hallo everyone, I have a question, how can I see that the hamiltonian H=p^2-x^4 is not hermitian, with p the momentum operator and x the position operator.

Hi there! After some years of physics studies I'm accustomed to the Hamiltonian principle but I sometimes still wonder why physicists tacitly assume that the eq.s of motion of any physical theory (no matter if quantized or not, relativistic or not, strings etc.) can be obtained as...

Homework Statement Using cartesian coordinates, find the Hamiltonian for a projectile of mass m moving under uniform gravity. Obtain Hamiltonian's equation and identify any cyclic coordinates. Homework Equations The Attempt at a Solution I think I will just have trouble...

Can anybody tell me what they mean by 2, 3, 6 or any band hamiltonians. What does it even mean?

Homework Statement Hi all. I have a Hamiltonian given by: H = H_x + H_y = -\frac{\hbar^2}{2m}(d^2/dx^2 + d^2/dy^2). Now I have a stationary state on the form \psi(x,y)=f(x)g(y). According to my teacher, then the Hamiltonian can be split up, i.e. we have the two equations: H_x...

Hi, I have a (maybe rather technical) question about the Hamiltonian formulation of gauge theories, which I don't get. With an infinitesimal symmetry on your space-time M one can look at the corresponding transformation of the canonical variables in phase-space PS. This can be done by a phase...

To which entitity (operators, wavefunctions etc) in quantum mechanics do the dynamical variables and the hamiltonian vector fields that they generate (through Symplectic structure of classical mechanics) correspond to?

Homework Statement Hi all. I am commuting the Hamiltonian (H = p2/(2m) + V(x)) with position. This is what I get: [H,x] = -\frac{i\hbar}{m}p, where p is the momentum operator. But here's my question: The momentum-operator contains d/dx, so does this mean that the commutator is zero, or do...

Have you got any clues how to prove q2=Acos(q2)+Bsin(q2)+C using hamiltonian H =(1/2)*(p12 q14 + p22 q22 - 2aq1) , where a,A,B,C=const. I've tried to solve hamiltonian eqations what let me to equations which I can't solved. How you got any ideas solving this problem?

We know if a force is conservative we can use a potential function. Assume there are non-conservative forces in our problem. For example the air resistance force exerting on a oscillating mass-spring system. How should we write the hamiltonian for this case?

Homework Statement The Lagrangian of a non-relativistic particle propagating on a unit circle is L=\frac{1}{2}\dot{\phi}^{2} where ϕ is an angle 0 ≤ ϕ < 2π. (i) Give the Hamiltonian of the theory, and the Poisson brackets of the ca- nonical variables. Quantize the theory by promoting...

dear members, My problem is... suppose take the spin Hamiltonian Hham=D[Sz2 -S(S+1)/3 +(E/D)(Sy2-Sy2)] +Hi\vec{S} (most often in EPR experiments, etc). here external magnetic field Hamiltonian Hi = \betagiBiext and i =x, y and z. Also gx=gy=gz=2 and the external magnetic field is...

Homework Statement Suppose that the Hamiltonian is invariant under time reversal: [H,T] = 0. Show that, nevertheless, an eigenvalue of T is not a conserved quantity. Homework Equations The Attempt at a Solution Using Kramer's Theorem. Consider the energy eigenvalue...

If the classical Hamiltonian is define as H = f(q, p) p, q is generalized coordinates and they are time-dependent. But H does not explicitly depend on time. Can I conclude that the energy is conserved (even q, p are time-dependent implicitly)? Namely, if no matter if p, q are time-dependent or...

Homework Statement The Hamiltonian of an electron with mass m, electric charge q and spin of \frac{\hbar }{2}\vec{\sigma} in a magnetic field described by the potential vector \vec{A}\left( \vec{r},t\right) and a scalar potential U\left( \vec{r},t\right) is given by...

Homework Statement Find the energy spectrum of a system whose Hamiltonian is H=Ho+H'=[-(planks const)^2/2m][d^2/dx^2]+.5m(omega)^2x^2+ax^3+bx^4 I gues my big question to begin is what exactly makes up the energy spectrum. I know the equation to the first and second order perturbations...

Homework Statement N points at distance a each other -> chain length L=Na. q_{}n is the n-point shift. Homework Equations q\stackrel{}{}.._{}n=\Omega^{}2(q_{}n+1+q_{}n-1-2q_{}n I must find the hamiltonian and lagrangian of the system. The Attempt at a Solution

Homework Statement if we have the particle ins free its hamiltonian has a continuous spectrum of eigen enegies and superposition of arbitrary initial state in eigenstates φ_k of H( hamiltonian oprator) becomes ∫_(-∞)^∞▒〖b(k) φ_k dk〗,what is the dimemension of b(k) (lb(k)l^2 is a...

I have a question..I am trying to solve a differential equation that arises in my research problem. Because the differential equation has no solution in terms of well known functions, I had to construct a series solution for the differential equation which is physical and agrees with the...

Homework Statement Given the Hamiltonian H=\vec{\alpha} \cdot \vec{p} c + mc^2 = -i \hbar c \vec{\alpha} \cdot \nabla + mc^2 in which \vec{\alpha} is a constant vector. Derive from the Schrödinger equation and the continuity equation what the current is belonging to the density \rho...

Homework Statement There are (m-1)n+1 people in the room. Show that there are at least n people who mutually do not know each other, or there is a person who knows at least n people Homework Equations The Attempt at a Solution I think this has to do with hamiltonian walks...

Homework Statement I have been given the Hamiltonian H = \sum_{k} (\epsilon_k - \mu) c^{\dag} c_k where c_k and c^{\dag}_k are fermion annihilation and creation operators respectively. I need to calculate the ground state, the energy of the ground state E_0 and the derivative...