Hamiltonian Definition and 833 Threads

Can anybody explain what is meant by positive definite Hamiltonian? All I know is that if a Hamiltonian can be factorized as H={Q}^{\dagger}Q then that Hamiltonian is one such example. But I am not sure if that is the definition of a positive definite Hamiltonian. In the quantum mechanical...

I did search for topics in this forum. but i could not find basics that deal with hamiltonian of the system Well I'm pretty new to the field of quantum mechanics. I just could not understand what exactly it means by the hamiltonian of a system? I was told that it describes the total energy of...

Matrix Hamiltonian? Homework Statement I have two non-identical spin 1/2 particles, which have vector magnetic moments S_1 and S_2. The interaction energy (Hamiltonian) is given by a constant times the dot product of S_1 and S_2. There is no external field present. I need to find the...

After several failures in the past (why does the universe have to be so complicated?!), I'm once again trying to learn to understand the basics of QM, out of sheer frustration with not knowing what the heck physicists are talking about all the time. I know, I still have a long way to go...

I'm using McQuarrie's "Quantum Chemistry" book for a little bit of light reading. He included a proof of a theorem that states that if a Hamiltonian function is separable, then the eigenfunctions of Schrodinger's equation are the products of the eigenfunctions of the simpler "separated"...

I'm preparing for an exam at the moment and in one of the past exams the is a question asking to prove that the hamiltonian operator can be expressed in terms of the ladder operators. The solution is this (The minus sign didn't come out in the last line, and obviously there is one more...

In the Anderson model, it cost an energy Un_{\Uparrow}n_{\Downarrow} for a quantum dot level to be occupied by two electrons. Here n_{\Uparrow} is the second quantized number operator, counting the number of particles with spin \Uparrow. I need the term Un_{\Uparrow}n_{\Downarrow} in first...

Hi all, could someone give me a quick answer on the exact conditions for the hamiltonian to be non degenerate, i.e. to have different eigenvalues? thanks in advance.

In the volume III of R Feynman series which is on Quantum Mechanics , please explain to me the eq.8.43 given on page 1529, i know how we got the equation but the 2nd part of 1st equation (H12)C2, what does it mean ?

Hello everyone! I'm trying to find the relation between the lagrangian density and the hamiltonian, does anyone know how they are related? I also need a reference where I can find the relation. Thanks!

Hi all! I'm starting to study the time evolution operator, and now i came up with this objective... i need a time dependant hamiltonian! since no fundamental interaction is time dependent i need to think of a system in such a configuration that i have a time dependency on H. Anyway, if anyone...

Hey does anybody have an idea of how to prove that \frac{d}{dt}\left\langle{XP}\right\rangle= 2\left\langle{T}\right\rangle-\left\langle{x\frac{dV}{dx}}\right\rangle for a hamiltonian of form H=\frac{P^2}{2M}+V(x) where X is the position operator, P is the momentum and T is the kinetic...

I was thinking of how to solve the single particle Hamiltonian H=...+\sum_i \frac{1}{\vec{r}-\vec{r}_i} where \vec{r}_i=i\cdot\vec{a} Transforming it into second quantization k-space I had terms like H=...+\sum_G...c^\dag_{k+G}c_k But it seems to me that for the method of trial wavefunctions any...

I’m trying to decipher this particular passage from a paper. Ref: FM Kronz, JT Tiehen - Philosophy of Science, 2002 “Emergence and Quantum Mechanics” Can you provide an example of a physical system which corresponds to a classical system with a nonseparable Hamiltonian (ex: the Milky Way...

I would guess that they would as every observable is a function of the q's and p's and as those commute with the hamiltonian I couldn't imagine an observable that wouldn't commute, however are there any other cases where an observable won't commute with the hamiltonian?

I was wondering if anyone knows of systems for which the Hamiltonian is not equall to the total energy? This is an interesting problem in analytic mechanics (e.g. Lagrangian and Hamiltonian dynamics) but is rarely, if ever, mentioned in forums and newsgroups. I'd love to see a large set of...

Hey I was just wondering what the differences between the three forms of mechanics were. I've only studied basic Newtonian mechanics so I'm not really sure about the other two. Could anyone elaborate?

I have to show that the hamiltonian for a homogeneous system can be simplified in scaled coordinates. The first two terms I can convert to scaled coordinates <T>+<V> whereas I have some trouble for the last term -½* \int d³r d³r' \frac{n²}{|r-r'|} where n is the density. The scaled...

[SOLVED] rotationally invariant hamiltonian Homework Statement Show that the Hamiltonian H = p^2/2m+V_0r^2 corresponding to a particle of mass m and with V_0 constant is a) rotationally invariant. Homework Equations Rotation operator: U_R(\phi ) = \exp (-i \phi \vec{J} / \hbar...

Does any willing to explain in detail from basics about what is this 'Spin Hamiltonian'.?? thanks

momentum operator in Hamiltonian Hello all. I'm in an introductory QM course as a physics major. As I understand it, to quantize a classical system, we just replace momentum in Hamiltonian with momentum operator? But why? One answer is that because it works. Is there any other reasons why it...

Eigenstates of interacting and non-interacting Hamiltonian Have multi-particle state of full Hamiltonian and one-particle state of free Hamiltonian non-zero scalar product? Intuitively one can say that scalar product of such states should be zero because each of these states mentioned above...

Problem : in the Attach file

I had encountered a problem for which I need to know how to proceed in order to solve it. Taking a particle m with box potential (one dimensional) where V(x) = 0 when mod(x) <=a and V(x) = infinity when mod(x) > a and where wave function phi(x) = A (phi1(x) + ph2(x)) where phi1(x) and phi2(x)...

hello all, i'm an EE student,and I've recently started studying quantum mechanics. most textbooks start with schrodinger's equation directly but a few others (like say Liboff) start with the concept of hamiltonian from hamiltonian mechanics. is a knowledge of the same i.e...

Homework Statement Show that the Hamiltonian of the Heisenberg model can be written as: H=\sum^{N}_{k=1}[H_{z}(k)+H_{f}(k)] where H_{z}(k)\equivS^{z}(k)S^{z}(k+1) H_{f}(k)\equiv(1/2)[S^{+}(k)S^{-}(k+1)+S^{-}(k)S^{+}(k+1)] Homework Equations As above The Attempt at a Solution I...

For an atom with one electron and nuclear charge of Z, the Hamiltonian is: H=-~\frac{\nabla^{2}}{2}~- ~\frac{Z}{r}~ 1) show that the wavefunction: \Psi_{1s}=Ne^{-Zr} is an eigenfunction of the Hamiltonian 2) find the corresponding energy 3) find N, the normalisation constant In...

Wht is the significance of an operator commuting with the hamiltonian?? Its definitely more than them being measurable simultaneously and precisely!

Hi, I'm just wondering if someone could explain to me exactly what the hamiltonian function is

i know this is sort of an obvious question but what is the difference between the hamiltonian and energy momentum tensor since they are both matrices and energy and momentum are equivalent? are they different in terms of the cicumstances in which they are used.

what is the difference between these two formalism and when are each used? also is it true the lagrangian formalism is used more in qft, if so i am curious to know why?

Is it possible to say that for a general Schrödinger equation i \hbar \frac{\partial}{\partial t} | \psi (t) \rangle = \hat{H} | \psi (t) \rangle one may obtain the general Hamiltonian operator \hat{H} = i \hbar \frac{\partial}{\partial t} Thanks!

Hi! I've been studying Dirac's programme for some time and I realized that there's something missing: Actually this is missing in every standard book on classical mechanics concerning how constraints are implemented in the lagrangian. They are usually inserted with some unknown variables...

Homework Statement Let V = V_r - iV_i, where V_i is a constant. Determine whether the Hamiltonian is Hermitian. Homework Equations H = \frac{-\hbar^2}{2m}*\Delta^2+V_r - iV_i The Attempt at a Solution I think you can distribute the Hamiltonian operator as follows: H^{\dag} =...

The equation for the Hamiltonian is H = T + V. Can someone explain how you can use this to get this equation for a free particle: i\hbar|\psi'> = H|\psi> = P^2/(2m)|\psi> The first part is obviously Schrodinger's equation but how do you get H = P^2/2m? Go to page 151 at the site below...

hey I am i have just finished my final yr from Bangalore.. did phy n maths..interested in pursuing physics in the future.Cld i get some help regarding which books cld b followed if i want to learn Hamiltonian n findin its eigen values. If u cld suggest a book which deals from the basics of the...

I was curious as to the form of the hamiltonian, whose energy eigenstate in the position basis is a gaussian distribution (or minimum uncertainty state, as I've heard from somewhere.) I haven't taken quantum for a few years, and remember studying the minimum uncertainty state as a wavefunction...

Suppose that we take the Klein-Gordon Hamiltonian to be of the form H = \int d^3x \, \mathcal{H}(x) = \frac{1}{2}\int d^3x\, (\pi^2(x) + (\nabla\phi(x))^2- m^2\phi^2(x)) If we want to compute, say, the evolution equation for \phi(x) we use the Poisson bracket: \dot{\phi}(x) = \{\phi(x),H\} =...

A charged particle of mass m is attracted by a central force with magnitude F = \frac{k}{r^2} . Find the Hamiltonian of the particle. I'm just wondering if I did this correctly because it seemed too easy. First I used the fact that -dU/dr = F = k/r^2, so the potential (with infinite...

URGENT x 10 DERIVE a 6x6 Hamiltonian for bulk semiconductors Okay here is a little challenge for you guys. Try and test your skill a little. First 10 people to properly derive a 6x6 Hamiltonian for bulk semiconductors will gain bragging rights in this forum.

if we define Z as: Z(s)=Tr[exp(-sH)] my 2 questions are.. a) is the trace unique and define the Hamiltonian completely? i mean if we have 2 Hamiltonians H and K then Tr[exp(-sH)]\ne Tr[exp(-sK) and if we use the 'Semiclassical approach' then Z(s)=Tr[exp(-sH)]\sim...

Say I have a canonical transformation Q(q,p), P(q,p). In the {q,p} canonical coordinates, the Hamiltonian is H(q,p,t)=p\dot{q}-L(q,\dot{q},t) And the function K(Q,P,t)=H(q(Q,P),p(Q,P),t) plays the role of hamiltonian for the canonical coordinates Q and P in the sense that...

Homework Statement Can a system have the total energy conserved but the hamiltonian not conserved? Homework EquationsIf the partial of the lagrangian w.r.t time is zero, energy is conserved. The hamiltonian is found by the usual method- get the generalized momentum from the lagrangian...

Homework Statement Using spherical coordinates (r, \theta, \phi), obtain the Hamiltonian and the Hamilton equations of motion for a particle in a central potential V(r). Study how the Hamilton equations of motion simplify when one imposes the initial conditions p_{\phi}(0) = 0 and \phi (0)...

Consider the time-dependent Hamiltonian H(q,p;t) = \frac{p^2}{2m \sin^2{(\omega t})} - \omega pq \cot{(\omega t)} - \frac{m}{2} \omega^2 \sin^2{(\omega t)} q^2 with constant m and \omega. Find a corresponding Lagrangian L = L(q,\dot{q};t) Ok, I know that the Hamiltonian is given by...

I have a problem that uses the QM Hamiltonian for the berylium atom, but I am having trouble finding this Hamiltonian using the Born-Oppenheimer approximation (leaving out the nuclear-nucler and nucler-electron terms). Any know how to get this?

Let's suppose we have a Hamiltonian of the form: \bold H =0 then it's obivious that if you want to get its energies you would get E_{n}=0 for every n, this is a non-sense since you must have positive energies (and a ground state) ...then if we "cheat" :rolleyes: and make: \bold H...

One more question. Sorry! Here's the problem: An electron moves in a straight line under the influence of a conservative force so that the Hamiltonian is H = \frac{p\wedge^2}{2m} + V(x), where p\wedge means the momentum operator and I think V(x) is the potential energy. I need to find an...

The Hamiltonian is given: H=Aâ†â + B(â + â†) where â is annihilation operator and ↠is creation operator, and A and B are constants. How can I get the eigenenergy of this Hamiltonian? The given hint is "Use new operator b = câ + d, b† =c↠+ d (c and d are constants, too) But...

Find the eigenvalues of the hamiltonian H=a(S_A \cdot S_B+S_B \cdot S_C+S_C \cdot S_D+S_D \cdot S_A) where S_A, S_B, S_C, S_D are spin 1/2 objects _________________________ I rewrite it as H=(1/2)*a*[(S_A+S_B+S_C+S_D)^2-(S_A+S_C)^2-(S_B+S_D)^2] then i define...