In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a Hamiltonian path that is a cycle. Determining whether such paths and cycles exist in graphs is the Hamiltonian path problem, which is NP-complete.
Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the icosian game, now also known as Hamilton's puzzle, which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. Hamilton solved this problem using the icosian calculus, an algebraic structure based on roots of unity with many similarities to the quaternions (also invented by Hamilton). This solution does not generalize to arbitrary graphs.
Despite being named after Hamilton, Hamiltonian cycles in polyhedra had also been studied a year earlier by Thomas Kirkman, who, in particular, gave an example of a polyhedron without Hamiltonian cycles. Even earlier, Hamiltonian cycles and paths in the knight's graph of the chessboard, the knight's tour, had been studied in the 9th century in Indian mathematics by Rudrata, and around the same time in Islamic mathematics by al-Adli ar-Rumi. In 18th century Europe, knight's tours were published by Abraham de Moivre and Leonhard Euler.
I'm reading through Hohenberg's seminal paper titled: "Inhomogeneous Electron Gas" that help lay the foundation for what we know of as Density Functional Theory (DFT) by proving the existence of a universal functional that exactly matches the ground-state energy of a system with a given...
Hi all
I was reading a certain paper that involves solving the Bohr-Mottelson Hamiltonian for a 5dimential square well potential, the B-M Hamiltoian reads:
my question is just how do I calculate the mass parameter "B" for a certain nuclei, and with a 5D infinite potential well how do I get the...
Hi,
I'm not sure to understand what ##| \phi_n \rangle = \sum_i \alpha_i |\psi_n^i## means exactly or how we get it.
From the statement, I understand that ##[U,H] = 0## and ##H|\psi_n \rangle = E_n|\psi_n \rangle##
Also, a linear combination of all states is also an solution.
If U commutes...
Hi,
In my book I have and expression that I don't really understand.
Using the definition of action ##\delta S = \frac{\partial L}{\partial \dot{q}} \delta q |_{t_1}^{t_2} + \int_{t_1}^{t_2} (\frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}}) \delta q dt##
Where L...
I am trying to reproduce the results from this paper. On page 10 of the paper, they have an equation:
$$ \frac{S}{T}=\int dt\sum _{n=0,1} (\dot{c_n}{}^2-c_n^2 \omega _n^2)+11.3 c_0^3+21.5 c_0 c_1^2+10.7 c_0 \dot{c_0}{}^2+3.32 c_0 \dot{c_1}{}^2+6.64 \dot{c_0} c_1 \dot{c_1} \tag{B12} $$
where they...
Greentings,
I've dealt with Quantum Theory a lot, but there's one thing I don't quite understand.
When deriving the Fermion-Propagator, say ##S_F##, all the authors I've read from made use of the Fourier-Transform. Basically, it always goes like
$$
\begin{align}
H_D S_F(x-y) &= (i \hbar...
On ***page 38*** of Becker Becker Schwarz, we're given ***equation 2.69*** which is the Hamiltonian for a string given as $$H=\frac{T}{2}\int_{0}^{\pi}(\dot{X}^{2}+X^{'2})$$
Considering the open string we have...
Someone says we can choose the new eigenstate: exp(-iλx/hbar)*ψ,and let the momentum operator p acts upon this new state. At the same time, so does p^2. Something miraculous will happen afterwards. My question is: how to image this point? Thank you very much.
To calculate the Hamiltonian of a charged particle immersed in an electromagnetic field, one calculates the Lagrangian with Euler's equation obtaining ##L=\frac{1}{2}mv^2-e\phi+e\vec{v}\cdot\vec{A}## where ##\phi## is the scalar potential and ##\vec{A}## the vector potential, and then we go to...
What I did was first noting that ##\hat{\vec{S}}_1\cdot\hat{\vec{S}}_2=\frac{1}{2}(\hat{\vec{S}}^2-\hat{\vec{S}}_1^2-\hat{\vec{S}}_2^2)##, but these operators don't commute with ##\hat{S}_{1_z}## and ##\hat{S}_{2_z}##, this non the decoupled basis ##\ket{s_1,s_2;m_1,m_2}## nor the coupled one...
Hello ! I require some guidance on this prove :I normally derive the Hamiltonian for a SHO in Hilbert space with a term of 1/2 hbar omega included. However, I am unsure of how one derives this from Hilbert space to Fock space. I have attached my attempt at it as an image below. Any input will be...
In Classical Mechanics by Kibble and Berkshire, in chapter 12.4 which focuses on symmetries and conservation laws (starting on page 291 here), the authors introduce the concept of a generator function G, where the transformation generated by G is given by (equation 12.29 on page 292 in the text)...
Dear everybody,
I am involved with a system of two spins and I ended up with the following Hamiltonian:
$$H_c(t) = W\sin(2J_+ t) \big( \mathbb{1} \otimes \sigma_z - \sigma_z \otimes \mathbb{1}\big) + W \cos(2J_+ t) \big( \sigma_y \otimes \sigma_x - \sigma_x \otimes \sigma_y \big) \: ,$$
where...
Hello!
I need some help with this problem. I've solved most of it, but I need some help with the Hamiltonian. I will run through the problem as I've solved it, but it's the Hamiltonian at the end that gives me trouble.
To find the Lagrangian, start by finding the x- and y-positions of the...
The hamiltonian ´for a free falling body is $$H = \dfrac{p^2}{2m} + mgy$$ and since we are using cartesian coordinates that do not depend on time and the potential only depends on the position, we know that ##H=E##. For this hamiltonian, using the Hamilton's equations and initial conditions...
The issue here is that I don't know how to operate the final equations in order to get the phase diagram. I suppose some things are held constant so I can get a known curve such as an ellipse.
I attach the solved part, I don't know how to go on.
Hello everybody, new here. Sorry in advance if I didn't follow a specific guideline to ask this.
Anyways, I've got as a homework assignment two cannonical transformations (q,p)-->(Q,P). I have to obtain the hamiltonian of a harmonic oscillator, and then the new coordinates and the hamiltonian...
Once I know the Hamiltonian, I know to take the determinant ##\left| \vec H-\lambda \vec I \right| = 0 ## and solve for ##\lambda## which are the eigenvalues/eigenenergies.
My problem is, I'm unsure how to formulate the Hamiltonian. Is my potential ##U(r)## my scalar field ##\phi##? I've seen...
Alright my idea is that, in order to show that ##f_i(q_i, p_i)## is a constant of motion, it would suffice to show that the Hamiltonian is equal to a constant.
Well, the Hamiltonian will be equal to a constant iff:
$$f(q_1, q_2, ..., q_N, p_1, p_2,..., p_N) = \text{constant}$$
Which is what...
I need help with part d of this problem. I believe I completed the rest correctly, but am including them for context
(a)Show that the hermitian conjugate of the hermitian conjugate of any operator ##\hat A## is itself, i.e. ##(\hat A^\dagger)^\dagger##
(b)Consider an arbitrary operator ##\hat...
In quantum mechanics, the Eigenfunction resulting from the Hamiltonian of a free particle in 1D system is $$ \phi = \frac{e^{ikx} }{\sqrt{2\pi} } $$
We know that a function $$ f(x) $$ belongs to Hilbert space if it satisfies $$ \int_{-\infty}^{+\infty} |f(x)|^2 dx < \infty $$
But since the...
I have always seen this problem formulated in a well that goes from 0 to L
I am confused how to use this boundary, as well as unsure of what a dimensionless hamiltonian is.
This is as far as I have gotten
Hi!
So this is my first homework ever of Hamiltonian dynamics and I am struggling with the understanding of the most basic concepts. My lecturer is following Saletan's and Deriglazov's and from what I have read and from my lectures, this is what I think I know. Please let me know if this is...
I am given this Hamiltonian:
And asked to diagonalize.
I understand how we do such a Hamiltonian:
But I don't understand how to deal with the extra term in my given Hamiltonian. Usually we use
To get
Hello!
I am stuck at the following question:
Show that the wave function is an eigenfunction of the Hamiltonian if the two electrons do not interact, where the Hamiltonian is given as;
the wave function and given as;
and the energy and Born radius are given as:
and I used this for ∇...
Hi, I hope this is in the right section. It's for EM which I guess is a relativistic theory but the question itself is not to do with any Lorentz transformations or anything similar.
I'm reading through Jackson with my course for EM and I'm on the section where he is generalising the Hamiltonian...
I have been attempting to modify a symplectic integrator that I wrote a while ago. It works very well for "separable" hamiltonians, but I want to use it to simulate a double pendulum.
I am using the Stormer-Verlet equation (3) from this source. From the article "Even order 2 follows from its...
Well I think it is cool anyhow ;)
Here is a dependency-free variable-order double pendulum simulation in about 120 lines of Python. Have you seen the equations of motion for this system?
As usual, this is based on code that I have provided here, but trimmed for maximum impact. Can you see...
hi all!
I’m trying to generalise the Caldeira-Leggett Hamiltonian (heat bath + particle) to the case of high velocities. Naturally, the multi-oscillator Hamiltonian needs to change and I have a gnawing suspicion that the multi-particle Hamiltonian is just the sum of single-particle hamiltonians...
in the Lagrangian mechanics, we assumed that the Lagrangian is a function of space coordinates, time and the derivative of those space coordinates by time (velocity) L(q,dq/dt,t).
to derive the Hamiltonian we used the Legendre transformation on L with respect to dq/dt and got
H = p*(dq/dt) -...
I was looking at the following proof of Louviles theorem :
we define a velocity field as V=(dpi/dt, dqi/dt). using Hamilton equations we find that div(V)=0. using continuity equation we find that the volume doesn't change.
I couldn't figure out the following :
1- the whole point was to show...
Hey, I have this chaotic system. It is a modified Hamiltonian Chaotic System and it is based on Henon-Heiles chaotic system. So I have this functions (as shown below). I want to know how can I make it as a discrete function. Like, how can I know the value for x dot and y dot.
1. Prefer to...
Hi guys! I am starting to study Hubbard model with application in DFT and I have some doubts how to solve the Hubbard Hamiltonian. I have the DFT modeled to Hubbard, where the homogeneous Hamiltonian is
$$ H = -t\sum_{\langle i,j \rangle}\sigma (\hat{c}_{i\sigma}^{\dagger}\hat{c}_{j\sigma} +...
Homework Statement
[/B]
A 1D spin chain corresponds to the following figure:
Suppose there are ##L## particles on the spin chain and that the ##i##th particle has spin corresponding to ##S=\frac{1}{2}(\sigma_i^x,\sigma_i^y,\sigma_i^z)##, where the ##\sigma##'s correspond to the Pauli spin...
Lagrangian Mechanics uses generalized coordinates and generalized velocities in configuration space.
Hamiltonian Mechanics uses coordinates and corresponding momenta in phase space.
Could anyone please explain the difference between configuration space and phase space.
Thank you in advance for...
A single particle Hamitonian ##H=\frac{m\dot{x}^{2}}{2}+\frac{m\dot{y}^{2}}{2}+\frac{x^{2}+y^{2}}{2}## can be expressed as: ##H=\frac{p_{x}^{2}}{2m}+\frac{p_{y}^{2}}{2m}+\frac{x^{2}+y^{2}}{2}## or even: ##H=\frac{p_{x}^{2}}{2m}+\frac{p_{y}^{2}}{2m}+\frac{\dot{p_{x}}^{2}+\dot{p_{x}}^{2}}{4}##...
Homework Statement
A particle moves on the ##xy## plane having it's trajectory described by the Hamiltonian
$$
H = p_{x}p_{y}cos(\omega t) + \frac{1}{2}(p_{x}^{2}+p_{y}^{2})sin(\omega t)
$$
a) Find a complete integral for the Hamilton-Jacobi Equation
b) Solve for ##x(t)## and ##y(t)## with...
In Lagrangian mechanics, both q(t) and dq/dt are treated as independent parameters. Similarly, in Hamiltonian mechanics q and p are treated as independent. How is this justified, considering you can derive the generalized velocity from the q(t) by just taking a time derivative. Does it have...
Homework Statement
I've constructed a 3D grid of n points in each direction (x, y, z; cube) and calculated the potential at each point.
For reference, the potential roughly looks like the harmonic oscillator: V≈r2+V0, referenced from the center of the cube.
I'm then constructing the Hamiltonian...
Homework Statement
I am given the Hamiltonian of the relativistic free particle. H(q,p)=sqrt(p^2c^2+m^2c^4) Assume c=1
1: Find Ham-1 and Ham-2 for m=0
2: Show L(q,q(dot))=-msqrt(1-(q(dot))^2/c^2)
3: Consider m=0, what does it mean?
Homework Equations
Ham-1: q(dot)=dH/dp
Ham-2: p(dot)=-dH/dq...
The Hamiltonian of the Qi, Wu, Zhang model is given by(in momentum space):
## H(\vec{k})=(sink_x) \sigma_{x}+(sink_y) \sigma_{y}+(m+cosk_x+cosk_y)\sigma_{z} ## .
What is the physical meaning of each component of this Hamiltonian?
Note: for the real space Hamiltonian(where maybe the analysis of...
Hello,
The hydrogen atom Hamiltonian is
$$H=\frac{p^2}{2m} -\frac{e^2}{r}\tag{1}$$
with e the elementary charge,m the mass of the electron,r the radius from the nucleus and p,the momentum. Apparently we can factorize H $$H=\gamma +\frac{1}{2m}\sum_{k=1}^{3}\left(\hat p_k+i\beta\frac{\hat...
Hello, I Have a non-Hermitian Hamiltonian, which is defined as an ill-condition numbered complex matrix, with non-orthogonal elements and linearily independent vectors spanning an open subspace.
However, when accurate initial conditions are given to the ODE of the Hamiltoanian, it appears to...
I have a question connected with the problem:
https://www.physicsforums.com/threads/continuity-equation-in-an-electromagnetic-field.673312/
Why don’t we assume H=H*? Isn’t hamiltonian in magnetic field a self-adjoint operator? Why? Why do we use (+iħ∇-e/c A)2 instead of (-iħ∇-e/c A)2 two times?
Homework Statement
I'm having some issues understanding a number of concepts in this section here. I attached the corresponding figure at the end of the post for reference.
Issue 1)
1st of all, I understand that a Hamiltonian can be written as such
$$H = T_2 - T_0 + U$$
whereby ##T_2##...
Hello, I am trying to "integrate into my understanding" the difference between Hamiltonian and Lagrangian mechanics.
In a nutshell: If Lagrange did all the work and formulated L = T - V, they why is Hamilton's name attached to the minimization principle?
YES; I KNOW about Hamilton's Second...
1. The problem statement
Consider a particle of mass m under the action of the one-dimensional harmonic oscillator potential. The Hamiltonian is given by
H = \frac{p^2}{2m} + \frac{m \omega ^2 x^2}{2}
Knowing that the ground state of the particle at a certain instant is described by the wave...