Questions tagged [hamiltonian]
The central term in the hamiltonian formalism. Can be interpreted as an energy input, or "true" energy.
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questions
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Conditions of separation of variable of wave function
In quantum mechanics as far as I know when the hamiltonian operator
can be written as
$$H = H_1 + H_2$$
then the wave function can be written as
$$\psi_1 \cdot \psi_2$$
but as we get further in ...
1
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0answers
59 views
Calculation of effective Hamiltonian
I'm stuck with calculating the effective Hamiltonian of a model system. The energy-dependence of the effective Hamiltonian is confusing me a lot. My question is how to treat this dependence ...
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1answer
26 views
Can energy eigenstates be in a superposition of quantum numbers?
I know that given say 4 quantum numbers $J^2$, $J_z$, $J_1^2$, $J_2^2$ (e.g. for the Hamiltonian $H=\lambda J_{1}.J{_2}$), the state |$J$,$J_z$,$J_1^2$,$J_2^2$>=|2,2,1,1> will be an energy eigenstate (...
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1answer
46 views
Energy degeneracy given a rigid-rotor Hamiltonian
I'm trying to work out the degeneracy of some energy levels in a Hamiltonian given by
$H = \frac{1}{2a} (L_x^2 + L_y^2) + \frac{1}{2b} L_z^2$.
Looking for a common base of eigenstates $Y_l^m$ ...
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2answers
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Numerical calculation of a quantum field's observables
Okay so QFT is definitely beautiful and elegant theory, its mathematics is rich and ingenious, but there is so much one can do with symbolic manipulations of mathematical objects only, how can I ...
3
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1answer
59 views
Unitary time evolution operator for hamiltonian
I have homework, which I've seen solution with which I have some problem.
The homework is to find time evolution of state $|\psi(t)\rangle$, if the hamiltonian of the state is $\hat{\textbf{H}}=\...
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39 views
Hamiltonian for the spin-orbit coupling in spinor-notation
After some research I can't find a good source which explains how to write the Hamiltonian of a spin-orbit coupling in spinor notation.
The following notation is commonly used $\hat H =\xi \hat L.\...
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1answer
49 views
Why does Hamiltonian no longer commutes with $\mathbf L$ and $\mathbf S$ in presence of spin orbit coupling?
Is this something due to change in Hamiltonian due to relativistic correction.
i.e. $H$ being $$(\frac{e^2}{4\pi {\epsilon _0}}) \mathbf{S \cdot L}$$ instead of $$(\frac{e^2}{8\pi {\epsilon _0}}) \...
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28 views
In the derivation of Kubo Formula for electrical conductivity, how we can calculate the perturbation term which is due to the total electric field?
The perturbed Hamiltonian for the system can be taken as $H + V$, where $V$ is interaction between the total electric field and the particles of the system. In my calculation I need to find a form of ...
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1answer
82 views
Functional Analytic Square Root of Hamiltonian Alternative to Dirac
I was thinking about the history of the Dirac equation and asked myself, what happens if one simply considers the Schrƶdinger equation $i\hbar\frac{\partial\phi}{\partial t}=\sqrt{-c^2\hbar^2\Delta+m^...
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1answer
24 views
Hamiltonian for a system of 3 interacting particles and the meaning of potential
I'd like to discuss some physics basic.
assume we have 3 particles $\{\vec{q_i},\vec{p_i}\}, \quad i=1,2,3$
whereas $\vec{q_i}$ is the position and $\vec{p_i}$ is the momentum.
We also have a "pair-...
3
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0answers
21 views
Projection operator (relative angular momentum) in FQHE Toy hamiltonian
I am working on Fractional Quantum Hall Effect and reading these lecture notes http://www.damtp.cam.ac.uk/user/tong/qhe/qhe.pdf. As all others sources I have found, none of them precisely define the ...
3
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1answer
44 views
Is it possible for a quantum system to evolve out of a determinate state of some observable before measurement is made?
On page 96 of his book, Griffiths explains that determinate states of some observable $Q$ are eigenfunctions of that operator. So if a particle starts out in that state it will continue to be in that ...
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In context of defining a hamiltonian of a system, can someone please explain the meaning of zero-point oscillations/vibrations?
I came across this term while studying electron delocalization in amorphous semiconductors. The delocalization of the electronics wavefunction is explained in terms of zero-point oscillations.
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Convert a Lindbladian time evolution operator to the Kraus operator sum representation
I try to understand how I can convert a Lindbladian time evolution operator to the corresponding Kraus operator sum. Let's assume we have a time independent Hamiltonian $H$ and a set of time ...
1
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1answer
30 views
Significance of sign of perturbation matrix
I understand that the off-diagonal elements of a Hamiltonian denote the interaction between different states. The magnitude of off-diagonal elements therefore tells us how strong the interaction is. ...
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1answer
41 views
In QM perturbation theory, is the system generally in an eigenstate of the perturbing Hamiltonian, $\hat H_1$?
In my notes the derivation of the second order energy correction we don't do the following:
$$\sum_k a_{nk}\langle\phi_n|\hat H_1|\phi_k\rangle=\sum_ka_{nk}E_k^{(1)}\langle\phi_n|\phi_k\rangle$$
...
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2answers
52 views
Why can a partial derivative be added to a hamiltonian in canonical transformations?
In canonical transformations, how come we allow hamiltonian to change by a partial derivative of time?
$$H'(P, Q, t) = H(p, q, t) + \frac{\partial F}{\partial t}.$$
Here $F$ is the generating function....
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1answer
23 views
Evolution of quantum state that cannot exist with accompanying Hamiltonian
So I am studying a certain Hamiltonian that has projection operators in its definition. To keep it simple, suppose our Hilbert space is a one particle system that can be spin up/spin down (excited, ...
1
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1answer
86 views
Eigenfrequencies of an Hamiltonian dynamical system in different bases
Consider the Hamiltonian
$$
H=H(x,y,p_x,p_y)
$$
which generates the dynamical system
$$
\dot{x}=+\frac{\partial H}{\partial p_x}
$$
$$
\dot{y}=+\frac{\partial H}{\partial p_y}
$$
$$
\dot{...
3
votes
2answers
131 views
How come the eigenvalues of the Hamiltonian represent energy levels when the Hamilton doesn't represent the energy of the system?
Like in the Hamiltonian for a particle in an electromagnetic field. This is not a conservative field so the Hamiltonian doesn't represent the energy of the system. And yet the time independent ...
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0answers
45 views
Effective hamiltonian method for the quantum Ising model
I am reading Subir Sachdevs book on quantum phase transitions (second edition). In chapter 5 (page 58) he defines a hamiltonian
$H=H_0+H_1$ where the eigenstates of $H_0$ are known and the influence ...
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2answers
68 views
How to argue on physical grounds that a function is the ground state of a Hamiltonian?
$u_l(r) = Ar^{l+1}e^{-\kappa r}$ is provided as a solution to the radial wave equation for the Coulomb potential $$-\frac{\hbar^2}{2m}\frac{d^2}{dr^2}u_l(r)+\Bigl[\frac{\hbar^2}{2m}\frac{l(l+1)}{r^2} -...
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24 views
Defining the number operator for a general Hamiltonian
I have a Hamiltonian of the following general form,
$$H=\int \frac{d^{3} k}{(2 \pi)^{3}}\left[A_{\vec{k}}(t) a_{\vec{k}}^{\dagger} a_{\vec{k}}+B_{\vec{k}}(t) b_{\vec{k}}^{\dagger} b_{\vec{k}}+C_{\vec{...
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1answer
73 views
Derive hamiltonian from equations of motion
Is there a method for deriving the hamiltonian given that you know the equations of motion?
For example given the equation (equation 5 in paper linked) they simply the derive the Hamiltonian in ...
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1answer
66 views
Spectrum of Dirac Hamiltonian
The Dirac Hamiltonian is given by,
\begin{aligned}
H &=\sum_s\int \frac{d^{3} p}{(2 \pi)^{3}} E_{\vec{p}}\left[b_{\vec{p}}^{s \dagger} b_{\vec{p}}^{s}+c_{\vec{p}}^{s \dagger} c_{\vec{p}}^{s}\right]...
2
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0answers
38 views
Electron-Phonon Hamiltonian and One-particle Approximation
Consider a hamiltonian with electron-phonon coupling, for instance, a very simple version of Holstein hamiltonian: $$t\sum_k \hat{c}^\dagger_k \hat{c}_{k+1}+\text{h.c.}+\hat{b}^\dagger \hat{b}+\alpha ...
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4answers
251 views
Hamiltonian of non-regular Lagrangian is well-defined on phase space
In section 1.1.3 of Quantization of Gauge Systems by Henneaux and Teitelboim, it is stated that the Hamiltonian
$$H=\dot{q}^np_n-L,\tag{1.8}$$
although trivially a function of $q$ and $\dot{q}$, can ...
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0answers
58 views
Inversion symmetry in Bloch bands
In Bloch bands, when we consider the energy degeneracy caused by translation symmetry and spatial inversion symmetry, we use the standard procedure like in Bloch waves under space inversion (parity) ...
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1answer
27 views
Hamiltonian in explicit Spin z basis
For a time independent Hamiltonian
$H = \frac{\mu}{\hbar}(\vec{S_1} + \vec{S_2})*\vec{B}$
and $B= (0,0,B_0)^T $, $\vec{S}= \frac{\hbar}{2}\vec{\sigma}$
I want to find the explicit Hamiltonoperator ...
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0answers
21 views
Exchange Momentum With Position Preserves Hamiltonian
I read a paper a while back that showed you can exchange the roles of p and q in a Hamiltonian and still retain its Hamiltonian nature. In some cases, this can simplify the problem. For the ...
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Minimal coupling to electric dipole form - II
In addition to link, may I ask you for details of the Hamiltonian transformation.
Knowing that:
$[\textbf{x},\textbf{p}]=i\hbar$, $[\textbf{x},\textbf{p}^2]=2i\hbar\textbf{p}$ and $e^{-i\alpha A}Be^{...
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60 views
This step in Griffiths' Introduction to Quantum Mechanics book
I am working my way through time-dependent perturbation theory at the moment. There is a derivation of the formula for determining the time-dependent coefficients, $c_a(t)$, $c_b(t)$, which I am stuck ...
3
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1answer
65 views
Extension of classical Liouville operator
Let us consider a classical Hamiltonian system described by the Hamiltonian
\begin{equation}
H(q,p) =\frac{p^2}{2m}+V(q)
\end{equation}
where we stick to the case of single particle for simplicity. I ...
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0answers
23 views
Scalar hamiltonian and electromagnetic transitions
How can a Hamiltonian be a scalar and allow transitions between states with different angular momentum at the same time? Electromagnetic induced transitions are usually represented as a perturbation ...
0
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1answer
41 views
The Hamiltonian and differentials
From Lifshitz and Landau Vol.$1$:
From the equation in differentials
$$
\mathrm{d} H=-\sum \dot{p}_{i} \mathrm{d} q_{i}+\sum \dot{q}_{i} \mathrm{d} p_{i}
$$
in which the independent variables ...
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0answers
37 views
Acting state on pauli matrix and problem with dimension
I define local hamiltonian as:
$ H_{\Omega_{k}}=\frac{\mathbb{1}- \sigma^{z}}{2} \frac{\mathbb{1}- \sigma^{z}}{2} \in M_{2{\times}2}
$
where $\sigma^{z}$ is pauli operator.
And prepare the state:
...
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vote
1answer
58 views
Chaotic Hamiltonian system poincare surfaces depend on the integrator
First question on StackOverflow so go easy on me. I have a Hamiltonian system that consists on the following Hamiltonian:
$H(p,x;\textbf{P,X})=\frac{p^2}{2m}-a\frac{x^2}{2}+b\frac{x^4}{4}+x\sum_{n=1}^...
0
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1answer
56 views
Poisson bracket of Hamiltonian with Hamiltonian always vanishes
Since Poisson bracket of Hamiltonian with Hamiltonian always vanishes then in case of explicit time dependence of Hamiltonian, how does Poisson bracket gives correct result?
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1answer
38 views
Validity of Linear Response Theory
Suppose the perturbation of the hamiltonian is some multiple of the free hamiltonian, that is
$$H=H_0+H_1=H_0+\lambda H_0=(1+\lambda)H_0.$$
Here, certain operators apparently have no response due to ...
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0answers
24 views
How to generate Hubbard Hamiltonian?
I am a beginner with Hubbard Hamiltonian and my question is very basic:
how can I generate the matrix form of the Hubbard Hamiltonian?
I know the theory but I don't know how to put it numerically.
$$\...
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votes
1answer
26 views
Balance the units of the following hamiltonian
The following image is taken from an article and shows the hamiltonian of a spin chain model. I knew that the dimensional units in an equation must balance. To ensure this, the author took a procedure ...
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1answer
77 views
What happens to the configuration manifold when one quantizes the Hamiltonian?
A system in classical mechanics can be described by a configuration manifold $Q$ and a Lagrangian
\begin{equation}
L:TQ\rightarrow \mathbb{R}
\end{equation}
where $TQ$ is the tangent bundle or a ...
2
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0answers
38 views
How to check whether Weyl field Hamiltonian is bounded below?
When constructing the Lagrangian for a two-component left-handed Weyl field $\psi$, in e.g. Srednicki, one rejects the choice of $\partial^\mu \psi \partial_\mu \psi+\partial^\mu \psi^\dagger \...
2
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0answers
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“Unnatural” Hamiltonian systems from a statistician's perspective
I would like to learn more about "unnatural" Hamiltonian systems, that is, systems whose energies cannot be written as
$$H(p,q) = K(q) + U(p).$$
I have seen the term "natural" applied to systems ...
6
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2answers
164 views
Are the classical hamiltonian and quantum hamiltonian different types of objects?
Context: I'm not a physicist.
I've come across the Hamiltonian in classical physics and in quantum physics, and I can't recognise why they have the same name. They seem very different. So I probably ...
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1answer
116 views
A non-Hermitian system whose “Hamiltonian” is the annihilation operator
Consider a notional quantum system whose "Hamiltonian" is the annihilation operator,
$$H=a .$$
Its initial state $|Ļ(0)\rangle$ is
$$|\psi(0)\rangle=\sum_{n=0}^{\infty} c_{n}| n\...
0
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2answers
102 views
Why is the Hamiltonian of a photon = 0?
I'm studying the motion of light near Schwarzschild black holes, and I was wondering why the Hamiltonian of the Schwarzschild metric $$H = - \left( 1-\frac{2M}{r} \right)^{-1} \frac{p_{t}^2}{2}+\left(...
3
votes
2answers
154 views
Can any sum of infinitesimal canonical transforms on phase space be obtained from evolution under a static Hamiltonian?
Suppose I have a canonical transformation on phase space, which is obtained by evolving a classical Hamiltonian system from time $t=0$ to $t=T$, with some arbitrary time-dependent Hamiltonian $H(t)$. ...
0
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1answer
52 views
Eigenfunctions of a quantum plane rotor?
I'm trying to determine the energetic levels of a system with Hamiltonian
$$H=-\frac{h^2}{2m}\frac{\partial^2}{\partial \phi^2}$$
And the border condition
$$\psi(0)=\psi(2\pi)$$
The eigenvalues ...
