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Questions tagged [hamiltonian]

The central term in the hamiltonian formalism. Can be interpreted as an energy input, or "true" energy.

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What is meant by unitary time evolution?

According to the time evolution the system changes its state the with the passage of time. Is there any difference between time evolution and unitary time evolution?
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1answer
29 views

Total angular momentum in QM

Dos the total angular momentum, $J=S+L$, commute with the hamiltonian of a general sistem, with no particularities?
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1answer
50 views

Perturbation theory: justifying expansion in terms of eigenstates of the basis Hamiltonian

I have been wondering why anyone ever thought that we could find an expansion for eigenstates of some perturbed Hamiltonian in terms fo those for the basis Hamiltonian. My lecturer insisted that this ...
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1answer
36 views

How to diagonalise a hamiltonian which posesses symmetry?

I have a large hamiltonian but I know that it posseses some symmetries. How do you reduce the hamiltonian in order to find the eigenenergies?
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Is there a unitary transformation such that the Hamiltonian in the time-dependent Schrödinger equation becomes real symmetric?

The time-dependent Schroödinger equation is given as (with $\hbar=1$): $$i\dfrac{d}{dt}\psi(t)=H(t)\psi(t)\ ,$$ where $\psi$ is some normalized column vector and $H(t)$ is a Hermitian matrix with time-...
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QM: Time evolution with $H = H(t)$

In order to calculate time evolution in QM we use Schrödinger equation \begin{align*} i \partial_t |\psi\rangle_t = H(t) | \psi\rangle_t. \end{align*} If $H\neq H(t)$ then \begin{align*} i \partial_t ...
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1answer
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Why does the Hamiltonian represent something different after plugging in the solution?

so I am beginning to learn Hamiltonian mechanics. We have learned that the Hamiltonian is a function of q, p, and t. Once we have a Hamiltonian, we can use the Hamiltonian equations to derive the ...
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1answer
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How to expand this equation? $H_{1}=\frac{e^{2}}{R}+\frac{e^{2}}{R+x_{1}+x_{2}}-\frac{e^{2}}{R+x_{1}}-\frac{e^{2}}{R+x_{2}}$ [closed]

$$H_{1}=\frac{e^{2}}{R}+\frac{e^{2}}{R+x_{1}+x_{2}}-\frac{e^{2}}{R+x_{1}}-\frac{e^{2}}{R+x_{2}}$$ in the approximation $ \left |x_{1}\right |,\left |x_{2}\right |\ll R $ we expand to obtain in lowest ...
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Definition of Hamiltonian in Quantum Mechanics [duplicate]

Is there any particular reason that the Hamiltonian operator was defined in quantum mechanics to be $$\hat H := \frac{\hat p^2}{2m} + V$$ as opposed to $$\hat H := i\hbar \frac{\partial}{\partial t}?$$...
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3answers
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Deriving or building a Hamiltonian from a Density Matrix

Is it possible to create a Hamiltonian if given a Density Matrix. If you already the the Density Matrix, then is the Partition Function (Z) even needed? This Q is not about physics. Its about an ...
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How to prove time reversal symmetry in a system (given a hamiltonian)

The generic hamiltonian for a particle that interacts with an electromagnetic field can be written as: $$H=\frac{1}{2M}\sum_{i}\left(P_i-\frac{q}{c}A_{i}(X_j)\right)^2+V(X_j)+q\phi (X_j)$$ Where $(\...
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29 views

Utility of the Magnus expansion (preserving symplectic form?)

There are (at least) two ways to perturbatively solve a matrix initial value problem: the Dyson expansion and the Magnus expansion To be explicit, suppose you're solving for a density matrix $\rho(t)$...
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2answers
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How to find the coordinate representation of the kinetic operator?

From my professor's notes on statistical mechanics. $\left|\bf{k}\right\rangle$ is eigenstate of the hamiltonian of the free particle with periodic boundary conditions: $$ \left\langle{\bf r}|{\bf k}\...
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0answers
47 views

Transforming the Hamiltonian of a free quantum field

I have been trying to explicitly Lorentz transform the Hamiltonian of a free quantum field between two inertial observers, instead of reading it off the manifestly Lorentz invariant action. My ...
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1answer
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Hamiltonian for a variable length pendulum

This question is taken from the book "Classical Dynamics of Particles and Systems" - Marion, problem 7.24. The problem is about a pendulum that is set into motion, it's length varies at a constant ...
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0answers
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Looknig for resources on finding periodic orbit and stability on multidimensional Hamiltonian systems

I am looking for resources (books, papers, algorithms, codes) that explicitly explain the computation and analysis (using the monodromy matrix) of periodic orbits of multidimensional Hamiltonian ...
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1answer
55 views

When is the Hamiltonian a function of momenta alone?

In my statistical mechanics course, I'm deriving the entropy for an ideal gas and I've come across a statement in the book by Pathria where it states that in the case of an ideal gas, the Hamiltonian ...
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1answer
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Difference between the energy and the Hamiltonian in a specific example

The problem is the following: Consider a particle of mass $m$ confined in a long and thin hollow pipe, which rotates in the $xy$ plane with constant angular velocity $\omega$. The rotation axis ...
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0answers
19 views

How to construct a minimal model based $\vec{k} \cdot \vec{p}$ method and symmetry arguments?

Currently, I am repeating the results of this famous paper written by Di Xiao. In this paper, the authors construct a minimal band model based symmetry arguments and $\vec{k}\cdot\vec{p}$ method. The ...
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4answers
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How come the energy $E$ appears in the Time-Independent Schrödinger Equation, if only energy differences $\Delta E$ are actually physical?

Consider the time-independent Schrödinger equation: $$\operatorname{\hat H}\vert\Psi\rangle=E\vert\Psi\rangle$$ Is it not true that $E$ doesn't factor into any physically meaningful relation, and ...
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4answers
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The definition of the hamiltonian in lagrangian mechanics

So going through the "Analytical Mechanics by Hand and Finch". In section 1.10 of the book, the Hamiltonian $H$ is defined as: $$H = \sum_k{\dot{q_k}\frac{\partial L}{\partial \dot{q_k}} -L}.\tag{1.65}...
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1answer
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$4\times4$ Dirac Hamiltonian in Graphene

When linearizing the Hamiltonian of Graphene in reciprocal space around $\vec{q} = \vec{k}-\vec{K}_\pm = \vec{0}$, where $\vec{K}_\pm$ are two independent Dirac points, one can get two Hamiltonians, ...
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1answer
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How can I say whether a Hamiltonian is integrable or not?

The transverse field Ising Hamiltonian $$ H = J\sum_{i=0}^{N}\sigma_{i}^{z}\sigma_{i+1}^{z}+h_{x}\sum_{i=0}^{N}\sigma_{i}^{x} $$ is integrable because it can be exactly solved using Jordan Wigner ...
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1answer
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Hamiltonian Matrix for XXZ Model

Given the XXZ model Hamiltonian, $H = -\frac{1}{2}\sum^{N}_{i}(\sigma_{i}^{x}\sigma_{i+1}^{x}+\sigma_{i}^{y}\sigma_{i+1}^{y}+\Delta\sigma_{i}^{z}\sigma_{i+1}^{z})$ The two-site Hamiltonian reads $H ...
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understanding electron-photon interaction hamiltonian

My apologies for a somewhat out-of-context question in advance. In equation 26 of the paper here , they have the following hamiltonian $\hat{H_i} = \sum_{p',p, \mathbf{k}} = (g_{p,p',\mathbf{k}})(u^...
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2answers
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$q\mathbf{A}\cdot\mathbf{v}$ term in potential energy

In the famous Goldstein mechanics book, there is an example about a single (non-relativistic) particle of mass m and charge q moving in an E&M field. It says the force on the charge can be ...
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Diagonalizing the free Hamiltonian $\hat{H}$ in terms of smeared wave packet operators

Consider the free real scalar field in (1+1)-dimensions. If the (normal-ordered) Hamiltonian is $$ \hat{H}=\int_{-\infty}^{\infty} dk\ \sqrt{ k^2 + m^2 } \hat{a}_{k}^{\dagger} \hat{a}_{k} $$ where $\...
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1answer
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Confusion about why deducing pointer observable from the structure of the Hamiltonians is not practical

I am trying to learn Zurek's theory of decoherence. Right now I am reading Decoherence, einselection and the existential interpretation (the rough guide) which seems like an easier read than his big ...
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3answers
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Why operator of kinetic energy has a double derivative instead of square of single derivative?

I know that operator for $p = {h\over i} {d\over dx}$. so $p = {h\over i} {d\psi\over dx}$ where $\psi$ is the wave function. So, $T$ (kinetic energy) $ = {p^2 \over 2m} = {-h^2\over 2m} {d\psi \over ...
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1answer
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Hamiltonian operating on a function of time

I've seen a few people claiming: $$\hat{H(t)}[\psi(x)T(t)] = \hat{H(t)}[\psi(x)]T(t)\tag{1}$$ i.e. an explicit function of t is not acted upon by H, even if H itself may be dependent on t. A more ...
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1answer
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Intuition for Hamilton-Jacobi equation derived from least action

I am trying to understand the Hamilton-Jacobi equation without the framework of the canonical transformations. Even on the case of a 1D free particle I'm getting stuck. The system starts at fixed ...
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0answers
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How are energy eigenvalues of a wave-function related to the eigenvalue of matrices?

Schrodinger equation is a second order DE. Given a potential, I wish to perform standard -find the ground state energy, plot the wavefunction, find the reduced mass exercises using finite difference ...
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1answer
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Strange question involving finding a relation between a commutator and the time derivative of an operator

In order to get to the parts I am stuck at, I will add the examiners' solutions to each subquestion, which is needed to get to the subquestion that I am querying. The following is a bizarre question ...
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In Monte Carlo integration for Molecular dynamics simulation, why is a Boltzmann distribution assumed?

In statistical physics, The calculation of partition function for an ensemble takes a Boltzmann's distribution of the Hamiltonian. Similarly, In Monte-Carlo integration of Molecular Dynamics ...
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2answers
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Functional integration for the order parameter in $XY$ model

In the continuum limit the Hamiltonian of the classical XY model is given by, ignoring the inessential constant: $$H=\int d\vec{r}\ (\nabla\theta)^2$$ and the x-component of the order parameter is ...
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1answer
60 views

Delta potential in terms of annihilation/creation operators

Let the Hamiltonian of a system on a discrete lattice be given by $$ \mathcal{H} = \gamma \sum_\vec{x} c^\dagger_\vec{x}c^\vphantom{\dagger}_{\vec{x}+\vec{y}} + \text{h.c.}, $$ where $\gamma$ is ...
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2answers
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Proving orthogonality of eigenstates of a Hamiltonian

Suppose we have $\Psi_{1}$ and $\Psi_{2}$ which are eigenstates of some (self-adjoint) Hamiltonian $\hat{H}$ with unequal eigenvalues. Could you explain me how can I prove that these arbitrary $\Psi_{...
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1answer
63 views

What is the physical meaning of expectation value of the Hamiltonian operator?

I've been studying David Griffiths' Introduction to Quantum Mechanics and int that, it was explained that the expectation value of position $x$ is the average of the positions of $N$ identically ...
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1answer
35 views

Hermitian conjugate in Hubbard model hopping term

I'm new to Hubbard model and I have a few questions about it. From the sources I can find on the internet, Fermionic-Hubbard model is often written as (please correct me if I'm wrong!) \begin{align} H ...
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Expectation value of time dependent Hamiltonian

Let us assume that we have a time dependent Hamiltonian, precisely a Hamiltonian of a harmonic oscillator with time dependent frequency. \begin{equation} \hat{H}=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\...
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Hamiltonian for particle moving in a sphere

Context: I know that if we have a particle (say, with unit mass) moving in the plane $\Bbb R^2$ subject to a spherically symmetric potential $V\colon \Bbb R^2 \to \Bbb R$, it will move along the ...
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1answer
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Hamiltonian diagonalisation using quantum Fourier transform [closed]

Here is a problem to solve: diagonalize the following hamiltonian using quantum fourier transform. The hamiltonian reads: $$ \sum_{i,j=1}^N e^{-\theta_{ij}} c_i^\dagger c_j + h.c. $$ Where $c_j$ are ...
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1answer
249 views

Is the Hamiltonian of a relativistic charged particle in an electromagnetic field only an approximation?

Consider a system of two relativistic charged point particles 1 and 2 which interact through their electric and magnetic fields. The equation of motion for the first particle is then given by the ...
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1answer
53 views

Time evolution of operators with explicit time dependence in case of time dependent Hamiltonian

In case of a time dependent Hamiltonian of the sort $$H=\frac{p^2}{2m}+\frac{1}{2}m \omega(t) x^2$$ I have solved for the time evolution operator using the Schrodinger equation and got $U(t,0)$. If, I ...
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2answers
373 views

How do contact transformations differ from canonical transformations?

From Goldstein, 3rd edition, section 9.6, page 399 after equation 9.101: [...] The motion of a system in a time interval $dt$ can be described by an infinitesimal contact transformation generated ...
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Time dependent Hamiltonian operator and $SU(1,1)$ generator method

In this screenshot of a paper I am reading, I have the following question: 1.What is a $SU(1,1)$ group and how do we find its generators? 2.From the expression for the Hamiltonian $\hat{H}$, how do ...
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Diagonalization of Quadratic Fermionic/Bosonic Hamiltonians

I'm currently reading Quantum Theory of Finite Systems by Blaziot and Ripka, and I have a question regarding the first few pages of chapter 3. In particular, the chapter takes on the task of ...
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1answer
59 views

A fundamental question about Time-dependent Hamiltonians

I have a fundamental question about Quantum Mechanics or even mechanics in general. I am aware that there are stationary solutions and non-stationary solutions. The stationary solutions solve ...
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1answer
56 views

Eigenfunctions of Hamiltonian (question about the book “Quantum Field Theory for the Gifted Amateur”)

In the book "Quantum Field Theory for the Gifted Amateur" by Blundell and Lancaster, (page 21) the Hamiltonian (when discussing the number operator) is given by $$ \hat{H} = \left(\hat{a}^{\dagger}\...
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2answers
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Multi-electron Atom

I'm reading the following text on multi-electron atoms: for a system of $n$ electrons the Hamiltonian is $$ \hat H = -\frac 1 2 \sum_{i=1}^n \nabla_i^2 - \sum_{i=1}^n \frac{Z}{R_i} + \frac{1}{2} \...