Questions tagged [hamiltonian]

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

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Symmetry of the second partial derivatives of the Hamiltonian

A dynamical system with generalised particle position $q$ and generalised momentum $p$, described by: $$\dot{q}=F_1(q,p)\quad\text{and}\quad\dot{p}=F_2(q,p)\tag{1}$$ is a Hamiltonian system if: $$\...
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Coherent States of a Harmonic Oscillator

I have used the definitions of the annihilation and creation operators to determine the coherent state of a harmonic oscillator. I have derived the equation $$|\alpha \rangle=e^{\frac{-\alpha^{2}}{2}}\...
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Master equation with a coherent bath

When we consider an oscillator $a$ acting with a bath of oscillators $b_i$ with the interaction Hamiltonian reads $$H_{int}=\sum_{i}g_ia b_i^{\dagger}+g_i^*a^{\dagger}b_i,$$ with the free Hamiltonian: ...
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Does this relativistic generalization of the Schrodinger equation make sense? [duplicate]

So I'm aware that the correct relativistic approach to quantum mechanics is through quantum fields, but I'm still interested in the question that follows. We know the Schrodinger equation in free ...
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What is the role of Hermitian Hamiltonians in relativistic QFT?

In single-particle quantum mechanics, the probability of finding the particle in all space is conserved due to the hermiticity of the Hamiltonians (and remains equal to unity for all times, if ...
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Derivation of Hamiltonian $H=T+V$ from Lagrangian $L=T-V$

I understand that the Hamiltonian is the Legendre transform of the Lagrangian: $$ \begin{split}H(q,p,t) &= \frac{\partial L}{\partial \dot{q}}\dot{q} - L(q,\dot{q},t) \\ \implies H&=p\dot{q} -...
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Factorization of the wavefunction in a central Hamiltonian problem

I am trying to understand the topic of the title. If I consider a central Hamiltonian, so an Hamiltonian of the form $H=\frac{p^2}{2m}+V(r)$ what are the logical steps that lead me to the known result?...
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Questionable Taylor expansion for Peierls substitution

In this paper, on page 3, the authors go from the tight binding model w the Peierls substitution $$ H = \sum_{i,j} \sum_{a,b} t_{a,b} \exp\left(i \int_{\textbf{R}_{j,b}}^{\textbf{R}_{i,a}} dr'_{\mu} ...
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How to find the fine structure constant? [closed]

Is given in the scalar field below the lagrangin. How to find the finite structure constant?
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Coherent state from particle creation by a classical source

On page 32 of Peskin and Schroeder, it is found that the Hamiltonian of a Klein-Gordon field after a source $j(x)$ has been turned on and off is $$H=\int\frac{\mathrm{d}^3p}{(2\pi)^3}E_{\boldsymbol{p}}...
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The Hamiltonian of Schrödinger field: symmetrized and anti-symmetrized form

I have troubles to prove the eq.(1.1) in the article of S.Kamefuchi & Y.Takahashi: "A generalization of field quantization and statistics" Nucl.phys 36. (1962) 177-206: $$ H = \frac{1}{...
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Finding a bound on ground state energy derivatives of molecule

Part of a project I'm working on involves exploring the smoothness of the ground state energy $E(x)$ of a molecule where $E$ is parameterized by the geometry of the molecule. In particular, my goal ...
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Symmetry relations of a Hamiltonian (Mirror, Chiral, Inversion, Rotation)

I'm reading a paper https://journals.aps.org/prb/abstract/10.1103/PhysRevB.104.235136. Regarding Eq. (4)~(6), I have the following questions: (A) Eq. (5) shows a mirror symmetry relation $M_z H \left( ...
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How can I explain the following optomechanical transmission sprectrum?

I am considering the following (standard optomechanical) Hamiltonian in the rotating frame: $$H = -\Delta a^{\dagger}a + \omega_m b^{\dagger}b + g_0 * a^{\dagger}a (b + b^{\dagger})$$ for a cavity ...
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What is the correct general form of Hamilton's equation?

Usually, Hamilton's equations of motion are given by: $$ (1)\;\; \frac{dp}{dt} = -\frac{\partial H}{\partial q} \;\;\; \text{ and } \;\;\;(2)\;\; \frac{dq}{dt} =\frac{\partial H}{\partial p}.$$ ...
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Time-dependent canonical transform step in Hamiltonian perturbation theory (Percival problem 8.20)

In Percival and Richards's great book, "Introduction to Dynamics", problem 8.20 asks the following question. Any insight on how to solve this would be appreciated: Consider a system with ...
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Ising Model without periodic boundary conditions (PBC)

I try to calculate the correlation function $<\sigma_i \sigma_j>$ with the method of transfer matrices. I do understand how to use this method with PBC. But how can I do it without PBC? My ...
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In canonical transformation, is there any rules or methods for finding the transformation $(q,p)\to(Q,P)$?

If we get two different Hamiltonian by using two methods of canonical formulation of theory and these two Hamiltonian are equivalent. How can I find the canonical transformation from which we can ...
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Unitary transform using displacement operator to get time-independent Hamiltonian?

I am considering a driven cavity field with Hamiltonian $$H = \hbar\omega a^{\dagger}a + f(t)(a + a^{\dagger})$$ where $f(t) = \epsilon e^{-i\omega_{d}t} + \epsilon^* e^{i\omega_{d}t}$ is a classical ...
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$S$-matrix commutation with Hamiltonian

I know from scattering theory that $S$-matrix and the free Hamiltonian $H_{0}$ commute due to energy conservation of incident and outgoing asymptotic states, but can the $S$-matrix and $H = H_{0} + V$,...
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How to calculate surface states in Weyl semimetals?

I'm reading an article https://journals.aps.org/prb/abstract/10.1103/PhysRevB.89.235127. Fig. 2 in this article shows band structures calculated from Eq. (9), (13), (14), (15), and (16). For example, ...
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Commutator of $V(\hat{\vec{r}})$ and $\hat{L_z}$

Can someone explain to me why this is true? to me I see that $$x[\hat{p_y},V(r)]- y[\hat{p_x},V(r)]$$ $$=x(\hat{p_y}V(r)- V(r)\hat{p_y} )- y (\hat{p_x}V(r)+ \hat{p_x}V(r)).$$ The only way this can ...
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Why don't we chose $\det(H)$ in winding number density?

Hello i have a short question on the winding number in chiral systems. If we have a chiral system described by a hamiltonian like this $$ H = \left [ \begin{array}{cc} 0 & K \\ K^{\dagger} & 0 ...
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Why does time reversal symmetry requires a real Hamiltonian?

I have some problems understanding the consequences of time reversal symmetry. If Hamiltonian $H$ is symmetric under time reversal, it satisfies: $$ \mathcal{T} H \mathcal{T}^{-1} = H \quad \mathrm{...
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Two identical fermions in a box and the spin-statistics theorem

Consider the case of two identical non-interacting spin 1/2 particles in a box, whose length is $L$ with walls at $x=0$ and $x=L$. The Hamiltonian in this case (setting $m=1/2$, $\hbar=1$) is $$ H=H_1+...
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The constraint commute with Hamiltonian in Gauge theory

When canonical quantizing gauge theory, we find that the canonical momentum corresponding to $A_0$ vanish since the Lagrangian contains no $\dot{A_0}$ . Thus we need to choose a gauge, for example, $...
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Meaning of the Eigenvalues of the Hessian Matrix in Polar Coordinates in a Ferromagnetic System

I have the Hamiltonian for a magnetostatic system (exchange, dipole-dipole, zeeman) which is in polar coordinates since the spins are confined to being in-plane. If I calculate the Hessian Matrix of ...
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Evaluating a commutation relation - (Mahan's book)

I am trying to replicate one of the equations from Mahan's Many-body theory book. First the Hamiltonian $H$ is defined as: $$ H = \sum_i h_i \quad;\quad \text{where} \quad h_i=\frac{t}{2}\sum_{a}[c_{...
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Are the 'wrong' states eigenstates of perturbed Hamiltonian?

Townsend quantum mechanics In our earlier derivation we assumed that each unperturbed eigenstate $\left|\varphi_{n}^{(0)}\right\rangle$ turns smoothly into the exact eigenstate $\left|\psi_{n}\right\...
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Expression of density operator for Microcanonical ensemble

Consider a quantum microcanonical ensemble,with a fixed energy $E$. In Greiner, the expression for its density operator is given as. $\displaystyle\hat\rho=\frac{\delta(\hat H-E.1)}{Tr(\delta(\hat H-E....
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Commutator of positive semidefinite Hamiltonians

I have the following questions about the commutator of positive semidefinite Hamiltonians. Under what condition, the commutator will be positive semidefinite? Under what condition, the commutator ...
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Solving the time-dependent Schrödinger equation with a time delta potential

Suppose I want to model a system described by a Hamiltonian $H_0$ to which I give a quick kick at time $t = 0$. I would use the time-dependent Hamiltonian $$ \mathcal{H}(t) = H_0 + \bar{V} \delta (t)$$...
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How to find the ground state of a system via its Hamiltonian density?

I am trying to find the ground state of the following Lagrangian (with $\lambda> 0 , g > 0$): $$\tag{1} \mathcal{L}= -\frac{1}{2}(\partial_\mu \partial^\mu \sigma + \partial_\mu \pi \partial^\mu ...
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What is the non-relativistic differential equation for the helium atom?

I understand that the Schrodinger Equation for the hydrogen atom is $$E\Psi=-\frac{\hbar^2}{2m}\nabla^2\Psi+V\Psi$$ with $$-\frac{\hbar^2}{2m}\nabla^2\Psi+V\Psi$$ being the Hamiltonian and I ...
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Why are the eigenkets of perturbed hamiltonian the eigenkets of the perturbation matrix?

I've just started studying perturbation theory, and of course have now encountered the case where degeneracy arises. I understand why we have to diagonalize the perturbation matrix, and the concept of ...
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How to understand Heisenberg time in random matrix theory?

Recently, from few papers, I have encountered the word 'Heisenberg time' $t_{\text{H}}$ which is an inverse of a mean level spacing $\Delta(\hat{\mathcal{H}})$ of a finite system Hamiltonian $\hat{\...
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Hamiltonian of a free particle in spherical coordinates

Why would the hamiltonian, expressed in spherical coordinates, give a free partucle a non-zero acceleration in the radial direction? I'm learning it from Liboff's quantum mechanics book and he says it'...
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Confusion about symmetry spontaneous breaking in a simple model

Hamiltonian and symmetry I was learning about the symmetry spontaneous breaking (SSB), but confused by this simple minima model$$H=E_0\sum_{i=1}^3|i\rangle\langle i|+J\left(|1\rangle\langle 2|+|2\...
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Klein-Gordon Hamiltonian in terms of Fourier transformed variables

The Klein-Gordon Hamiltonian density is a function of four complex variables $\psi , \psi ^* , \pi , \pi ^*$. Suppose we make the change to Fourier transformed variables. Then the Fourier expansions ...
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Unlinearity of second order Stark effect in comparison to diagonalized Hamiltonian

I am extremely confused right now; please forgive the long title. From H.W. Kroto's Molecular Rotation Spectra (ISBN: 9780486672595) pp. 166 I read that in the second order perturbation theory, the ...
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I'm getting the wrong Hamiltonian of the quantum oscillator

First, I generalised the oscillator's Hamilton's equations to complex variables: $$\frac{dz_1}{dt}=\frac{\partial (z_1^2+z_2^2)}{\partial z_1}=2z_1$$ $$\frac{dz_2}{dt}=2z_2$$ So the real world ...
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Multi-mode Jaynes-Cummings Model

The Jaynes Cummings Model describes that a qubit coupled to a harmonics oscillator. The Hamiltonian of this model can be written as $$H_1=\omega_c a^{\dagger}a+\omega_a\sigma_z+\Omega(a^{\dagger}\...
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How to find wavefunction in this case?

We have a two interacting particle system, with Hamiltonian as: $$H_{s y s}=\frac{\mathbf{p}_{1}^{2}}{2 m_{1}}+\frac{\mathbf{p}_{2}^{2}}{2 m_{2}}+V\left(\mathbf{r}_{1}, \mathbf{r}_{2}\right).$$ we ...
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Term in Hamiltonian when electric field is turned on

In David Tong's notes on Quantum Hall Effect, in his derivation of the Kubo formula, he says that turning on the electric field has the effect of adding a $- \mathbf{J} \cdot \mathbf{A}$ to the ...
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Schwartz book: Heisenberg equation of motion for quantum fields (equation 7.28)

In his book "Quantum field theory and the standard model", section 7.2 "Hamiltonian derivation" (of the Feynman rules), Schwartz states that the equations of motion $i\partial_t\...
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When is the motion of the perturbative term resonant with the Hamiltonian?

I am given the following Hamiltonian, $H$, which is a perturbed version of $H_0$, $$ H(\theta,I) = H_0(I) -\epsilon \cos(\theta-\Omega t)$$ where $H_0 = \frac{I^2}{2}$, $\epsilon << 1$ and $(I,\...
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Why is this hamiltonian not the energy? [duplicate]

Let a pendulum of length $\ell$ be connected to a rod that rotates with constant angular velocity $\omega$. $\theta$ is the angle of the pendulum wrt $z$ axis ($z$ axis is parallel to the rod). I ...
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Is an Hamiltonian like this non-degenerate? [closed]

If I have a system of only two energy levels $E_0$ and $E_1$ and an Hamiltonian $$H=\begin{pmatrix} E_0 & 0 \\ 0 & E_1 \\ \end{pmatrix}$$ can the Hamiltonian be both degenerate and non-...
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Why does a symmetry operator commute with the Hamiltonian?

Suppose that a symmetry operator $O$ leaves the Hamiltonian $H$ unchanged. From books I know that there should be the relation $OH=HO$. But I don't understand why it is not that $H=HO$ since when the ...
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Energy eigenvalues for time evolution operator

I have to solve this question for a homework: Suppose the state of a single mode of the electromagnetic field is give at time $t = 0$ by: $$|\psi(0)\rangle=\frac{1}{\sqrt{2}}(|n\rangle+e^{i\phi}|n+1\...
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