Questions tagged [hamiltonian]

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

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Quasi-periodic motion of $N$-particle systems [closed]

My question is about the time evolution of multi-particle systems in QFT. There are such systems evolving a-periodically. I struggle with the treatment of them, always obtaining periodic or quasi-...
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Do different Hamiltonians result in different ground states?

I'm learning density functional theory. In the proof of Hohenberg–Kohn theorem I, we assume that different Hamiltonians result in different ground states. Is it true? In general, for example, we can ...
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Using variation principle on quantum oscillator with general potential

Consider a general bounding potential $V(x)$. The hamiltonian is $$H = \frac{p^2}{2m} + V(x).$$ We want to apply the variation principle in equation $$F\leq F_0+\langle H-H_0\rangle_0.$$ $\langle\...
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Lindblad equation from microscopic principles for free particles with momentum interactions

I'm rather familiar with the formalism of quantum master equations, but I'm struggling with deriving from microscopic principles the collapse operators for a particular case I need. I consider two ...
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Chemical potential terms in Hamiltonian

In the derivation of grand canonical ensemble, which assumes that the physical system (with Hamiltonian $H$) has an average energy $E$ and an average number of particles $\bar{N}$, the density matrix ...
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1 answer
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Non-abelian magnetic field interaction

Consider the magnetic part of a non-Abelian field. I want to know if we can define Hamiltonian for the interaction of this sector with the spin (something such as Hamiltonian of interaction of ...
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Why does the Eigen-Energy-Shift for the AC-Stark-Shift, calculated in the "rotating frame", actually matter?

As is neatly described in this answer, the AC-Stark Shift is the observation that the energy-eigenvalues of stationary states in the "rotating frame" of a two-level-system behave as (Adam ...
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Hamiltonian in a semiconductor quantum well

I am currently studying for a photonics exam, and in particular the topic of this lecture is the energy dispersion of a quantum well in a semiconductor heterojunction. My professor wrote a particular ...
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Magnon spectrum with in-plane Dzyaloshinski-Moria interraction

I'm attempting to derive a magnon spectra using first principles. In-plane DMI is considered in my system. $z$ axis is out of plane. First of all I wrote a Hamiltonian for interacting spins with two ...
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Hamiltonian of the BEC in 2nd quantization [closed]

If I have $N$ non-interacting particle (bosons) forming a BEC that is trapped to $x = 0 $ (assume the system to be 1D) by an applied harmonic potential $V=\frac{1}{2}m\omega^{2}x^{2}$ How can I write ...
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When are two quantum descriptions / models equivalent?

I am occupied with different types of quantization methods for constrained systems. I start with a constrained phase space and then follow two different paths to get rid of the constraints. In the end,...
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Is the QFT Hamiltonian on an eternal Schwarzschild black hole background unbounded below?

Consider the $t=0$ Cauchy slice of the maximally extended Schwarzschild black hole. Let the parts of the slice to the left and right of the bifurcation surface have Hilbert spaces $\mathcal{H}_L $ and ...
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The relation of time-evolution operators from Schwartz's textbook

In the section 7.2.2 of Schwartz's QFT textbook, it says: define the generation definition of time-evolution operators: $$U_{21}\equiv U(t_2,t_1)=T{\exp[-i\int^{t_2}_{t_1} dt'V_I(t')]}\tag{7.46}$$ ...
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Trace of a Hamiltonian and zero-point energy

In finite-dimensional quantum mechanics, we are free to assume that our Hamiltonian is traceless. We can define $H' = H - \mathrm{tr}(H)\cdot I$, and since \begin{equation} \exp(-iH't) = \exp(-iHt)\...
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How does a general operator in a tensor product space act on operators explicitly written as the tensor product of operators in each subspace?

Suppose I model an atom as a two-level system with states $|g \rangle$ and $|e\rangle$, with eigenvalue equations $\hat{H_1}|g\rangle = g|g \rangle$ and $\hat{H_1}|e\rangle = e|e \rangle$, and an ...
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Derivative of parameterized time-evolution operator

Suppose we have a parameterized time-dependent Hamiltonian $$H(t; \theta) = H_0 + f(t; \theta)V_0$$, is there any way to obtain the derivative of time-evolution operator $\frac{d}{d\theta} U(t, t_0; \...
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Why the first-order derivative is missing when composing a Hamiltonian of simple harmonic oscillator by the lowering and the raising operators? [closed]

Given the lowering operator ($a$) and the raising operator ($a^\dagger$) $$\begin{align*} a &= \frac{1}{\sqrt{2m \hbar \omega}}\left(-i \hbar \frac{\partial}{\partial x} - i m \omega x\right) \\ a^...
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How to conceptually understand the symmetries of the Heisenberg Antiferromagnet

Consider the Heisenberg antiferromagnet's Hamiltonian: $$ \hat{H} = -\frac{1}{2}\sum_{n = 1}^L\bigg(J_x \hat{\sigma}^x_n \hat{\sigma}^x_{n+1} + J_y \hat{\sigma}^y_n \hat{\sigma}^y_{n+1} + J_z \hat{\...
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Derivative term in chain of LC coupled oscillators Hamiltonian

I am taking quantum superconducting circuits course and I cannot recover a formula provided by the lecturer. I want to calculate the Hamiltonian of the following distributed element model of coplanar ...
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How to derive the formal solution of Heisenberg's equation? [closed]

In the book Introductory to Quantum Optics https://ostad.hormozgan.ac.ir/ostad/UploadedFiles/386042/386042-1758823246346514.pdf, we have that for an arbitrary operator $\hat{O}$ having no explicit ...
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When solving the Schrodinger Equation, where do we add the condition that $E$ is real?

I'm reading through Griffiths, and I noticed two seemingly contradictory facts: In Chapter 1, it is proved that for any square-integrable function solving the Schrodinger Equation, $$\frac{d}{dt}\...
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How do we know that the operator $U(t) = e^{-itH/\hbar}$ corresponds to time?

By Stone's theorem on one-parameter unitary groups we know that there is a one-to-one correspondence between strongly continuous one-parameter unitary groups and self-adjoint operators. Hence, if $H$ (...
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Justification for Dropping Cross Terms in Transmon Hamiltonian

I am working through the "Introduction to Transmon Physics" page on Qiskit, and there is a part of the transmon Hamiltonian derivation that I am stuck on. The initial form of the transmon ...
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Commutation $[H_0, \phi_0(\vec{x},t)]$ in the Heisenberg picture [closed]

Studying from Schwartz "Quantum Field Theory and the Standard Model" p. 23, I got to the part where he discusses time dependence of the field operator $\phi$ and the annihilation/creation ...
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Why is the sign negative in time evolution operator in Quantum Mechanics? [duplicate]

Disclaimer: The question has been answered in related posts, example this one which addresses LSZ reduction formula without explaining the minus sign issue. We consider a unitary operator $U(t+dt,t):\...
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Adiabatic theorem for a 3-level system

If I have a 2-level system, with the energy splitting between the 2 levels $\omega_{12}$ and an external perturbation characterized by a frequency $\omega_P$, if $\omega_{12}>>\omega_P$ I can ...
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Understanding (non-degenerate and degenerate) Perturbation theory

I was learning the basics of perturbation theory (PT) in QM, and I had quite a bit of trouble with it, especially (time independent) degenerate PT. My reference books are primarily Sakurai (became too ...
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Discretized derivation of Majorana path integral

Shankar's QFT book gives an overview for deriving a path integral representation for Majorana fermions. In the derivation, he works directly in continuous imaginary time, sweeping issues of ...
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Confusion about the first-order correction term of an eigenstate (Degenerate Perturbation Theory)

I'll try my best to put into words what I want to know. But I have the feeling that it is hard to properly explain and correctly articulate the different notations and formulas for this particular ...
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Hamiltonian as a matrix and its elements [closed]

Let us consider an electron in an infinitely deep one-dimensional potential well of thickness L with zero potential energy at the bottom of the well. The normalised eigenfunction solutions to this can ...
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Status of Approach of constructing Hamiltonians from Transfer Matrix

I am studying this old paper from J.B.Kogut on lattice gauge theories and spin systems [Rev. Mod. Phys. 51, 659(1979)]. This paper discusses about the way of constructing a quantum Hamiltonian using ...
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General form of a fermionic Hamiltonian in second quantization

I am a new to quantum chemistry. I've been reading this paper and it seems like the equation (1) in the paper which is, $H = \sum_{pq} T_{pq} a_p^\dagger a_q + \sum_p U_pn_p + \sum_{pq} V_{pq} n_p n_q$...
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Interpretation of the Hamiltonian of a real scalar field in FLRW background

In spatially flat FLRW spacetime, using conformal time, the action of a real free massless scalar field is $$S=\int d \eta d^3 x \frac{a^2}{2}\left[\left(\phi^{\prime}\right)^2-(\nabla \phi)^2\right]$$...
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Numerically approximating eigenstates and energies of a particle subject to some potential. Question about dirac orthonormality and infinite terms

Context I am attempting to approximate the eigenstates and energies of a particle over an interval $[-a,a]$ subject to some potential. The goal is then to approximate the wavefunction $\Psi$ with ...
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Boundedness of a Hamiltonian and when does a Hamiltonian have a spectrum?

In the context of Quantum Field Theory we put restrictions on the potentials we can use. One argument is boundedness. If the potential is unbounded, for example $V(\phi) = \phi^3$, then `the field can ...
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Hamiltonian density is time-independent, so how does energy get transported in QFT?

In a QFT on $d+1$-dimensional Minkowski spacetime, the Hamiltonian is: $$\tag{1} \hat{H} = \int d^{d}\vec{x} \hat{\mathcal{H}}(\vec{x}), $$ where $\hat{\mathcal{H}}(\vec{x})$ is the Hamiltonian ...
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Gauge Invariance and Observables [duplicate]

We know that observables are gauge invariant. We know that energy is an observable. Now, the Hamiltonian of a charged particle in an electromagnetic field is not gauge invariant as: $$ H=\frac{1}{2 m}\...
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On time-evolution of a quantum state

Suppose I have a quantum system governed by a time-independent Hamiltonian $H$. Its eigenvectors $\{|\varphi_n\rangle\}_{n\in\mathbb{N}}$ form a complete orthonormal set (or basis) for the Hilbert ...
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Time evolution operator of a two-level system with a completely general Hamiltonian

Using the identity ($\sigma_{0}$) and the Pauli operators ($\sigma_{i}$) as a basis, the Hamiltonian of any two-level system can be expressed as follows $$ H =\alpha \cdot \sigma _{0}+\textbf{r}\cdot\...
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Kinetic term of the Hamiltonian constructed from the action of perturbative string motion is not positive definite

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....
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Condition for stationary density matrix

I have a question about section 5 in 'Statistical mechanics' (Pathria). According this book, the density matrix (operator) should satisfy the following identity, which describes the time evolution of ...
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Why is the full Hamiltonian used instead of the approximate Hamiltonian for determining the effective nuclear charge using the variational principle?

My question is in regards to the variational principle in approximating the wavefunction of Helium. Some Background: $$\hat{H}=-\frac{\hbar^2}{2m_{e}}\nabla_{1}^{2}-\frac{\hbar^2}{2m_{e}}\nabla_{2}^{2}...
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Why do we ever use the Dyson series?

In dealing with interacting Hamiltonians, it's common to do expansions of some sort to make the problem tractable. Two common methods are the Dyson and Magnus expansion. Of the two, the Magnus ...
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Observables commuting with the Hamiltonian [duplicate]

Why is it that observables which commute with the Hamiltonian are constants of the motion? I can't understand why that should be true.
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Meaning of the transition amplitudes in time dependent perturbation theory

Consider a time dependent Hamiltonian $$\hat{H}(t)=\hat{H}_0+\hat{V}\tag{1}\label{1}(t)$$ where $\hat{H}_0$ is the unperturbed Hamiltonian, for which the eigenvalue problem has been solved $$\hat{H_0}\...
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3 votes
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Factorization of 1d Ising model partition function

If I'm studying a 1-dimensional Ising model such that $\mathcal H = \sum_k J_k\sigma_k\sigma_{k+1}$, where $$J_k=\begin{cases}J&k \in2\mathbb N\\2J&k\in2\mathbb N+1 \end{cases}$$ can I ...
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Propagator for radial force field?

The propagator $K(x,y;t)$ is well known for the (1D) harmonic oscillator: $$H = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + \frac{m}{2}\omega^2 x^2$$ is there a simple closed form solution ...
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1 answer
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Hamiltonian and Thermal Energy

I was reviewing my lessons when I read this definition on Thermal Energy: The sum of potential energy and kinetic energy is equal to Thermal Energy But isn't this the same as the Hamiltonian? So, ...
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How does $[H,H] = 0$ imply $dH/dt = 0$?

I have a conceptual question about how $[H,H] = 0$ implies $dH/dt = 0$. Does this relation work for both time-dependent and -independent Hamiltonians, or a general observable $H$?
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Time Independent Schrödinger Equation meaning

Working on some QM and we realised we don't understand the simple equation is for the wavefunction. $H \psi(x) = E \psi(x)$ We know $H$ is the hamiltonian, the sum of the kinetic and potential ...

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