Questions tagged [hamiltonian]

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

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Confused about AC Stark effect

Can someone explain to me in an intuitive way (or a nice mathematical demonstration) or point me towards some accessible papers about the AC Stark effect (Autler-Townes effect)? I am mainly confused ...
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Are partial derivatives in the context of Action-Angle variables different from partial derivatives of functions?

Let's say I have a system with two degrees of freedom and I can find two independent action variables. One action variable is total energy expression, such as is often used in classical mechanics. $$...
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Reference for mathematical time-independent perturbation theory

I was wondering if anyone knew of a reference which treat time-independent perturbation theory in a mathematical fashion. All of the mathematical textbooks in quantum mechanics that I know of do not ...
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Why is there a chemical potential term in the Bose-Hubbard Hamiltonian?

When looking at the Bose-Hubbard Hamiltonian $$H_{BH}=-t\sum_{\langle i,j\rangle}a^\dagger_ia_j+\frac{U}{2}\sum_ia^\dagger_ia_i(a_i^\dagger a_i-1)-\mu\sum_ia_i^\dagger a_i,$$ I'm wondering why there ...
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Isn’t the Hamiltonian always hermitian?

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.118.130201 If the analysis presented here can be made rigorous to show that ^ H is manifestly self-adjoint, then this implies that the ...
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Correspondence between quantum operators and classical formulas

Background From what knowledge of quantum mechanics I have so far, it is a postulate that Hermitian operators corresponding to a certain observable act on a quantum state $\psi$ to produce a new ...
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Connection between Hamiltonian and density matrix

I found this equation in my notes from a seminar on BCS theory $$ H_{m,n} = \frac{dE}{d\rho_{n,m}} $$ where $H_{m,n}$ are elements of the Hamiltonian in matrix representation, $\rho_{n,m}$ are ...
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Hamiltonian of charged particle in an EM field and magnetic field does no work on charged particles

I am trying to understand a part of I.E.P.'s answer here. I.E.P. states that one can see from the following Hamiltonian, $$ H = \frac{1}{2m}|{\bf p}+q{\bf A}|^2 +q \phi \tag{8.35} $$ that the magnetic ...
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Is perturbation/interaction hamiltonian in interaction theory time-dependent?

Reading quantum field theory text, I am confused on regard to whether perturbation (or interaction equivalently) hamiltonian added to free-field hamiltonian is time-dependent or not. In Heisenberg ...
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How is Hamilton's first equation useful in solving mechanics problems? [duplicate]

Here is the first Hamilton equation: $\frac{\partial H}{\partial {p}_q} = \dot{q}$ Let's use it. Imagine a ball rolling down a frictionless hill (ignore the friction vector in the image). As time goes ...
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Hamiltonian of a particle in magnetic field squared

I'm trying to follow Tong lectures about Gauge Theories, but I think I'm doing some really stupid mistake. At one point he takes the Hamiltonian for a spin $1/2$ particle in a potential as the usual \...
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What is a Hamiltonian of a System?

What is a Hamiltonian of a System? When learning about Hamiltonian for the first time it is an object introduced as Legendre Dual Transform of Lagrangian of the same system. And we learn further that ...
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How do I determine the zero elements of a Hamiltonian in a 4 ket space?

The Hamiltonian matrix of particle subject to a central potential is described by $$ H=\begin{pmatrix} H_{11} & H_{12} & H_{13} & H_{14}\\ H_{21} & H_{22} & H_{23} & H_{24}\\ ...
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Are there non-constant potentials that result in eigenstates of the Hamiltonian that are all plane waves?

It is commonly known that the eigenstates to the Hamiltonian of a constant potential are plane waves, aka $$ V(r) = V_0 \Rightarrow H\psi = n \text{ with } \psi = \exp\left(\frac{ip}{\hbar}x\right)\...
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Finding the unitary transformation associated with a symmetry

Suppose we have a Hamiltonian that has a symmetry. Let's consider a simple example where the Hamiltonian depends on a vector $ H(\vec r)$ such that a rotation $R \vec r$ is a symmetry. So, $H(r)$ and $...
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Transverse field Ising in 2 dimensional lattice - kronecker product

Assume we have a transverse field Ising chain (1D): $\hat H =-J\sum_{i=1}^{N}\sigma^z_i\sigma^z_{i+1}-h\sum_{i=1}^{N}\sigma^x_i$, where $\sigma^{\alpha}_i$ are the local spin operators at site i ...
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Decoupling theory by diagonalising the Hamiltonian

I have a Hamiltonian of the form $H = 2k(\alpha \alpha^* -\beta \beta^*) -2\lambda (\alpha\beta^* + \beta \alpha^* )$ and I'd like to decouple the $\alpha$'s and $\beta$'s if possible. I know I need ...
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The Hamiltonian for Josephson Junctions

In the Feynman Lectures on physics, Feynman says the amplitudes across a Josephson Junction should be related by the following, where the subscript denotes the side of the junction that the amplitude ...
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Deriving the Hamiltonian of the Klein-Gordon field in terms of ladder operators (Peskin and Schroeder 2.31)

In Peskin and Schroeder's QFT book they give \begin{align*} H &= \int d^3x\int \frac{d^3p d^3 p'}{(2\pi)^6}e^{i(\mathbf{p+p'})\cdot \bf x}\left\{-\frac{\sqrt{\omega_{\bf p}\omega_{\bf p'}}}{4} (a_{...
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Derivative of a potential when deriving Boltzmann equation

Consider a system with $N$ identical particles of mass $m$, whose coordinates and momenta are $(q_i,p_i)$, $i = 1,\ldots,N$, and with Hamiltonian $$ H=\sum_{j=1}^N \frac{p_j^2}{2m} + \sum_{1\leq j <...
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Time dependence of ladder operators in QFT

I'm currently going through Matthew D. Schwartz book Quantum Field Theory and the Standard Model, p. 23. For free (non interacting) field theories we are able to quantise the field by expanding our ...
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Scattering, 4 point correlator, #distinct Feynman diagrams

In order to compute the scattering probability that two particles of type 1 (associated to $\phi_1(x)$) which come from the far past with the momenta $p_1$ and $p_2$, to scatter and evolve into two ...
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Construction of propagator for time-dependent hamiltonian

In deriving a general propagator to the time-dependent ($H = H(t)$) Hamiltonian problem, Shankar works to first order in $\Delta = T/N$ (a small time interval for large $N$) and argues that by ...
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Generator of Time Shift in Classical and Quantum Mechanics

The time evolution of a point in phase space in classical mechanics can be described as \begin{equation}\label{eq:TmeShift} ( q_i(t + \Delta t),p_i(t + \Delta t) ) = \left( 1 - i\Delta t \hat{L}\...
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Why is the Legendre transformation the correct way to change variables from $(q,\dot{q},t)\to (q,p,t)$?

I always found Legendre transformation kind of mysterious. Given a Lagrangian $L(q,\dot{q},t)$, we can define a new function, the Hamiltonian, $$H(q,p,t)=p\dot{q}(p)-L(q,\dot{q}(q,p,t),t)$$ where $p=\...
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Commutation of Hamiltonian with momentum

In which case does the Hamiltonian $H$ commutes with the momentum $P$? Can anybody help me? With an example? (No particular or strange Hamiltonians and no particular momenta are involved). How can I ...
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A priori properties of classical Hamiltonian (unitarity in Classical Mechanics)

A canonical equation of motion has form: \begin{equation} \dot{p}_i = -\frac{\partial H}{\partial q_i} = \left\{ p,H\right\}, \quad \dot{q}_i = \frac{\partial H}{\partial p_i} = \left\{q,H\right\}....
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One-dimensional Ising Model in a three spin chain

I have a system of three aligned spins with $S=\frac{1}{2}$. There are interactions between nearest neighbors, and each spin has a magnetic moment. The Hamiltonian of the system is: $$H=J[S_z(1)S_z(2) ...
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How does symplectic geometry relate to classical hamiltonian mechanics?

I just found out about symplectic geometry in the context about this question on volume preservation in phase space. It seems somewhat complicated and I am not sure what to do with the notation $\...
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Is Hamiltonian a linear operator on Hilbert space?

I believe this question may seem silly, every student who has studied quantum mechanics in school must has been told that Hamiltonian is a linear operator on Hilbert space. However, today I think this ...
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Do Hamiltonian operators preserve square integrability? [closed]

Is Hamiltonian operator an operator that preserves the property of square integrability?
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Help on Hamiltonian tensor equations in Einstein's original General Relativity papers

Studying Einstein's original Die Grundlage der allgemeinen Relativitätstheorie published in 1916's Annalen Der Physik, I came across Equations 47b) regarding the gravity contribution to the stress-...
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Energy of Free-electron Gas - Landau Levels in 3D

so i am looking into Landau Diamagnetism and am reading Dupre's paper. I am slightly confused at where he has got a term in his value of E from. He states that: $$ E=(n+1/2)\hbar\omega+\hbar^2k_z^...
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Unitary Transformation of an Interfering Beam Splitter

I was reading this research paper Quantum interference enables constant time quantum information processing and was confused by one particular expression involving the Hamiltonian of a beam splitter. ...
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Getting the eigenvalues of a quadratic boson Hamiltonian numerically

I have the following quadratic Hamiltonian (of boson type): $$\hat{H}=\epsilon b^\dagger b -v(b^\dagger b^\dagger + b b )$$, where both $\epsilon$ and $v$ are real parameters. The operators $(b,b^\...
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Eliminating an eigenvalue from the Hamiltonian

I have a momentum space Hamiltonian $H(\vec k)$ for a Kagome lattice and I want to find its eigenvalues which may be dependent on $\vec k$. Now, I'm told that one of the eigenvalues for such ...
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How the Hamiltonian of a classical system expressed in quantum mechanics?

I was dealing with a problem, which said that, Supposedly Hamiltonian of a conservative system in classical mechanics is $\omega xp$, where $\omega$ is a constant, and $x$ and $p$ are the position ...
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Is this called an operator?

Consider the Hamiltonian: $$H=D\bigg(S_z-\frac{1}{3}S(S+1)\bigg)$$ Where $S_z$ is the spin-$z$ operator (one half the Pauli matrix for a doublet state) and the matrix representation of $S$ is the unit-...
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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 (interaction-picture) ...
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Why are time derivatives of states in QFT equal to zero?

In equation 6-38 on page 176 of the book "Student Friendly QFT" by Robert D. Klauber it is said that the partial derivative w.r.t. time of a multi-particle state is equal to zero since we ...
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Identifying the relevant directions in the Ising model renormalization

I'm reading the chapter about the renormalization group in Yeoman's book "Statistical mechanics of phase transitions" and I'm puzzled about how the author relates the scaling of the RG with ...
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Bethe ansatz wavefunction vs plane waves

I am reading Negele & Orland's "Quantum many-particle systems". In problem 1.9 you show that the (Bethe ansatz) wave function $$ \psi(\{x \}) = \exp \left( - \alpha \sum_{i < j}^N |...
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Electronic component of the Hamiltonian operator and uncertainty principle

This question has to do with the concept of uncertainty principle. The Hamiltonian operator has the electronic component that takes the inverse of the distance between any two electrons. My question ...
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Is there a physical interpretation to invariant random matrix ensembles?

Disclaimer. I am a graduate student in pure mathematics, so my knowledge of physics more advanced than basic 1st/2nd year undergraduate physics is very limited. I welcome corrections on any ...
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Predictability in decoherence theory to find the classical states: at which time must we evaluate?

I have read Decoherence, einselection, and the quantum origins of the classical, end a way to quantify the classicality of states is the following. We have the system $S$ and its environment $E$. The ...
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$T$-odd vs $T$-violation

I am a bit confused by the difference between $T$-odd and $T$-violation. For example, I read that the existence of a fundamental particle EDM is a violation of time symmetry. However, placing an ...
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How to write the time-dependent Schrödinger equation from generic functions?

Given the initial state: $$\Psi(x,t=0)=c_1 \psi_1(x)+c_2\psi_2(x)+c_yy(x)$$ where $\psi_1$ and $\psi_2$ are eigenstates of $\hat{H}$ and $y(x)$ is a normalizable function but is not eigenstate of $\...
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Book about the measurement energy of the Hamiltonian

I am searching for a book about the measurement energy of hamiltonian in adiabatic quantum computing. Have you ever seen good resources? I need good references for my work.
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From spins to fields

In statistical field theory, one usually considers the so-called Landau Hamiltonian: $$\beta H = \int d^{d}x\bigg{[}\frac{t}{2}m^{2}(x) + \alpha m^{4}(x)+\frac{\beta}{2}(\nabla m)^{2}+\cdots+ \vec{h}\...
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Treat stochastically non-Hamiltonian perturbations

Let us consider a classical dynamical system whose obserbvables $A$ evolve according to the following equation of motion \begin{align} \dot A &= -\{H,A\}+f(q) \end{align} $f(q)$ is a non-...

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