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 happens to energy if you add an extra term in front of the momentum operator in the Hamiltonian of the harmonic oscillator?

What happens to Energy if you add an extra term in front of the momentum operator in the Hamiltonian of the harmonic oscillator. Suppose We have the usual Hamiltonian for the 1D Harmonic Oscillator: $$...
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Two-level systems under time dependent radiation - Hamiltonian interpretation

We have a two-level system, with ground state $|g\rangle$, and excited state, $|e\rangle$. I made a figure for this (sry some word are in portuguese on the scheme, but the down arrow means emission ...
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On minimal coupling in Coulomb gauge

The Hamiltonian for a non-relativistic particle in a uniform external magnetic field is given in its simplest form by: $$ \mathcal{H} = \frac{\left|\mathbf{\hat{p}}-\frac{q}{c}\mathbf{\hat{A}}\right|^...
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Hubbard model diagonalization in $k$-space

I am trying to diagonalize Hubbard model in real and k-space for spinless fermions. Hubbard model in real space is given as: $$H=-t\sum_{<i,j>}(c_i^\dagger c_j+h.c.)+U\sum (n_i n_j).$$ I solved ...
3 votes
1 answer
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Why do we "normal-order" instead of just subtracting off vacuum energy?

The Hamiltonian is arbitrary upto a constant anyway. Why don't we just subtract off the vacuum energy? The Hamiltonian was always observable only upto a constant. Instead, we do normal-ordering, in ...
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Why inverse flow of separable Hamiltonian with even kinetic energy can be written like this?

Why is it true that the inverse of the flow of a separable Hamiltonian with even kinetic energy can be written as $\phi_N \circ \varphi_t \circ \phi_N$ where $\varphi_t$ is the flow of the Hamiltonian ...
5 votes
2 answers
626 views

What is a symmetry of a physical system?

If I understand correctly, in many context in physics (quantum mechanics?), a physical system is specified by giving its Hamiltonian. I also hear that symmetries are rather essential. As far as the ...
1 vote
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394 views

Free Scalar Field Hamiltonian derivation

From David Tong's notes: How do we derive $$ H = \int\frac{d^3p}{(2\pi)^3}\omega_p[ a_p^\dagger a_p + \frac{1}2(2\pi)^3\delta^{(3)}\ (0) ] $$ from $$H = \frac{1}2\int d^3x\ [\ \pi^2\ +\ (\nabla\...
3 votes
1 answer
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A contradiction in Nonrelativistic Quantum Field Theory

Reference : "Field Quantization" by W.Greiner & J.Reinhardt, Edition 1996. In the above reference as concerns the Hamilton density $\:\mathcal H\:$ and the Hamiltonian $\:H\:$ of the ...
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Time reversal symmery and spectrum statistics of generic Hamiltonians

From Random-Matrix Theory, Hamiltonians are classified in three different ensembles depending on the spectrum statistics (Gaussian Orthogonal (GOE) , Gaussian Unitary (GUE), Gaussian Simplectic (GSE))....
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How would a free particle with known spin evolve?

I searched a lot for a Hamiltonian of a pauli spinor with no potential energy but got no luck, so I tried deriving one my own. I took an overkill shortcut and used pauli's equation: $$i\hbar \frac{\...
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Separability of Hamiltonian and Factorization of Wavefunction

In Shankar's QM book Chapter 10 pg. 274, it was said that quantum mechanically, the separability of the hamiltonian $$H=H_1(x_1, p_1)+H_2(x_2,p_2)$$ leads to the factorization of the wave function: $$\...
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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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Un-equal time correlation via non-interacting tight-binding Hamiltonian

Let's assume we have a model, which is initially defined by the tight-binding Hamiltonian with a random on-site energy $f_n$, as follows: $$H^i=-J\sum_n^{L-1}\left(a_n^\dagger a_{n+1}+h.c\right)+\...
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1 answer
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How to find ladder operators that diagonalize a Hamiltonian in QFT?

I have some trouble understanding how one can, in the context of QFT, diagonalize a Hamiltonian $H$ by the introduction of ladder operators $a$ and $a^\dagger$ (I have trouble understanding how one is ...
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1 answer
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Time-reversal symmetry for spin Hamiltonian

In the topology online course by TU Delft, the time-reversal operator acting on a system of spin-1/2 particles is introduced as $$ \mathcal T = i\sigma_y\mathcal K. $$ I understand this acts on the ...
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Why I cannot write the time evolution operator $e^{-i(T+V)t}$ as the product of operators $e^{-iTt}e^{-iVt}$

To calculate the wave equation of a time-independent Hamiltonian we use: $$ \Psi_{i}(r,t)=e^{-iH^{0}t}\psi_{i}(r,0). $$ We also know that the time-independent Hamiltonian $H^{0}=T+V$ is given to the ...
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Question on quantum field theory

I saw on Ryder's "Quantum Field Theory" book in chapter 7 where he writes down the hamiltonian of a real scalar field as \begin{equation} \begin{split} H&=\int d^3x[(1/2)(\partial_t \phi)...
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For a quantum system with two contacts, how does one construct the contact hamiltonian for NEGF transport?

For context, I have been going through Supriyo Datta's NEGF course, slides, lecture material, and have been learning how to simulate quantum transport in his formalism. For simplicity, consider a 1D-...
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In an $n$ particle system, why is the Hamiltonian summed over $n$?

Suppose I am working in a system consisting of $n$ particles. Thus the phase space will be $\mathbb{R}^{6n}$, and both the momentum and position space will be $\mathbb{R}^{3n}$ each. Then, for some ...
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The Hamiltonian of a system under only the effect of an electric field

I have a maybe silly doubt: in quantum mechanics, we have the Hamiltonian as kinetic energy + potential energy. Now kinetic energy is obtained from the integral of force and displacement. Potential ...
3 votes
1 answer
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Liouville's theorem for systems with dissipation described by a single hamiltonian

Following this link, one can treat dissipation in the lagrangian by using a factor $e^{\frac{t \beta}{ m}}$ in addition to the Lagrangian $L_0$ of a system without disspation: $ L_0[q, \dot{q}] = \...
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4 answers
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How the quantum Hamiltonian changes under a transformation?

Let's say that I have an Hamiltonian $H(k)$ in momentum space and I consider a transformation (to be concrete let's say time reversal $\mathcal{T}$). We say that this is a symmetry, if $$\mathcal{T} ...
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How to find the expectation value of $P^2$ for the hydrogen atom? [closed]

I tried using spherical coordinates, the wavefunction for the hydrogen atom and the Laplacian in spherical coordinates but I just ended up with a very long integral and I don't know what to do next. I ...
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On the completeness of solutions of a quantum particle on the surface of a sphere

The Hamiltonian of a particle of mass $m$ on the surface of a sphere of radius $R$ is $$H=\frac{L^2}{2mR^2}$$ where $L$ is the angular momentum operator. I want to solve the TISE $\hat{H}\psi=E\psi$ ...
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Solving Operators in Normal-Ordering

I am studying Appendix A in this paper https://doi.org/10.1364/JOSAB.28.001964 in which there are some calculations that I am having trouble solving. For a certain part of the calculations, I have ...
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Commutation relations when calculating Hamiltonian

I am reading Topics on Superfulidity of Walter Greiner Book Titled "Quantum:Mechanics Special Chapters" In Exercise page no 200, Hamiltonian has been discussed and derived throughly using ...
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Derivation of transmon hamiltonian in charge basis

In Girvin lectures on superconducting qubits I found this statement Is this formula really phenomenological? I didn't find any books with its derivation.
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Hamiltonian of Klein-Gordon Field

The Hamiltonian of the Klein-Gordon Field may be written $$H=\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{2\omega_{\mathbf{p}}}\omega_{\mathbf{p}}\left(a^{\dagger}(p)a(p)+\frac{1}{2}(2\pi)^{3}2\omega_{\...
3 votes
1 answer
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Modified quantum harmonic oscillator spectrum and eigenstates

I am trying to find the eigenstates/eigenvalues of the following Hamiltonian $$ \hat{H} = \hbar \omega \Big(\hat{a}^{\dagger}\hat{a}+\frac{1}{2}\Big)+A\big(\hat{a}^{\dagger}\hat{a}^{\dagger}+\hat{a}\...
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Hamiltonian eigenstates in Weiss-Heisenberg model

Reading about paramagnetism and ferromagnetism i've seen this formula: $$ \mu=g \mu_B J$$ where $g$ is the Landé g-factor $$g=1+\frac {j(j+1)+s(s+1)-l(l+1)}{2j(j+1)}$$ From the answer to this question ...
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Time Derivative of quantum moment of inertia

The moment of inertia is given as in terms of the wave function $I=\frac{m}{2}\int d\textbf{r}r^2|\psi|^2 $ The time derivative of it is basically $\dot{I}=\frac{m}{2}\int d\textbf{r}r^2 \partial_t(\...
3 votes
1 answer
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Why do "good" quantum states remain stationary under perturbation?

I've been reading the degenerate perturbation theory section of Griffiths QM. He introduces the idea that, if we can find an operator $\hat A$ which commutes with $\hat H^0$ and $\hat H'$, then ...
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Half-Filling Hubbard Model

How do I calculate the matrix elements of a 4x4 matrix following the Hubbard model? I am assuming half filling. I have the following states $$\lvert 1 \rangle = \begin{bmatrix}1 \\ 0 \\ 0 \\ 0\end{...
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How can I show that applying Hamiltonian dynamics recovers the original wave equation?

Problem Consider the wave equation: $$ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}\tag{1}$$ with $ u = u(t, x)$ over domain $x \in [0, l] = \Omega$. This can be ...
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Is refraction unpredictable?

Let us take a simplified version of refraction, where light slows down because of atom absorption and re-emission. Let us denote this time as $\tau_{emission}$, do we have a function $\tau_{emission}(\...
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5 answers
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Hamiltonian is conserved, but is not the total mechanical energy

I wondering about the interpretation for the energy difference between the Hamiltonian and the total mechanical energy for systems where the Hamiltonian is conserved, but it is not equal to the total ...
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Calculation and interpretation of creation/annihilation with basis vector

The two site Fermion Hubbard toy model has the following equation: \begin{equation} {H}^2_F=-t({c}^{\dagger}_{1\uparrow}{c}_{2\uparrow}) \end{equation} Using the basis vector for one electron, \begin{...
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1 answer
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Hamiltonian symmetry Lie algebra

What is the connection between complete set of commuting observables and generators of the Lie group? I have a Hamiltonian written down in second quantized formalism and I also checked that it ...
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Energy of Free-electron Gas - Landau Levels in 3D

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^2/2m ...
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Eigenkets of a two-state hamiltonian

I have a question related to this other question: Eigenenergies and eigenkets given the Hamiltonian. In it, OP is given the following hamiltonian: $$ H=a(|1\rangle \langle1|-|2\rangle\langle2|+|1\...
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Criterion for stationary density matrix

A density matrix $\rho$ is time independent iff it commutes with the Hamiltonian $H$. I am wondering if there is a criterion to test whether $[\rho, H] =0$ using some trace condition. Specifically, I ...
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Why are Eigenvectors of a 1D quantum ising hamiltonian real

I was modelling the 1D transverse quantum Ising model and made a Kronecker product loop to find the Hamiltonian of the system, for a given magnetic field configuration. Now, my question is that when I ...
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Separability of an Hamiltonian with spin

I'd like to know if this Hamiltonian $\hat{H}=\frac{p^2}{2m}+\frac{1}{2}m\omega^2r^2+\frac{A}{\hbar^2}(J^2-L^2-S^2)$ is separable into two parts $H_1=\frac{p^2}{2m}+\frac{1}{2}m\omega^2r^2$ and $H_2=\...
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Finding a new hamiltonian from a given canonical transformation

Let us suppose we have a given Hamiltonian $$H = \frac{P_1^2}{4}+\frac{P_2^2}{2}+V(Q_2)$$ and $q_1 = Q_1 + Q_2/2$ with $q_2 = Q_1 - Q_2/2$. I need to find the new hamiltonian by using these canonical ...
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Is there a way to understand which variable is more influential in the dynamics of a system?

Is there any known way to identify which variable has the most impact in the dynamics of a system given its lagrangian or hamiltonian formulation? Let's say i have a system with 3 variables, two ...
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In Quantum Mechanics is it possible to apply time evolution operator to wavefunction?

If I consider a wavefunction that is the superposition of Hamiltonian eigenfunctions, for example like: $$\psi(x)=\frac{1}{\sqrt{2}}\psi_1(x)+\frac{1}{\sqrt{2}}\psi_2(x)$$ with $\hat{H}\psi_1(x)=E_1\...
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Adding an Impurity as a Perturbation to a Simulation

I am working on some plots corresponding to superconductors (of many kinds). If we insert a vortex, we can view this as an impurity and observe the Quasi-Particle-Interference (QPI). How I go about ...
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Volume element in system phase space

Consider $N$ particles in $3D$, with coordinates $q_i$ and momenta $p_i$, so $\{q_1,p_1,q_2,p_2,...,q_{3N},p_{3N}\}$ are variables. Construct a phase space of the system, with axes $(q_1,p_1,q_2,p_2,.....
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Generalised Gell-Mann and Low theorem for explicitly time-dependent hamiltonians

Is there a generalisation of the Gell-Mann and Low theorem that applies to the case of explicitly time-dependent hamiltonians? (Not on the original proof which is for $H=H_{0}+e^{-\epsilon|t|}V$, but ...

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