Questions tagged [hamiltonian]
The central term in the hamiltonian formalism. Can be interpreted as an energy input, or "true" energy.
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Connection between dispersion relation and symmetries of the Hamiltonian
I am having trouble understanding intuitively the connection between the dispersion relation and the symmetries of the Hamiltonian. For example, suppose we have a lattice and there are four sub-...
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Momentum space representaion of an electron-phonon coubling Hamiltonian
I am facing a problem transforming the following Hamiltonian into momentum space:
\begin{align}\hat{H} = -\gamma \sum_\alpha\sum_{i=1}^2 \hat{X}_{i,\alpha} \hat{c}_{i,\alpha}^+\hat{c}_{i,\alpha} +t\...
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Hamiltonian of a system? [closed]
I have a three level lambda system with two ground states $\lvert1\rangle$ and $\lvert3\rangle$ and an excited state $\lvert2\rangle$, interacting with a classical field and the SPP mode of the ...
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Relationship between unitaries generated by a Hamiltonian and its negative sign
Consider two unitary operations $U_1$ and $U_2$ defined by:
$\partial_t U_1 = -iH_1U_1$ and $\partial_t U_2 = iH_1U_2$
Here, $U_1$ is generated by $H_1$ and $U_2$ is generated by $-H_1$, with the ...
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Time-evolution operator in QFT
I am self studying QFT on the book "A modern introduction to quantum field theory" by Maggiore and I am reading the chapter about the Dyson series (chapter 5.3).
It states the following ...
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How to get $ H=\int\widetilde{dk} \ \omega a^\dagger(\mathbf{k})a(\mathbf{k})+(\mathcal{E}_0-\Omega_0)V $ in Srednicki 3.30 equation?
We have integration is
\begin{align*}
H =-\Omega_0V+\frac12\int\widetilde{dk} \ \omega\Big(a^\dagger(\mathbf{k})a(\mathbf{k})+a(\mathbf{k})a^\dagger(\mathbf{k})\Big)\tag{3.26}
\end{align*}
where
\...
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How to get a lower bound of the ground state energy?
The variational principle gives an upper bound of the ground state energy. Thus it is quite easy to get an upper bound for the ground state energy. Every variational wave function will provide one.
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When is the derivative of Hamilton flow respect to initial conditions independent of time?
Consider a Hamiltonian system with coordinates $\Gamma^A=(q^i,p_i)$ and let $X^A(s,\Gamma_0)$ be the Hamiltonian flow (i.e. a solution to Hamilton's equations) parametrized by time $s$ and initial ...
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Examples of systems with infinite dimensional Hilbert space, whose energy is bounded from above
We often encounter (and love to!) deal with systems whose energy is bounded from below, for good reasons like stability, etc. But what about systems whose energy is bounded from above? In finite ...
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In degenerate perturbation theory why can we assume that matrix elements above and below the degenerate subspace disappear?
The picture shows some original Hamiltonian H which has some degeneracies. Suppose I have some perturbation V to the system and I want to find the new energies and eigenstates of the system. Then from ...
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Solving for unitary operation using perturbation theory
Let the time-dependent Hamiltonian be
\begin{equation}
H(t) = H_0(t) + \lambda H_1(t),
\end{equation} where $\lambda$ is a small parameter. In the interaction picture (i.e. rotating frame w.r.t ...
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Mean energy measurement in an arbitrary quantum state
I've gone through many papers looking for a way to measure a mean energy in an arbitrary state $\langle \psi | H | \psi \rangle$. I am interested in a theoretical protocol or an exemplary experimental ...
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Are "good" states in perturbation theory eigenstates of both the unperturbed and perturbed Hamiltonian?
In my quantum course, my professor asked us the true/false question:
"Are 'good' states in degenerate perturbation theory eigenstates of the perturbed Hamiltonian, $H_0 + H'$?"
I was ...
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What are the similarities and differences between the Magnus expansion and the Schrieffer-Wolff transformation?
The Magnus expansion and the Schrieffer-Wolff transformation are both methods used to get certain effective Hamiltonians. I know that at a basic level, the Schrieffer-Wolff transformation eliminates ...
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Derivation of Dirac Hamiltonian
In Minkowski spacetime with signature $(-,\;+,\;+,\;...,\;+)$ the Dirac Lagrangian reads
$$
L=\int d^dx\;\mathcal{L}=\int d^dx\;\psi^\dagger\left(i\gamma^0\gamma^\mu\partial_\mu-im\gamma^0\right)\psi.
...
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Energy and momentum operators using Hamilton's equations
The energy operator is:
$${\displaystyle {\hat {E}}=i\hbar {\frac {\partial }{\partial t}}}\tag1$$
and the momentum operator is
$${\displaystyle {\hat {p}}=-i\hbar {\frac {\partial }{\partial x}}}.\...
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Why the kinetic term of the Hamiltonian has to be positive definite for well-posed time evolution?
I was going through this paper on QCD chaos, where in Appendix B (page 10), for equation B12:
$$\frac{\mathcal{S}}{\mathcal{T}}= \int \mathrm{d}t\sum _{n=0,1} \left(\dot{c}_n^2-c_n^2 \omega _n^2\right)...
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Eigenstates of the Laplacian and boundary conditions
Consider the following setting. I have a box $\Omega = [0,L]^{d} \subset \mathbb{R}^{d}$, for some $L> 0$. In physics, this is usually the case in statistical mechanics or some problems in quantum ...
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Math in Hamiltonian of the hyperquantization of EM field
1. Background: I encounter this when looking into the hyperquantization of EM field.
We have the secondly quantized field as below:
$$\hat{E}^{(+)}(t)=\mathscr{E} e^{-iwt+i\vec{k}\cdot\vec{r}}\hat{a}=\...
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AC Stark shift in the non-perturbative regime
I am trying to simulate the following situation. I have a 2 level system, with the energy spacing $\omega_0$. I have a laser, with Rabi frequency $\Omega_1$ and frequency $\omega_1$, which I can scan ...
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Are $\mathcal{PT}$-symmetric Hamiltonians dual to Hermitian Hamiltonians?
I was reading this review paper by Bender, in particular section VI where they show that, despite $\mathcal{PT}$-symmetric Hamiltonians not being hermitian, they can have a real spectra. They go on ...
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Hamiltonian in Non-Linear Optics
I want to know why we add an additional term known as hermitian conjugate in the hamiltonian of many non-linear optical processes like SPDC. For example the in the equation below,
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Existence of eigenstates in the context of continuous energies in the Lippmann-Schwinger equation
In the book QFT by Schwartz, in section 4.1 "Lippmann-Schwinger equation", he says that:
If we write Hamiltonian as $H=H_0+V$ and the energies are continuous, and we have eigenstate of $H_0$...
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How do I formulate a quantum version of Hamiltonian flow/symplectomorphisms in phase space to have a "geometric", quantum version of Noether's theorem
I'm currently exploring how Noether's theorem is formulated in the Hamiltonian formalism. I've found that canonical transformations which conserve volumes in phase space, these isometric deformations ...
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Why is the time derivative of the wavefunction proportional to a linear operator on it? [closed]
I am currently trying to self-study quantum mechanics. From what I have read, it is said that knowing the wave function at some instant determines its behavior at all feature instants, I came across ...
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Why is the "decision" version of the local Hamiltonian problem promised to have a positive gap?
The Wikipedia article on the local Hamiltonian problem is ungrammatical and unclearly written. I think that this is what it is supposed to say:
The decision version of the $k$-local Hamiltonian ...
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How Can I find Free Hamiltonian for this Problem?
I have got an Open Quantum System in which two two level atoms (two identical qubits) interact separately with two independent environments in the presence of the ...
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Discrete to continuous quantum operator
Let's say that we have a discrete lattice with $N$ sites. Let's label the site by the index $i$.
Let's say that we have the operators $a_i$ and $a_i^\dagger$ which correspond to the creation and ...
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Quantum harmonic oscillator meaning
Imagine we want to solve the equations
$$
i \hbar \frac{\partial}{\partial t} \left| \Psi \right> = \hat{H}\left| \Psi \right>
$$
where $$\hat{H} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial ...
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Derivation of the number operator in the energy basis of a qubit
I am trying to model the capacitive coupling of two transmon qubits. I would like to write the number operator in the energy basis, currently, I am working on using
$$
\hat H = \hat H_1 + \hat H_2 + \...
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Form of the Hamiltonian at Half-filling
I am trying to understand why chemical potential $= U/2$ is considered to be at half-filling in the case of the Hubbard Model Hamiltonian. So when I substitute this in its Hamiltonian, this is the ...
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Constant of Motion in Quantum Mechanics for explicit time-dependent Operators
I was studying constants of motion in quantum mechanics, and at first, I don't understand the condition to be a constant of motion. Generally, the temporal variation of an operator $A$ is given by the ...
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How to deal with explicit time dependence in the Heisenberg picture?
I am studying for my test in Quantum Mechanics, and there is something I don't quite understand about the Heisenberg picture and Heisenberg's equation of motion. In the lecture, we derived Heisenberg'...
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Where does the complex conjugate term generally come from in a Hamiltonian?
I find myself stumbling across Hamiltonians which go like
$$
\hat{H}\sim\alpha\hat{a}+\alpha^*\hat{a}^\dagger
$$
How does this form of Hamiltonian actually come about?
To my knowledge, the Hamiltonian ...
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Is the overall (distinguishble-particle) ground state for a many-body identical particle Hamiltonian also immediately the bosonic ground state?
Consider the following many-body Hamiltonian of $N$ particles in an external trapping potential with inter-particle interactions:
\begin{align}
\hat{H}= \sum_{i=1}^{N} \left[-\frac{\hbar^2}{2m} \...
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Time evolution using non-Hermitian (not a PT symmetric) Hamiltonian
I am currently dealing with non-Hermitian hamiltonian and dynamics using it. In general the diagonalizable non-Hermitian matrix might have complex eigenvalues and the eigenvectors may not be ...
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Does the Hamiltonian always commute with the Time Evolution Operator?
The time evolution operator $U(t, t_0)$ is given as the solution of the equation
$$ i\hbar \dfrac{\text{d}}{\text{d}t} U(t, t_0) = HU(t, t_0)$$
whether or not the system is conservative. When the ...
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Going to momentum space from Hamiltonian equation 17.4 chapter 17 in Schwartz
I'm reading the chapter 17 on the anomalous magnetic moment in QFT and the SM (Schwartz).
In the section "17.1 Extracting the moment" he says
"Going to momentum space,the Dirac equation ...
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Understanding equation for eigenvalues of a Hamiltonian
I'm reading the paper Hamiltonian Truncation Study of Supersymmetric
Quantum Mechanics. I'm not understanding a claim they make about the eigenvalues of a certain Hamiltonian. In particular, how eqn 3....
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Exercise on self-adjointness of Hamiltonian [closed]
I am struggling with some exercise I have to solve for my quantum mechanics class.
PROBLEM:
Suppose $|\psi\rangle, |\phi\rangle$ are normalised and linearly independent (but not necessarily ...
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Wigner's formula for the kinetic energy density in QM
In the Schroedinger equation the kinetic energy is represented by the operator $T = -\frac {\hbar^2} {2m} \Delta$ which acts on a wavefunction $\Psi$. If we multiply this by the complex conjugate of ...
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Uncertainty due to assuming a variable is constant - Adiabatic Invariance
I am studying classical mechanics from Goldstein and I ran into a confusing equation in the textbook. In the third edition of the book, equation (12.92) calcucates the average change of the action ...
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Obtaining Eigenvalue Expression
Say we have an excitonic system with a creation operator operator:
$$ |\Psi_{ex}\rangle=\sum_{\vec{k}} \phi(\vec{k})c^\dagger_{\vec{k}+\vec{Q}}b_{\vec{k}}|GS\rangle$$
And the Hamiltonian of the system ...
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Is the initial state the eigenstate of a Hamiltonian?
Solutions to the Schrödinger equation can take the form $ \psi(r,t)=\psi(r)f(t) $, where $f(t) = e^{\frac{-iEt}{\hbar}}$,
$$ H \psi(r) = E \psi(r) ,$$ where $\psi(r)$ is the eigenstate of a ...
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Fourier transform of spinless $t$-$V$ model for $t=0$
I am trying to compute the Fourier transform of the 2D $t$-$V$ model for the case $t=0$.
\begin{equation}
\hat H = -t \displaystyle \sum_{\langle i,j\rangle} ( \hat c_i^{\dagger} \hat c_j + \hat c_j^{...
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Property of the Hamiltonian's discrete spectrum
I have found a statement online saying that there must be an eigenvalue of the Hamiltonian inside the range $(E-\Delta H,E+\Delta H)$. Where the mean value and variance are defined for a random (...
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Schrodinger equation with $\hbar =1$
The Schrodinger equation is given by:
$$i \hbar \frac{d}{dt}|\psi(t)\rangle = H(t)|\psi(t)\rangle.$$
Sometimes, physicists set $\hbar=1$. I suppose that they achieve this by changing the scaling and ...
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If $H$ anniliates a state, must $Q$ and $Q^\dagger$ also annihilate the state?
Suppose we have a a Hamiltonian, $H$. And suppose also we have some operator $Q$ such that $\{Q, Q^{\dagger}\} = H$, and $Q^2 = 0$.
If we find a state $|\psi \rangle$ such that $Q|\psi \rangle = Q^{\...
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Is it possible to formulate classical Hamiltonian mechanics without reference to a Lagrangian? [duplicate]
The typical way to arrive at Hamiltonian mechanics is through Lagrangian mechanics, defining canonical momentum and the hamiltonian itself in reference to the Lagrangian and its derivatives, but I'm ...
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Why are these unbounded operators (essentially) self-adjoint?
Can anyone provide exact mathematical reasoning as to why the following fundamental unbounded symmetric operators are essentially self-adjoint? I.e. on, their natural domains, they admit a unique ...
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