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