Questions tagged [hamiltonian]

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

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The formal solution of the time-dependent Schrödinger equation

Consider the time-dependent Schrödinger equation (or some equation in Schrödinger form) written down as $$\tag 1 i \partial_{0} \Psi ~=~ \hat{ H}~ \Psi .$$ Usually, one likes to write that it has a ...
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Is there a valid Lagrangian formulation for all classical systems?

Can one use the Lagrangian formalism for all classical systems, i.e. systems with a set of trajectories $\vec{x}_i(t)$ describing paths? On the wikipedia page of Lagrangian mechanics, there is an ...
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Find the Hamiltonian given $\dot p$ and $\dot q$

I have these equations: $$\dot p=ap+bq,$$ $$\dot q=cp+dq,$$ and I have to find the conditions such as the equations are canonical. Then, I have to find the Hamiltonian $H$. To answer to the first ...
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What does it mean for a Hamiltonian or system to be gapped or gapless?

I've read some papers recently that talk about gapped Hamiltonians or gapless systems, but what does it mean? Edit: Is an XX spin chain in a magnetic field gapped? Why or why not?
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How does non-Hermitian quantum mechanics (PT-symmetric QM) fit in physics?

In the late nineties Bender has started a research program on what is called PT symmetric QM, or non hermitian QM, in which he has shown that if the hamiltonian enjoys a PT symmetry then the spectrum ...
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Why particle hole symmetry and chiral symmetry are called symmetries?

$PHP^{-1}=-H$ (particle-hole symmetry) and $\Gamma H \Gamma^{-1}=-H$ (chiral symmetry) I understand why we get the negative signs but im just a bit confused as to why such equalities mean $H$ is ...
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Is the Ground State in QM Always Unique? Why?

I've seen a few references that say that in quantum mechanics of finite degrees of freedom, there is always a unique (i.e. nondegenerate) ground state, or in other words, that there is only one state (...
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Why do excited states decay if they are eigenstates of Hamiltonian and should not change in time?

Quantum mechanics says that if a system is in an eigenstate of the Hamiltonian, then the state ket representing the system will not evolve with time. So if the electron is in, say, the first excited ...
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Detailed derivation and explanation of the AKLT Hamiltonian

I am trying to read the original paper for the AKLT model, Rigorous results on valence-bond ground states in antiferromagnets. I Affleck, T Kennedy, RH Lieb and H Tasaki. Phys. Rev. Lett. 59, 799 (...
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Is there a Hamiltonian for the (classical) electromagnetic field? If so, how can it be derived from the Lagrangian?

The classical Lagrangian for the electromagnetic field is $$\mathcal{L} = -\frac{1}{4\mu_0} F^{\mu \nu} F_{\mu \nu} - J^\mu A_\mu.$$ Is there also a Hamiltonian? If so, how to derive it? I know how ...
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What is a “picture” in quantum mechanics?

One of the basic ingredients of quantum mechanics is the possibility of working in different "pictures". Thus, while we normally work in the Schrödinger picture, in which states evolve according to ...
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Is it possible to reconstruct the Hamiltonian from knowledge of its ground state wave function?

Is it possible to "construct" the Hamiltonian of a system if its ground state wave function (or functional) is known? I understand one should not expect this to be generically true since the ...
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How to include Berry connection in Hamiltonian?

When we calculate Berry connection, $A(R)=i<\psi(x,y)|\frac{d}{dR}|\psi(x,y)>\hat{R}$ corresponding to the Berry phase of any system, the gauge potential is related to the $R$ of the parameter ...
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On Groenewold's Theorem and Classical and Quantum Hamiltonians

I recently encountered Groenewold's Theorem or the Groenewold-Van Hove Theorem which shows that there is no function which can satisfy the following mapping $$\{A,B\} \to \frac{1}{i\hbar}[A,B].$$ ...
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How can an inverted anharmonic potential $V(x)=-x^4$ have discrete bound states?

I've been watching the lectures on mathematical physics by Carl Bender on youtube where he uses the non-Hermitian Hamiltonian methods to prove that the inverted anharmonic potential $V(x)=-x^4$ has a ...
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Quantum Stat-Mech Proof of an Inequality for the Partition Function

I have the following problem that I was unable to solve for class, but I had a couple first steps that I started with that I am unable to finish. I know I can't get this since it's already been ...
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Hamiltonian and Energy of a charged particle in an Electromagnetic field

The Lagrangian of a charged particle of charge $e$ moving in an electromagnetic field is given by $$L=\frac{1}{2}m\dot{\textbf{r}}^2-e\phi-e\textbf{A}\cdot \textbf{v}$$ where $\phi(\textbf{r},t)$ is ...
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Angular momentum squared and Hamiltonian?

I'm trying to show, for the Hamiltonian $H = \vec{P}^2/2m + V(\vec{X})$, that $[\vec{L}^2,H]=0$ if $V(\vec{X}) = V(|\vec{X}|)$, and I pretty much almost have it, there's just one thing I'm getting ...
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Proof that rotational symmetric potential operators are scalar operators

Defintion: A scalar operator B is an operator on a ket space that transforms under rotations \begin{equation}\left| \xi ' \right >=\exp{(\frac{i}{h} \mathbf{\phi \cdot J})}\left| \xi \right >\...
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Heisenberg equation with time-dependent Hamiltonian

It is the root of quantum mechanics that Heisenberg picture and Schrödinger picture are equivalent? In most textbooks and wikipedia, the equivalence is proved with a time-independent Hamiltonian. ...
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Perturbation theory in quantum mechanics: assumptions on eigenvectors [duplicate]

In a course that I follow, we use the perturbation method to find the eigenvectors and energies to an Hamiltonian written $H_0 + V$ where $V$ is a weak perturbation. It is written that as $V$ is a ...
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Does the poisson bracket $\{f,g\}$ have any meaning if neither of $f$ or $g$ is the system's Hamiltonian?
Say one has a mechanical system with hamiltonian $H$, and two other arbitrary observables $f,g$. $H$ is super useful since $\{H, \cdot\} = \frac{d}{dt}$. But does $\{f,g\}$ give any useful information ...