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Questions tagged [hamiltonian]

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

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When is the Hamiltonian of a system not equal to its total energy?

I thought the Hamiltonian was always equal to the total energy of a system but have read that this isn't always true. Is there an example of this and does the Hamiltonian have a physical ...
Noah's user avatar
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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 $$ \tag 1 i\hbar \partial_{t} \Psi ~=~ \hat{H} \Psi . $$ Usually, one likes to write that it has ...
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Example where Hamiltonian $H \neq T+V=E$, but $E=T+V$ is conserved

I'm looking for an example of a Hamiltonian $H$, where $H\neq T+V$, but the total energy in the system, $E=T+V$, is still conserved. While I'm at it, I might as well add that I'd be most interested ...
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Evolution operator for time-dependent Hamiltonian

When I studied QM I'm only working with time independent Hamiltonians. In this case the unitary evolution operator has the form $$\hat{U}=e^{-\frac{i}{\hbar}Ht}$$ that follows from this equation $$ i\...
Oiale's user avatar
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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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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 $$ \{A,B\} \to \frac{1}{i\hbar}[A,B].$$ ...
Jake Xuereb's user avatar
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Bead on a rotating hoop: Hamiltonian is conserved, but is not the total mechanical energy

Consider a bead (mass $m$) on a frictionless hoop (radius $R$) in the presence of gravity. The hoop is spun around an axis parallel to the gravitational acceleration at constant angular speed ($\omega$...
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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?
Jordan's user avatar
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What is meant by unitary time evolution?

According to the time evolution the system changes its state the with the passage of time. Is there any difference between time evolution and unitary time evolution?
herry's user avatar
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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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Why are Hamiltonian Mechanics well-defined?

I have encountered a problem while re-reading the formalism of Hamiltonian mechanics, and it lies in a very simple remark. Indeed, if I am not mistaken, when we want to do mechanics using the ...
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Where does the $i$ come from in the Schrödinger equation?

I am currently trying to follow Leonard Susskind's "Theoretical Minimum" lecture series on quantum mechanics. (I know a bit of linear algebra and calculus, so far it seems definitely enough to follow ...
EelkeSpaak's user avatar
7 votes
2 answers
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Gauge Invariance of the Hamiltonian of the electromagnetic field

The Hamiltonian for an electron of mass $m$ and charge $e$ in an exterior electromagnetic field is $$H=\frac{1}{2m}(p-(e/c)A)^2+e\varphi.$$ The corresponding (via canonical quantization) quantum ...
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When does Hamiltonian equals to energy of the system?

In classical mechanics, the Hamiltonian is well defined by the Lagrangian. Whereas, energy is a very ambiguous term. We just say $E=T+U$, and usually it equals to Hamiltonian. Does there exist a way ...
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Why does Quantum Field Theory use Lagrangians rather than Hamiltonians? [duplicate]

Why does Quantum Field Theory use usually Lagrangians rather than Hamiltonains? I heard many reasons, but I'm not sure which is true. Some say it's just a matter of beauty, so Lagrangians are more ...
The Quantum Physicist's user avatar
20 votes
4 answers
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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 ...
Physics Moron's user avatar
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Hamilton's characteristic and principal functions and separability

Just hoping for some clarity regarding Hamilton's characteristic function $W$. When we take a time independent Hamiltonian we can separate the Principal function $S$ up into the characteristic ...
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Lower bound on energy is potential minimum

Suppose we have a particle of mass $m$ that is in an eigenstate $|\psi\rangle$ of the Hamiltonian $\hat{H}=\hat{T}+\hat{V}$, where $\hat{T}$ is the kinetic energy operator and $\hat{V}=V(\hat{r})$ is ...
Arturo don Juan's user avatar
21 votes
1 answer
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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 ...
Blackwidow's user avatar
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6 answers
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Why does time evolution operator have the form $U(t) = e^{-itH}$?

Let's denote by $|\psi(t)\rangle$ some wavefunction at time $t$. Then let's define the time evolution operator $U(t_1,t_2)$ through $$ U(t_2,t_1) |\psi(t_1)\rangle = |\psi(t_2)\rangle \tag{1}$$ and ...
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Translationally invariant Hamiltonian and property of the energy eigenstates

If the Hamiltonian of a quantum mechanical system is invariant under spatial translation, then the linear momentum is a constant of motion. Apart from that, can we make some comment about the nature ...
SRS's user avatar
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Lagrangian of a relativistic free massive particle

Lagrangian for a relativistic free particle can be written as $$L=-m_0c^2\sqrt{1-\frac{v^2}{c^2}} .\tag{1}$$ It gives correct expression of Hamiltonian which is $$H=\sqrt{p^2 c^2+m_0^2c^4}.\tag{2}$$ ...
physics101's user avatar
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3 answers
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Constructing Lagrangian from the Hamiltonian

Given the Lagrangian $L$ for a system, we can construct the Hamiltonian $H$ using the definition $H=\sum\limits_{i}p_i\dot{q}_i-L$ where $p_i=\frac{\partial L}{\partial \dot{q}_i}$. Therefore, to ...
SRS's user avatar
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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 ...
Revo's user avatar
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Validity of Bogoliubov transformation

In condensed matter physics, one often encounter a Hamiltonian of the form $$\mathcal{H}=\sum_{\bf{k}} \begin{pmatrix}a_{\bf{k}}^\dagger & a_{-\bf{k}}\end{pmatrix} \begin{pmatrix}A_{\bf{k}} &...
leongz's user avatar
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13 votes
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The position-representation matrix elements of the propagator for a particle in a ring

I have a question about obtaining matrix elements of time evolution operator. I have the following Hamiltonian for a particle in a ring with magnetic field $$H=\dfrac {\hbar ^{2}} {2mR^{2}}\left[ -i\...
lafahi's user avatar
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Energy operator

Does the Hamiltonian always translate to the energy of a system? What about in QM? So by the Schrodinger equation, is it true then that $i\hbar{\partial\over\partial t}|\psi\rangle=H|\psi\rangle$ ...
hmm's user avatar
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How to explain the different forms of the Hamilton-Jacobi equation?

In Arnold's Mathematical Methods of Classical Mechanics, he derives the Hamilton-Jacobi equation (HJE) using a generating function $S_1(Q, q)$ to get $$ H\left(\frac{\partial S_1(Q, q)}{\partial q}, ...
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3 answers
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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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34 votes
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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 (...
Lior's user avatar
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27 votes
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Why is the ground state important in condensed matter physics?

This might be a very trivial question, but in condensed matter or many body physics, often one is dealing with some Hamiltonian and main goal is to find, or describe the physics of, the ground state ...
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Can one write down a Hamiltonian in the absence of a Lagrangian?

How can I define the Hamiltonian independent of the Lagrangian? For instance, let's assume that i have a set of field equations that cannot be integrated to an action. Is there any prescription to ...
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Off-diagonal elements of Hamiltonian matrix $H_{12}$ & $H_{21}$: energy of transition from $|1\rangle$ to $|2\rangle$ or amplitude of transition?

$ \newcommand{\k}[1]{\left| #1 \right\rangle} \newcommand{\dd}[1]{\frac{d #1}{dt}} $In a Hamiltonian Matrix like this: $$H = \begin{pmatrix} E_{11} & E_{12} \\ E_{21} & E_{22} \end{pmatrix}$$,...
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11 votes
3 answers
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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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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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Heisenberg equation of motion

In the Heisenberg picture (using natural dimensions): $$ O_H = e^{iHt}O_se^{-iHt}. \tag{1} $$ If the Hamiltonian is independent of time then we can take a partial derivative of both sides with respect ...
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3 answers
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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 ...
Emilio Pisanty's user avatar
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2 answers
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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. ...
atbug's user avatar
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5 votes
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Deriving the quantum Hamiltonian from the expression of classical energy

I am currently learning about the Dirac formalism in quantum mechanics, but don't quite understand how we derive the expression of the quantum Hamiltonian, given the value of energy in classical ...
liteplane's user avatar
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3 answers
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Why is time-evolution operator unitary?

When we shift the system's time from $t=0$ to $t = t$, we can define the following operator $\hat{U}$. $$\hat{U} = e^{- i \hat{H} t / \hbar} \, .\tag{1}$$ So many (as far as I read, almost all of) ...
ynn's user avatar
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1 answer
361 views

Why do such different Lagrangians give the same Euler-Lagrange equations?

The Lagrangians $$L_\pm:=\frac{1}{2}\left( \dot{q}_1^2\pm\dot{q}_2^2\right)-\frac{1}{2}m^2\left( q_1^2\pm q_2^2\right)$$ each have Euler-Lagrange equations $\ddot{q}_i=-m^2q_i,\,i\in\left\{ 1,\,2\...
J.G.'s user avatar
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2 votes
1 answer
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When is a quantum state stationary?

If a quantum state is an eigenstate of the Hamiltonian, then it is stationary. But can a state be stationary if it is not an eigenstate of the Hamiltonian? If yes, how can one prove whether a state is ...
Tendero's user avatar
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2 votes
1 answer
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Difference between the energy and the Hamiltonian in a specific example

The problem is the following: Consider a particle of mass $m$ confined in a long and thin hollow pipe, which rotates in the $xy$ plane with constant angular velocity $\omega$. The rotation axis ...
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Why can the Schroedinger equation be used with a time-dependent Hamiltonian?

I have a puzzle about Schroedinger equation with time-dependent hamiltonian, which is usually used in time-dependent quantum systems. However, one of the axioms in quantum mechanics postulates that ...
user28662's user avatar
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3 answers
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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 ...
Abhay Srivastav's user avatar
30 votes
1 answer
6k views

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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Does Hamilton Mechanics give a general phase-space conserving flux?

Hamiltonian dynamics fulfil the Liouville's theorem, which means that one can imagine the flux of a phase space volume under a Hamiltonian theory like the flux of an ideal fluid, which doesn't change ...
Peter's user avatar
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2 answers
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Dealing with tensor products in an exponent

I am looking at the following problem and I am struggling to follow the steps involved. Consider the non-interacting Hamiltonian $$H_{AB}=H_A\otimes I_B+I_A\otimes H_B$$ So I'm trying to prove that ...
ChrisM's user avatar
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9 votes
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566 views

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 ...
user avatar
8 votes
2 answers
439 views

Numerical calculation of a quantum field's observables

Okay so QFT is definitely beautiful and elegant theory, its mathematics is rich and ingenious, but there is so much one can do with symbolic manipulations of mathematical objects only, how can I ...
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