Questions tagged [hamiltonian]

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

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2answers
2k views

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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1answer
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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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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\...
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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 ...
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741 views

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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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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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 ...
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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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1answer
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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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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?
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1answer
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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 ...
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When Hamiltonian and the total energy are the same

In which condition, the Hamiltonian is the same as the total energy of the system, or say $H=T+V$?
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1answer
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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}} &...
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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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1answer
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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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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\...
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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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1answer
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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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1answer
667 views

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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First exposition of non-Hamiltonian system

Not all equations of motion admit a Hamiltonian. Several questions and answers on this site concern this correspondence, for example Hamiltonian or not?, When can an autonomous system be written using ...
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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 ...
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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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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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1answer
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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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Bogoliubov transformation with a slight twist

Given a Hamiltonian of the form $$H=\sum_k \begin{pmatrix}a_k^\dagger & b_k^\dagger \end{pmatrix} \begin{pmatrix}\omega_0 & \Omega f_k \\ \Omega f_k^* & \omega_0\end{pmatrix} \begin{...
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Hamiltionian spectrum in unstable systems

I have heard that the eigenvalue of Hamiltonian in an unstable system can contain an imaginary part corresponding the tunneling. Is that true? If it is the case, then I am very confused about it. Let ...
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1answer
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How do I show that the eigenstates of a Hamiltonian can be made orthonormal?

I've been tearing my hair out over this all evening. It should be simple but I must be missing something somewhere. Can someone show me how to prove that the eigenstates of a Hamiltonian can be made ...
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1answer
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Can all canonical transformation be obtained through generation function approaches?

The question can be formulated as following: Suppose $$\delta \int_{t_1}^{t_2}{[p\cdot \dot{q} - H(p,q,t) ]dt} = 0$$ $$\delta \int_{t_1}^{t_2}{[P\cdot \dot{Q} - K(P,Q,t) ]dt} = 0$$ in which $$P = P(...
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1answer
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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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Particle with spin in uniform magnetic field

In the text "Introduction to Quantum Mechanics" by Griffiths the following is stated: The magnetic dipole moment $\vec{\mu}$ is proportional to its spin angular momentum $\vec{S}$: $$\vec{\mu} = \...
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1answer
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Why do such different Lagrangians give the same ELEs?

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\right\}...
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1answer
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The relation between Hamiltonian and Energy

I know Hamiltonian can be energy and be a constant of motion if and only if: Lagrangian be time-independent, potential be independent of velocity, coordinate be time independent. Otherwise $$H\...
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1answer
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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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0answers
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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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2answers
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General solution of states of time dependent Hamiltonian

Given a time dependent Hamiltonian which commutes at different times, we have the time evolution operator given by $$\mathcal{U}(t,0) = \text{exp}\bigg[-\bigg(\frac{i}{h}\bigg)\int_{0}^{t}dt' H(t')\...
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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{...
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1answer
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Does a constant of motion always imply a Hamiltonian formulation?

If a continuous dynamical system has a constant of motion that is a function of all its variables, and is not already evidently Hamiltonian, is it always possible to use a change of variables and ...
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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 ...