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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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A short question regarding Hamiltonian [duplicate]

Can any one please tell me in what cases the hamiltonian is not Equal to Total energy. My guess, albeit educated, is if the potential is either a function of time explicitly or a function of velocity, ...
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Can the Dirac Hamiltonian accommodate a variable speed of light?

The Dirac Hamiltonian has the form1 $$\left[\beta m c^2+c\sum_{n=1}^3\alpha_np_n\right]$$ where $\alpha_n$ and $\beta$ are Hermitian matrices, and $c$ is the speed of light. My question: Is there a ...
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Diatomic Partition Function

Given the following Hamiltonian: $H = \frac { 1 } { 2 m } \left( \left| \mathbf { p } _ { 1 } \right| ^ { 2 } + \left| \mathbf { p } _ { 2 } \right| ^ { 2 } \right) + \frac { \kappa } { 2 } \left| \...
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Energy of Free-electron Gas - Landau Levels in 3D

so i am looking into Landau Diamagnetism and am reading Dupre's paper. I am slightly confused at where he has got a term in his value of E from. He states that: $$ E=(n+1/2)\hbar\omega+\hbar^2k_z^...
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Hamiltonian of charged particle in magnetic field in momentum space

How can we write the Hamiltonian of a charged particle in magnetic field $A(r,t)$ in momentum space? The conventional representation in position space is $$H=\frac{1}{2m}(p-qA(r,t))^2+V(r,t).$$ My ...
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One-dimensional Ising Model in a three spin chain

I have a system of three aligned spins with $S=\frac{1}{2}$. There are interactions between nearest neighbors, and each spin has a magnetic moment. The Hamiltonian of the system is: $$H=J[S_z(1)S_z(2) ...
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QFT Path integral for Harmonic Oscillator derivation [closed]

I am currently working through Srednicki's QFT, and am stuck on a step he uses in Eq. 7.3 to derive the path integral for the harmonic oscillator. He writes $$H = \frac{1}{2m}P^2 + \frac{1}{2}m \...
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Logarithm of Operators in Quantum Mechanics

In an operators algebra $\mathcal{A}$ one can consider a self-adjoint (i.e. real) operator $H$ and note that $$U=e^{iH}$$ exists and is unitary. A mathematical question will be whether any unitary ...
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Expressing the Hamiltonian of two coupled quantum harmonic oscillators as two independent oscillators [closed]

Shankar's Principles of Quantum Physics describes separating the Hamiltonian of two oscillators in the form $$H={p_1^2\over 2m}+{p_2^2\over 2m}+{m \omega^2 \over 2}(x_1^2+x_2^2-(x_1-x_2)^2) \, .$$ ...
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Building temperature into the Hamiltonian

Given a quantum Hamiltonian $H$ (e.g. the quantum Ising Hamiltonian $H= -h\sum_{i}X_i-\sum_{\langle i,j\rangle}Z_iZ_j$): we know that at temperature $T$, the system is in the state: $$\rho(T) = e^{-H/...
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How to calculate the ground state of Ising model at non-zero temperature

I'm studying the quantum Ising model, i.e. with Hamiltonian $H= -h\sum_{i}X_i-\sum_{\langle i,j\rangle}Z_iZ_j$. I know conceptually how to compute the ground state of the Ising model at zero ...
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How does the ground state of the quantum Ising model relate to Schrodinger equation?

The Hamiltonian $$H = -\sum_{i\in V} h_i \sigma_i^z -\sum_{(i,j)\in E} J_{ij} \sigma_i^z\sigma_j^z - \Gamma\sum_{i\in V} \sigma_i^x$$ is kind of the cost function of the quantum annealing optimization ...
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Projection of multiple band Hamiltonian onto four (or any number) of sub bands

First of, I think I should clarify that I'm probably lacking a lot of regular terminology. Sorry about that. I'm working on a model of a crystal that results in a N-band band structure. To get this ...
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Why don't the corrections to energy disappear in perturbation theory?

The corrections to the wavefunctions and energies depend on $<\psi_m^0\,| \,H'|\psi_n^0>$ to some order. I would've thought that $<\psi_m^0\,| \,H'|\psi_n^0> \, =\, <H' \psi_m^0\,|\...
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Calculating expectation value of the Hamiltonian squared

So the main idea of the problem was to find the error in the argument, which I think I have a good grasp of. Basically, the Hamiltonian of the wavefunction is a constant non-zero value inside the box ...
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1answer
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Solving Schrödinger equation by neural networks - trial function explanation

I'm reading this paper about solving Schrödinger equation using the combination of genetic algorithm and neural networks. But one part confuses me - the author defines his trial function, i.e. the ...
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1answer
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Hamiltonian for a magnetic field

An atom has an electromagnetic moment, $\mu = -g\mu_B S$ where S is the electronic spin operator ($S=S_x,S_y.S_z$) and $S_i$ are the Pauli matrices, given below. The atom has a spin $\frac{1}{2}$ ...
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What is the Hamiltonian in the “energy basis” for a simple harmonic oscillator?

My textbook says that for a simple harmonic oscillator the Hamiltonian can be expressed in the "energy basis" in this way: $$\hat H=\hbar\omega\bigg(\hat a^{\dagger}\hat a + {1\over 2}\bigg).$$ I ...
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Quantum mechanics problem about Bound states

Show that for a particle in the ground state in one dimension, the bound state energy decreases as mass increases. Also potential at infinities is 0 I used all the machinery I know. I think the ...
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Boltzmann equation derivation for $H=v\sigma \cdot p$ hamiltonian

I am trying to write the Boltzmann equation for $$H=v_{F}\vec{\sigma}\cdot(\vec{p}-e\vec{A}).$$ This is a free charged particles gas. The velocity for this hamiltonian is $$\vec{v}=v_{F} \vec{\sigma}.$...
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In quantum mechanics, why is $\langle T\rangle=\frac{\langle p^2 \rangle}{2m}$ rather than $\langle T\rangle=\frac{\langle p \rangle^2}{2m}$?

I'm a newbie reading quantum mechanics from "Inroduction to Quantum Meachanics" by Griffiths and in the early pages of the book the author defines: $$\langle x\rangle =\int_{-\infty}^{\infty} x|\Psi(...
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How to decouple the isotropic interaction Hamiltonian (nonzero trace) in NMR?

In NMR, one use dynamical coherent control to decouple the effective dipole-dipole interaction. Here, a spin system is subject to a strong magnetic field so that the local coupling is truncated and ...
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Do outside factor affect the Hamiltonian?

This is probably a fairly obvious question, but I'm unsure of whether the environment outside a system effects the Hamiltonian. Clearly temperature should affect it, since it is a measure of the total ...
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How to generate ladder operators for an arbitrary Hamiltonian?

How to generate ladder operators for an arbitrary Hamiltonian? i.e. for a power-law potential.
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Intro QM representation of Abraham-Lorentz Force

What does the Schrodinger equation look like if you add some term for the Abraham-Lorentz force? I get a self reference term I'm not sure how to handle. I realize this is probably addressed by QED, ...
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Why should the generating function of a canonical transformation involve one original and one new coordinate?

In canonical transformations,we use generating functions to transform from one set of variables to some another set. Why should such generating functions contain one original coordinate and one new ...
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1answer
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Hamiltonian for a 1D spin chain

I am trying to implement the Lanczos algorithm to tridiagonalize the Hamiltonian for a 1D spin chain of length $L$, but I am unable to decipher from my professor's notes (here's a link), what the ...
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1answer
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Is the ground state energy always larger for the system with higher potential energy?

Say we have two Hamiltonians $\hat{H}_1$ and $\hat{H}_2$ that differ only in their potential energies and $$V_2(x) > V_1(x)$$ for all $x$. Is the energy of the ground state of system 2 necessarily ...
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Eigenstates of a Hamiltonian [closed]

For a particle with a spin of 1/2, which was exposed to both magnetic fields $B_{0}=B_{z}e_z$ and $B_1=B_xe_x$ I already found the eigenvalues of its Hamiltonian which is given by \begin{...
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How to distinguish two different systems which have the same Hamiltonian in the Schrodinger equation?

Suppose we have 4 hydrogen atoms and 2 oxygen atoms. If we write the Hamiltonian containing all the possible interactions for the Schrodinger equation, how can we distinguish the system is two ...
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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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Total angular momentum in QM

Dos the total angular momentum, $J=S+L$, commute with the hamiltonian of a general sistem, with no particularities?
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Perturbation theory: justifying expansion in terms of eigenstates of the basis Hamiltonian

I have been wondering why anyone ever thought that we could find an expansion for eigenstates of some perturbed Hamiltonian in terms fo those for the basis Hamiltonian. My lecturer insisted that this ...
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How to diagonalise a hamiltonian which posesses symmetry?

I have a large hamiltonian but I know that it posseses some symmetries. How do you reduce the hamiltonian in order to find the eigenenergies?
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Is there a unitary transformation such that the Hamiltonian in the time-dependent Schrödinger equation becomes real symmetric?

The time-dependent Schroödinger equation is given as (with $\hbar=1$): $$i\dfrac{d}{dt}\psi(t)=H(t)\psi(t)\ ,$$ where $\psi$ is some normalized column vector and $H(t)$ is a Hermitian matrix with time-...
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QM: Time evolution with $H = H(t)$

In order to calculate time evolution in QM we use Schrödinger equation \begin{align*} i \partial_t |\psi\rangle_t = H(t) | \psi\rangle_t. \end{align*} If $H\neq H(t)$ then \begin{align*} i \partial_t ...
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Why does the Hamiltonian represent something different after plugging in the solution?

so I am beginning to learn Hamiltonian mechanics. We have learned that the Hamiltonian is a function of q, p, and t. Once we have a Hamiltonian, we can use the Hamiltonian equations to derive the ...
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How to expand this equation? $H_{1}=\frac{e^{2}}{R}+\frac{e^{2}}{R+x_{1}+x_{2}}-\frac{e^{2}}{R+x_{1}}-\frac{e^{2}}{R+x_{2}}$ [closed]

$$H_{1}=\frac{e^{2}}{R}+\frac{e^{2}}{R+x_{1}+x_{2}}-\frac{e^{2}}{R+x_{1}}-\frac{e^{2}}{R+x_{2}}$$ in the approximation $ \left |x_{1}\right |,\left |x_{2}\right |\ll R $ we expand to obtain in lowest ...
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Definition of Hamiltonian in Quantum Mechanics [duplicate]

Is there any particular reason that the Hamiltonian operator was defined in quantum mechanics to be $$\hat H := \frac{\hat p^2}{2m} + V$$ as opposed to $$\hat H := i\hbar \frac{\partial}{\partial t}?$$...
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Deriving or building a Hamiltonian from a Density Matrix

Is it possible to create a Hamiltonian if given a Density Matrix. If you already the the Density Matrix, then is the Partition Function (Z) even needed? This Q is not about physics. Its about an ...
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How to prove time reversal symmetry in a system (given a hamiltonian)

The generic hamiltonian for a particle that interacts with an electromagnetic field can be written as: $$H=\frac{1}{2M}\sum_{i}\left(P_i-\frac{q}{c}A_{i}(X_j)\right)^2+V(X_j)+q\phi (X_j)$$ Where $(\...
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Utility of the Magnus expansion (preserving symplectic form?)

There are (at least) two ways to perturbatively solve a matrix initial value problem: the Dyson expansion and the Magnus expansion To be explicit, suppose you're solving for a density matrix $\rho(t)$...
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How to find the coordinate representation of the kinetic operator?

From my professor's notes on statistical mechanics. $\left|\bf{k}\right\rangle$ is eigenstate of the hamiltonian of the free particle with periodic boundary conditions: $$ \left\langle{\bf r}|{\bf k}\...
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Transforming the Hamiltonian of a free quantum field

I have been trying to explicitly Lorentz transform the Hamiltonian of a free quantum field between two inertial observers, instead of reading it off the manifestly Lorentz invariant action. My ...
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1answer
66 views

Hamiltonian for a variable length pendulum

This question is taken from the book "Classical Dynamics of Particles and Systems" - Marion, problem 7.24. The problem is about a pendulum that is set into motion, it's length varies at a constant ...
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Looknig for resources on finding periodic orbit and stability on multidimensional Hamiltonian systems

I am looking for resources (books, papers, algorithms, codes) that explicitly explain the computation and analysis (using the monodromy matrix) of periodic orbits of multidimensional Hamiltonian ...
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1answer
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When is the Hamiltonian a function of momenta alone?

In my statistical mechanics course, I'm deriving the entropy for an ideal gas and I've come across a statement in the book by Pathria where it states that in the case of an ideal gas, the Hamiltonian ...
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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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How to construct a minimal model based $\vec{k} \cdot \vec{p}$ method and symmetry arguments?

Currently, I am repeating the results of this famous paper written by Di Xiao. In this paper, the authors construct a minimal band model based symmetry arguments and $\vec{k}\cdot\vec{p}$ method. The ...