Questions tagged [hamiltonian]

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

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68 views

Discretization of Hamiltonian with first derivative

In a particular 1D system, the Hamiltonian can be writen as $$H=\mathrm{i}\left(f(r)\frac{\partial}{\partial r}+\frac{1}{2}f'(r)\right)\; ,$$ wher $\mathrm{i}$ is the imaginary unit, and $f(r)$ is a ...
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46 views

How to interpret overlap in Hamiltonian if it is not a degeneracy?

In Fruchart et al.'s An Introduction to Topological Insulators, the Bloch Hamiltonian for a two-band insulator is given in the general form $ H(k)= $ \begin{bmatrix} h_0+h_z & h_x-i h_y \\ ...
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119 views

Eigenvalues of the Hamiltonian

Is every eigenvalue of the Hamiltonian a form of energy? If not are there values of the Hamiltonian that do not correspond to the energy of the system?
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90 views

Is the tight binding model an effective free fermion model?

The tight-binding Hamiltionian has the form $$H=-t\sum_i\left(c_i^\dagger c_{i+1} + c_{i}c_{i+1}^\dagger\right)$$ But does this mean that it can be represented in the form of free fermion modes?
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229 views

How does a Hamiltonian 'generate' a unitary?

I know that the unitary (propagator) is given by $$U=e^{iHt}\tag{1}.$$ But I actually never saw how a Hamiltonian translates into a unitary. For example when I consider a two-level rotation in a ...
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75 views

Hamiltonian of a quantum heat bath

I have seen the Hamiltonian for a heat bath written as: $$ H_B = \hbar \int_0^\infty \omega b(\omega)^\dagger b(\omega) d\omega $$ I was hoping to understand this equation better. This suggests that ...
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29 views

What is the 4x4 matrix for the charge inversion operator and how do you construct it?

I have a 4x4 Hamiltonian describing a part of my system. To get a holistic view of the situation I need to do a charge inversion on the matrix. What is the 4x4 charge inversion operator? And what is ...
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72 views

Wilson Sommerfeld Method to solve for Energy

I have an example in my notes to find the quantum energy levels when the Hamiltonian is $H(p,q)={p^2}/{2m}+(mw^2q^2)/2$. However when given the Hamiltonian $H(p,q)={p^2}/{2m}$, I'm having ...
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60 views

Mathematical representation of Symmetry Transformation

Consider a general Hamiltonian that is made up of three terms $\mathcal{H}$ = term I + term II + term III . Suppose the combination of charge conjugation and parity (CP) is a symmetry of this ...
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65 views

What people mean by “state evolving with the interacting/free theory”?

This is a quite basic question but I confess it is something I didn't get up to this point. When defining the Moller operators and hence the $\cal{S}$-matrix one usually considers "states $\Psi$ ...
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97 views

Potential must be real for Hamiltonian to be Hermitian?

I have seen a few proofs specify for finite wells, step functions, and harmonic oscillators, that $V$ must be real for $H$ to be Hermitian. Why is that? If we're solving the Schrodinger equation, we ...
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120 views

Good /not good quantum numbers in spin-orbit coupling

Given that the Hamiltonian associated with the spin-orbit interaction can be expressed in terms of the total orbital angular momentum and total spin operators as: $$ H_{SO} = -\frac{e}{2m_e ^2 c^2} \...
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58 views

Time reversal of a QM Hamiltonian

I'm interested in the time reversal properties of a term in the non-relativistic QM Hamiltonian proportional (up to a true scalar) to $$ H \propto (\vec S_1 \times \vec S_2) \cdot \vec L $$ The ...
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1answer
105 views

Quantum walk and the Hamiltonian operator

The Hamiltonian operator is defined on the graph $G$ as $H_{A}(t) = \exp(itA)$ where $A$ is the adjacency matrix of the graph $G$. It is said that this operator is a transition matrix and represents ...
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172 views

Time translation invariance of Hamiltonian

I am learning about the time translation invariance of the Hamiltonian. I read that the time translation invariance is already manifest in the fact that our Hamiltonian is chosen an ...
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103 views

‘Supersymmetrizing’ an arbitrary quantum-mechanical potential

To my understanding, it is not possible to $``\text{supersymmetrize}"$ an arbitrary quantum-mechanical system unless one knows how to represent the corresponding Hamiltonian in the form $$ H = A^\...
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66 views

Hamiltonian time-independent, partial derivative always zero?

For conceptual simplicity, let's restrict the discussion to systems with a two-dimensional phase space $\mathcal P$ with generalized coordinates $(q,p)$. Hamiltonian is a function that maps a pair ...
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45 views

Quantizing the double pendulum hamiltonian

So, just for kicks, I thought it might be fun to try to see what happens if you have a "quantum double pendulum". Take a simple point pendulum with mass $m$ and length $l$, and hang a second identical ...
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71 views

Time dependence of the momentum operator for a free particle

I was studying Modern Quantum Mechanics by Sakurai, and at the page 85, it is given the analysis of a free particle. There, the author assumes that Hamiltonian is $$\hat H = \frac{ \hat p ^2}{ 2m},$$...
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62 views

Two ways to define wave function in Heisenberg picture

I found two ways to define a wave function in Heisenberg picture, $| \psi(t) \rangle_\mathrm{H}=\mathrm e^{\mathrm i H t/\hbar} | \psi(t) \rangle_\mathrm{S}$ which further gives $|\psi(t) \rangle_\...
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109 views

How do we find a time-independent Hamiltonian that generates a given unitary transformation?

I know that for time independent Hamiltonians we can make the statement $$U = e^{-iHt}\tag{1}$$ where $H$ is a time-independent Hamiltonian (divided by $\hbar$) and $U$ the unitary, also known as ...
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4answers
76 views

Why we neglect the $\hbar ω/2$ in the Hamiltonian of the the Electromagnetic Field?

After the quantization of the electric and the magnetic field, we get the Hamiltonian of the electromagnetic field: $$H= \hbar ω(a^{\dagger}a +1/2) .$$ with $\hbar$ the planck constant and $a^{\...
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89 views

Action of complex conjugation on Hamiltonian

Consider a finite-dimensional non-relativistic QM system with hamiltonian $H$. Let $K$ denote the complex conjugation operator. What does $K H K$ simplify to, if the system is: (a) spin-zero; (b) spin-...
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108 views

Derivation of the BCS Hamiltonian

$$ \hat{H}_{\mathrm{BCS}}=\sum_{k \sigma} \varepsilon_{k} c_{k \sigma}^{\dagger} c_{k \sigma}-\sum_{k k^{\prime}} G_{k k^{\prime}} c_{k \uparrow}^{\dagger} c_{-k \downarrow}^{\dagger} c_{-k^{\prime} \...
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51 views

False solution of Landau Hamiltonian

The Landau Hamiltonian in 2D is given (in natural units $q=c=2m=1$) by $$ \hat{H} = (\hat{\vec{p}}-\vec{A}(\hat{\vec{x}}))^2 \,,$$ where $\vec{A}$ is the magnetic vector potential field. We know that ...
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70 views

How can we prove that correlation function depends only on the spatial difference if Hamiltonian is translationally invariant?

If $H$ is a translationally invariant Hamiltonian, how can I convince myself that the correlation function (on the ground state $\left|G\right\rangle$) $\left\langle G|\psi(x)\psi(x’)|G\right\rangle$ ...
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1answer
59 views

What if we set Hamilton-Jacobi mechanics as an axiom?

We postulate principle of least action then we get Lagrange mechanics, after we can get Hamilton mechanics either from postulate or lagrange mechanics. Then we get HJE. But what if we have HJE as ...
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39 views

Hamilton-Jacobi equation and method of solving it [duplicate]

So, this equation we get when we find canonical transformation that makes new hamiltonian=0. There are 4 main transformations: F1(q,Q,t), F2(q,P,t), F3(p,Q,t), F4(p,P,t). On practice and in every book ...
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163 views

How does the Berry curvature relate to the hopping strengths in the Haldane model?

Take Haldane's Hamiltonian, as quoted from Fruchart et al.'s An Introduction to Topological Insulators: 3.5.3. Haldane's Hamiltonian The first quantized Hamiltonian of Haldane's model can be ...
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27 views

$p$-spin spherical spin glass

Consider the $p$-spin spherical spin glass model with Hamiltonian $$H_{N,p}(\sigma)=\frac{1}{{N}^{\frac{(p-1)}{2}}} \sum \limits_{i_1,...i_p} J_{i_1,...i_p} \sigma_{i_1} \sigma_{i_2} .. \sigma_{i_p} $$...
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79 views

What is the symmetry of a Hamiltonian?

Recently while I was reading a paper on integrability of Rabi model by D Braak. In this paper there is a discussion about the symmetry of a model and that in the case of the Rabi model it is $\mathbb{...
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30 views

Finding the potential for a given Hamiltonian

Background: So I know from my lecture that if I am given a wave-function, which describes a particle and a potential V(x) in which the particle is in, I can use V(x) in the time-independent ...
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32 views

Expectation value of coordinate mixed operator with ground state

I have a Hamiltonian of the Hydrogen atom: $H=H_0+H_1+H_2$ , when: $H_0 $ is the hamiltonian from central force and from electron momentum , $H_1$ is the relativistic kinetic fixing, and $H_2$ is the ...
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60 views

Energy in spherically symmetric space times

In deriving the equations of motion for geodesics in spherically symmetric spacetimes through Hamiltonian formalism, we can find some constants of motion, namely, $E$ and $L$, the energy per unit of ...
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81 views

What is the physical meaning of Hamiltonian eigenstates for a single particle?

Let us assume we have one 2-dimensional quantum system with a Hamiltonian $$H = \sum_{n=1}^2 n \omega \mid n\rangle\langle n\mid$$ Do I understand it correctly when I assume that the eigenstates of ...
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109 views

Commutation relations

Given that the Hamiltonian for Muonium spin in zero magnetic field is $$\hat{H} = a \vec I \cdot \vec J$$ where $\vec I$ is the spin of a muon, and $\vec J$ is the spin of the electron, what is the ...
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52 views

Alternate definitions of Thermal states

The definition of thermal states I'm used to is: $$\tau_{\beta} = \frac{1}{Z}\,e^{-\beta H}$$ where $Z$ is the partition function defined as $Z= \mathrm{Tr}(e^{-\beta H})$, $\beta$ the inverse ...
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86 views

Harmonic oscillator energy difference between $(n+\frac{1}{2})h \omega$ and $(n+\frac{1}{2})\hbar \omega$

When I was studying the Harmonic Oscillator using the Schrödinger equation, I was told in lectures to pay attention to the units. There were 2 different equations given for the Energy of a Harmonic ...
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1answer
28 views

How to construct a 2-partite matrix

Let's assume we have an internal hamiltonian $H_0 = \mid 1\rangle \langle 1\mid$. Now let's assume we have two systems with identical Hamiltonians $H^1_0$,$H^2_0$ and I want to compute the joint ...
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How to calculate Hamiltonian when Lagrangian has higher order derivatives? [duplicate]

If we have a Lagrangian density $\mathcal{L}$ for a scalar field $\phi$ depending on $\phi$, $\partial _{\mu} \phi$, and $\partial _{\mu} \partial _{\nu} \phi$, what is the Hamiltonian? Additionally, ...
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1answer
77 views

Constructing a Hamiltonian for $N$-qubits

Let us assume we have a qubit with an internal Hamiltonian $H_0 = \sum_i \varepsilon_i |i\rangle\langle i|$. Now let's assume we have 2 such qubits. How would their joint Hamiltonian look like? I ...
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1answer
149 views

How to derive the retarded Green's function matrix for a quadratic Hamiltonian?

Start with the quadratic Hamiltonian for fermion: $$\hat{H}=\sum_{ij}H_{ij}\hat{c}_i^\dagger \hat{c}_j$$ and the definition of retarded Green's functon in time domain: $$G_{i,j}^r(t_1,t_2)=-i\theta(...
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Can we abstractly deduce the $L^2$ is conserved assuming only rotational symmetry of Hamiltonian?

Here $L^2$ is defined as $$ L^2=L_x^2+L_y^2+L_z^2 $$ representing the observable of the magnitude of the angular momentum. There are a lot of proofs showing the $z$-projection of the angular ...
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83 views

How to define the Hamiltonian properly in quantum field theory

In a rigorous fashion, how does one define the Hamiltonian of QFT as $$\hat{H}(t) = \int d^3x \hat{\mathcal{H}}(x, t)$$ For now I'm ignoring the fact that $\hat{\mathcal{H}}$ itself may be ill-...
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1answer
38 views

Hamiltonian as differential manifold

I know that phase space $(q^i, p^i)$ can be treated as manifold. But for me defining hamiltonian as a function also leads to new manifold $(q^i, p^i, H(q^i,p^i))$. Like map from open set $(q^i, p^i) \...
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28 views

Treating $-E_J \vert{\cos{(\hat{\theta}/2)}}\vert$ in the charge basis

I'm trying to write down the Hamiltonian for a Cooper pair box in the charge basis in the scenario where the Josephson potential term is not $-E_J \cos{\hat{\theta}}$, but $-E_J \vert{\cos{(\hat{\...
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1answer
210 views

Exponential of a Hamiltonian Matrix

I am trying to understand a problem which involves a two-level system given by: $$ \begin{pmatrix} C1(t) \\ C2(t) \\ \end{pmatrix}=e^{-i\hat{H}t/2} \begin{pmatrix} 0 \\ 1\\ \end{pmatrix} $$ I am ...
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3answers
141 views

Domains of $H$ and $U(t) = \exp(-iH t )$

I am not so familiar with functional analysis. But in my impression, the Hamiltonian $H$ is often not defined everywhere on the Hilbert space. On the other hand, the time evolution operator $U(t)$, ...
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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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77 views

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 ...