Questions tagged [hamiltonian]

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

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39 views

Physical meaning of gapped path between Hamiltonians in the same phase

I'm reading this famous paper about the classification of quantum phases, and I'm wondering about the physical meaning of the definition of phases the authors use. They say that two Hamiltonians $H_0$ ...
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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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Dispersion law for a tight binding Hamiltonian and particle states at $t$$\rightarrow$$\infty$

A spinless fermion (possessing an electric charge) can move across the sites of the discrete (translationally-invariant) lattice. The structure features three kinds of sites: $α_n$, $β_n$, $γ_n$ with ...
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1answer
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Hamiltonian with identity operator: how to visualize the (time-evolution) rotation?

For the Hadarmard Hamiltonian, $\hat H = (\hat X+\hat Z)/\sqrt 2$, where $\hat X$ and $\hat Z$ are Pauli matrices. The time evolution of a state under this Hamiltonian could be visualized by a ...
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1answer
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Interpreting Hamiltonian of single-mode squeezing

Hamiltonian represents energy. I can understand this when considering about harmonic oscillator, whose Hamiltonian is expressed as: $$ \hat{H} = \frac{1}{2m}\hat{p}^2 + \frac{m\omega^2}{2}\hat{q}^2$$ ...
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Translation operator and parity operator

(This is taken from Introduction to Quantum Mechanics by D. Griffiths, 3rd edition, Problem 6.18 .) If a system has inverse symmetry, we know that [$\hat{H},\hat{\Pi}] =0$ where $\hat{\Pi}$ is the ...
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How to make a $2\times 2$ Hamiltonian using any $2$ levels of an $N$-level Hamiltonian?

Is there a standard way for me to isolate 2 of N bands of a general $N\times N$ Hamiltonian? That is, I want to make a $2\times 2$ Hamiltonian given a larger one. I was told that there is a general ...
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Deriving Klein-Gordon from Hamilton's equations for fields using functional derivatives

I have found two different ways of doing this and I am seeking commentary on the fine nuance. Suppose there is a Hamiltonian $$ H=\frac{1}{2}\int\!d^3x \left[ \pi^2+(\nabla\varphi)^{\!2}+m^2\varphi^...
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A derivation of an equation involving singular Hamiltonian

I was trying to follow the derivation of an adiabatic theorem in the Appendix F.1 of Jordan, S. P. (2008). Quantum computation beyond the circuit model. The author claims that, for the sake of this ...
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3answers
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Finding matrix representation of Hamiltonian operator

Let the quantum system composed by an orthonormal base with the states $|1\rangle, |2\rangle$ and $|3\rangle$ with all being degenereted states of the observable D with eingenvalue $\delta$. So, being ...
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Commutator of Hamiltonian and momentum [duplicate]

I was solving an assignment on the Galilean group and we were ask to compute the commutators of its generators. So, the Hamiltonian is the generator of time translations and momentum is the generator ...
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How do we perform the time derivative of the perturbation series for the time-evolution operator?

The following image is from Greiner's book, Field Quantisation, where he carried out the derivative in question. The only way I could make sense of it, was that the derivative acts only on the last ...
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1answer
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Energy in hamiltonian formalism from phase space evolution

The hamiltonian for a free falling body is $$H = \dfrac{p^2}{2m} + mgy$$ and since we are using cartesian coordinates that do not depend on time and the potential only depends on the position, we know ...
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1answer
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Finding the probability of measuring a particular eigenvalue of an operator for a system after time evolution

Consider a quantum system with Hamiltonian H and consider the measurement of an observable $a_n$ associated with a different operator A. Initially the system is an eigenstate $|\phi_n \rangle$ with ...
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Phase density representation two-site Hubbard Hamiltonian with Fermions

I'm looking for the two-site Fermi-Hubbard Hamiltonian in phase density representation in linearized form and I don't know how to derive it. I then want to derive the equations of motion from that. ...
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1answer
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Can the hamiltonian be derived from phase space evolution?

Given the phase space evolution of a system, $x(t)$ and $p(t)$, is there any way of getting the hamiltonian to make a later study of the system under the hamiltonian formalism? My first thought was ...
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Effective Hamiltonian in Heisenberg model

How can we divide the whole matrix into submatrices that we can write effective Hamiltonian on the Heisenberg model Based on Fundamentals of the Physics of Solids book (Volume I) written by Jen˝o ...
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1answer
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Proving the Feynman-Hellmann Theorem in quantum mechanics

Concerning the Feynman-Hellmann theorem can someone point me on how solve this: If $H E = E |E\rangle$ and assuming $H$ is depending on a variable $\lambda$ eg., $H = H(\lambda)$ then $\langle \frac{\...
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Particle in a potential well in mathematica

For a three qubit-chain in a connected state barrier tunneling captured in the Hamiltonian below with J=1, U=3. How do I fin J' when sites 1 and 3 decouple from each other?
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Pertubation Theory

For long-range barrier tunneling, consider three qubits governed by the Hamiltonian $H$. $$ H = -J(\sigma_1^+\sigma_2^-+\sigma_1^-\sigma_2^++\sigma_3^+\sigma_2^-+\sigma_3^-\sigma_2^+)+\frac{U}2\sigma^...
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1answer
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Energy measures and probability of measuring them

We have the following Hamiltonian $\hat{H}=a|u_{1}\rangle\langle u_{2}|+a|u_{2}\rangle\langle u_{1}|$ with a $\in \mathbb{R}$ and $|u_{1}\rangle,|u_{2}\rangle$ an orthonormal system The matrix ...
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1answer
39 views

Expanding the quantum mechanical propagator in terms of the (non-degenerate) eigenvalues of the Hamiltonian

Could anyone please help me with this derivation? I am struggling to see how the Propagator Can be expanded out into the form This is a non-degenerate two-level system. Any help would be greatly ...
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4answers
159 views

Is $U^\dagger(R)\hat{H}U(R)=\hat{H}$ always true?

Consider a Rotation transformation on momentum state, $$U^\dagger(R)\hat{\mathbf{p}}U(R)=R\hat{\mathbf{p}}$$ Now the question is whether, $$U^\dagger(R)\hat{H}U(R)=\hat{H}\,?$$ Here, $\hat{H}$ is the ...
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Diagonalising the Hamiltonian phonon

I've been trying to derive the eigenvalues of the two mass atomic chain to get out both the acoustic and optic phonon dispersion curves. This is easy in classical physics but I wanted to see if I ...
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1answer
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Why is the energy function not always equal to total energy? [duplicate]

Why is the energy function $h = \dot{q_i}\frac{\partial L}{\partial \dot{q_i}} - L $ not always equal to total energy $E = T + V$? Here $T$ is Kinetic Energy and $V$ is Potential Energy. I've read ...
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1answer
45 views

A quadratic Hamiltonian is a model of independent particles - why?

I'm reading some notes on the Anderson Hamiltonian: $$ H=\sum h_i c_i^\dagger c_i -q\sum_{\langle i,j\rangle}(c_i^\dagger c_j+c_j^\dagger c_i)$$ Where the $c_i/c_i^\dagger$ are fermionic annihilation/...
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1answer
58 views

Expectation value of time-evolved number operator for ground state Coulomb system

I'm going through "Advanced Quantum Mechanics" by Franz Schwabl, and he calculates the electron energy levels from the Coulomb interaction in a perturbative way (section 2.2.3). In the ...
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2answers
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Can spin operator expectation value be time-independent while commutator with Hamiltonian is non-zero?

Considering the following (magnetic field) hamiltonian: $\hat{H}=-\gamma B_z \hat{S}_z$ ($\gamma$ is a constant). Suppose an electron is in an eigenstate of $S_x$, and we ask ourselves the question ...
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Regularization choices and problems in effective Lagrangian (Hamiltonian) derivation

I am trying to derive the explicit form of the effective interaction Lagrangian (Hamiltonian) for fermions interacting via a scalar particle (Yukawa's potential). For that, I am using the Lagrangian $$...
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Two spin block Ising-Hamiltonian eigenvalue in 1D (transverse ising model)

This problem is regarding to the SDRG (strong disorder renormalization group) around the critical point considering the transverse Ising model. The Hamiltonian is given, but I can't find a way to ...
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1answer
49 views

Tilting a water glass so that you can run faster without spilling water (counter-diabatic driving Hamiltonian)

In this paper, there is an interesting figure: Every attempt I've made to search online to confirm whether or not waiters/waitresses actually do this, has been unsuccessful. Is there really an ...
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1answer
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Any known way to diagonalize the Hamiltonian of a charge particle in a EM field?

Consider the Hamiltonian of a free (charged) particle, i.e., $$ H = \frac{p^2}{2m}. $$ This is easily "diagonalized" by wave functions $e^{ikx}$ (where I'm speaking loosely of ...
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1answer
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Combination of 'transposition operators': do they commute?

Suppose I have the Hamiltonian defined as $H =\hat A\hat B+\hat C\hat D$, where the operator $A,B,C and D$ are square matrices. If I label the positions of $A,B,C,D$ as $1,2,3,4$. Now I want to apply ...
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Adding λI to the Hamiltonian has no impact? [closed]

Show that If we add λΙ in H, where I is the identical operator and $λ\in\mathbb{R}$, it won't affect any measurement.
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Simplifying a Bra-Ket Expression

Consider the following relations $$H_0|\psi_a\rangle = E_a|\psi_a\rangle$$ $$H_0|\psi_b\rangle = E_b|\psi_b\rangle$$ I am struggling then to understand why the following identity holds (its probably ...
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1answer
71 views

Hamiltonian of relativistic particle in Coulomb field

In my problem we look at a relativistic particle of charge $q$ and mass $m$ in the presence of a second particle of unit charge and mass $M\gg m$, which is fixed at the origin. I need to find an ...
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1answer
91 views

Shared eigenbasis of commuting Operators

Suppose I have two Hamiltonian pieces $H_1$ and $H_2$ such that $[H_1,H_2]=0$. Then we know that the two pieces have shared eigenbasis. Assume both $H_1$ and $H_2$ have eigenvalues 2 and -2. Let $|\...
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2answers
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Hamiltonian of a quantum circuit including a diode?

The LC circuit has a Hamiltonian: $$\hat{H}={E_L\over2} \hat{\varphi}^2 + 4E_C \hat{n}^2$$ where $\hat{\varphi}$ is the magnetic flux and $\hat{n}$ is the number of charge. What is the Hamiltonian ...
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1answer
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How the matrix representation of a Hamiltonian affects the eigenvalues?

Suppose we're given the following Hamiltonian: $$\hat{H}=\frac{\omega}{\hbar} \left(\hat{S}_+^2+\hat{S}_-^2\right)$$ Suppose also that we measure $\vec{S}^2$ and get $6\hbar^2$, i.e. reduced to the $s=...
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1answer
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Quantum Mechanics Spectral decomposition misunderstanding

My notes state that the spectral decomposition formula is of the form: $$ \hat{A} = \hat{A}\hat{1} = \sum{\hat{A} } |A_i\rangle\langle A_i | = \sum{A_i } |A_i\rangle\langle A_i | $$ Now consider the ...
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1answer
51 views

Magnetization subspace and Hamiltonian representation

A follow-up question of the subspaces of 4-electrons: assume the magnetization of the system is conserved (the number of total spin-up $(\uparrow)$ particles is conserved), say 1, for example. Then ...
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Diagonalization of Time dependent Hamiltonian using ZHEEVR

I need to calculate all the eigen values of the time dependent Hamiltonian using ZHEEVR, but I don't know how to define the time dependent Hamiltonian matrix. Please help in this regard
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Continuous non-relativistic bound states

Consider a group of charged point(at least considered as such in this non-relativistic limit) particles such electrons,protons , nucleii alone in an empty infinite universe and NOT considering any ...
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1answer
86 views

Commutators with Hamiltonian of the form $H=\frac{p^2}{2m} + V(x)$ [closed]

Consider a one-dimensional problem with a Hamiltonian \begin{equation*} H=\frac{p^2}{2m} + V(x) \end{equation*} where $x$ and $p$ are the position and momentum operators, $m$ is the mass and $V(x)$ ...
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1answer
154 views

Heisenberg equation of motion — why is $\vec{\sigma}_H=\vec{\sigma}$?

Trying to obtaining the Heisenberg EOM ( "for $\vec{\sigma}$" ) for the following Hamiltonian $$ H = - \mu \vec{B}\cdot\vec{\sigma} $$ where the magnetic field $\vec{B}$ is generic for now, $...
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2answers
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Time ordering operator if commutator is $c$-number function

I have a question concerning the time ordering operator. Let's suppose we have a time evolution generated by some Hamiltonian $H(t)$ given by $$ U(t)=T_\leftarrow\exp\left(-\mathrm{i}\int_0^t\mathrm{d}...
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2answers
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From where does the Ising Hamiltonian come?

So in my Stat Mech course, we were introduced to the classical Ising Model: $$H = -J\Sigma _{<ij>}S_iS_j - K\Sigma_i S_i$$ But from where does this come from? Is there any rationale behind this? ...
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1answer
75 views

Heisenberg Hamiltonian matrix and subspaces

I'm dealing with a 4-site Heisenberg's model with no external field: $$ \begin{align*} H = \sum_{i<j=0}^3h_{ij}, \quad where\ h_{ij} \equiv J_{ij}(\vec{\sigma_i}\cdot\vec{\sigma_j}) = J_{ij}(X_i\...
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Is the tensor product structure $\hat{H}_0 = \hat{h}_0 \otimes \mathbb{I} + \mathbb{I} \otimes \hat{h}_0$ wrong when interactions are included?

First, consider two uncoupled harmonic oscillators $x_1(t)$ and $x_2(t)$ with classical Lagrangian $$ L_0 = \frac{1}{2} m_1 \dot{x}_1^2 - \frac{1}{2} m_1 \omega_1^2 x_1^2 + \frac{1}{2} m_2 \dot{x}_2^...
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Many-worlds interpretation and the Hamiltonian

In Everett's Many-worlds Interpretation of QM, Schrodinger's equation is never violated (unlike in Copenhagen Interpretation). But how is this even possible? When you measure a system with respect to ...

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