Questions tagged [hamiltonian]

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

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1answer
182 views

Confusion about why deducing pointer observable from the structure of the Hamiltonians is not practical

I am trying to learn Zurek's theory of decoherence. Right now I am reading Decoherence, einselection and the existential interpretation (the rough guide) which seems like an easier read than his big ...
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346 views

Why operator of kinetic energy has a double derivative instead of square of single derivative?

I know that operator for $p = {h\over i} {d\over dx}$. so $p = {h\over i} {d\psi\over dx}$ where $\psi$ is the wave function. So, $T$ (kinetic energy) $ = {p^2 \over 2m} = {-h^2\over 2m} {d\psi \over ...
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81 views

Hamiltonian operating on a function of time

I've seen a few people claiming: $$\hat{H(t)}[\psi(x)T(t)] = \hat{H(t)}[\psi(x)]T(t)\tag{1}$$ i.e. an explicit function of t is not acted upon by H, even if H itself may be dependent on t. A more ...
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250 views

Intuition for Hamilton-Jacobi equation derived from least action

I am trying to understand the Hamilton-Jacobi equation without the framework of the canonical transformations. Even on the case of a 1D free particle I'm getting stuck. The system starts at fixed ...
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83 views

Strange question involving finding a relation between a commutator and the time derivative of an operator

In order to get to the parts I am stuck at, I will add the examiners' solutions to each subquestion, which is needed to get to the subquestion that I am querying. The following is a bizarre question ...
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55 views

In Monte Carlo integration for Molecular dynamics simulation, why is a Boltzmann distribution assumed?

In statistical physics, The calculation of partition function for an ensemble takes a Boltzmann's distribution of the Hamiltonian. Similarly, In Monte-Carlo integration of Molecular Dynamics ...
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2answers
150 views

Functional integration for the order parameter in $XY$ model

In the continuum limit the Hamiltonian of the classical XY model is given by, ignoring the inessential constant: $$H=\int d\vec{r}\ (\nabla\theta)^2$$ and the x-component of the order parameter is ...
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183 views

Delta potential in terms of annihilation/creation operators

Let the Hamiltonian of a system on a discrete lattice be given by $$ \mathcal{H} = \gamma \sum_\vec{x} c^\dagger_\vec{x}c^\vphantom{\dagger}_{\vec{x}+\vec{y}} + \text{h.c.}, $$ where $\gamma$ is ...
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349 views

Proving orthogonality of eigenstates of a Hamiltonian

Suppose we have $\Psi_{1}$ and $\Psi_{2}$ which are eigenstates of some (self-adjoint) Hamiltonian $\hat{H}$ with unequal eigenvalues. Could you explain me how can I prove that these arbitrary $\Psi_{...
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1answer
182 views

What is the physical meaning of expectation value of the Hamiltonian operator?

I've been studying David Griffiths' Introduction to Quantum Mechanics and int that, it was explained that the expectation value of position $x$ is the average of the positions of $N$ identically ...
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82 views

Hermitian conjugate in Hubbard model hopping term

I'm new to Hubbard model and I have a few questions about it. From the sources I can find on the internet, Fermionic-Hubbard model is often written as (please correct me if I'm wrong!) \begin{align} H ...
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414 views

Hamiltonian for particle moving in a sphere

Context: I know that if we have a particle (say, with unit mass) moving in the plane $\Bbb R^2$ subject to a spherically symmetric potential $V\colon \Bbb R^2 \to \Bbb R$, it will move along the ...
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155 views

Hamiltonian diagonalisation using quantum Fourier transform [closed]

Here is a problem to solve: diagonalize the following hamiltonian using quantum fourier transform. The hamiltonian reads: $$ \sum_{i,j=1}^N e^{-\theta_{ij}} c_i^\dagger c_j + h.c. $$ Where $c_j$ are ...
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1answer
476 views

Is the Hamiltonian of a relativistic charged particle in an electromagnetic field only an approximation?

Consider a system of two relativistic charged point particles 1 and 2 which interact through their electric and magnetic fields. The equation of motion for the first particle is then given by the ...
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158 views

Time evolution of operators with explicit time dependence in case of time dependent Hamiltonian

In case of a time dependent Hamiltonian of the sort $$H=\frac{p^2}{2m}+\frac{1}{2}m \omega(t) x^2$$ I have solved for the time evolution operator using the Schrodinger equation and got $U(t,0)$. If, I ...
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761 views

How do contact transformations differ from canonical transformations?

From Goldstein, 3rd edition, section 9.6, page 399 after equation 9.101: [...] The motion of a system in a time interval $dt$ can be described by an infinitesimal contact transformation generated ...
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61 views

Time dependent Hamiltonian operator and $SU(1,1)$ generator method

In this screenshot of a paper I am reading, I have the following question: 1.What is a $SU(1,1)$ group and how do we find its generators? 2.From the expression for the Hamiltonian $\hat{H}$, how do ...
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1answer
78 views

A fundamental question about Time-dependent Hamiltonians

I have a fundamental question about Quantum Mechanics or even mechanics in general. I am aware that there are stationary solutions and non-stationary solutions. The stationary solutions solve ...
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1answer
74 views

Eigenfunctions of Hamiltonian (question about the book “Quantum Field Theory for the Gifted Amateur”)

In the book "Quantum Field Theory for the Gifted Amateur" by Blundell and Lancaster, (page 21) the Hamiltonian (when discussing the number operator) is given by $$ \hat{H} = \left(\hat{a}^{\dagger}\...
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Multi-electron Atom

I'm reading the following text on multi-electron atoms: for a system of $n$ electrons the Hamiltonian is $$ \hat H = -\frac 1 2 \sum_{i=1}^n \nabla_i^2 - \sum_{i=1}^n \frac{Z}{R_i} + \frac{1}{2} \...
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1answer
319 views

What is the unitary matrix that diagonalizes the Hamiltonian?

If $H = H_0 + g H_1$ is our (free + interaction) Hamiltonian, and we assume that we have a basis of states $\{ | i \rangle \}$ under which $H_0$ is diagonal, then we may diagonalize $H$ by some ...
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56 views

Variational principle, functional gradient

Given the energy functional $$E[\Psi] = \frac{\langle \Psi \vert H \vert \Psi \rangle}{\langle \Psi \vert \Psi \rangle},$$ its functional gradient is $$\frac{\delta E[\Psi]}{\delta \langle \Psi \vert}...
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Frequencies associated with boson/fermion operators

For a Hamiltonian like, $$\hat{H}=\sum_{k}\hbar\omega_{k}b_{k}^{\dagger}b_{k}$$ What does it mean to say that the frequencies $\omega_{k}$ must be positive if $b_{k}$, $b_{k}^{\dagger}$ are boson ...
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197 views

Integral of Schrodinger equation for a time-dependent Hamiltonian

I am given the following Hamiltonian, $H=H_1=\frac{p^2}{2m}+\frac{1}{2}m\omega_1^2x^2$ for $t<0$ and $H=H_2=\frac{p^2}{2m}+\frac{1}{2}m\omega_2^2x^2$ for $t\geq 0$. Now I want to integrate my ...
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38 views

State after energy measurement in NRQM

Let $H$ be the Hamiltonian of a system with a set of degenerate energies, so that $H |1⟩ = E_1 |1⟩$, $H |2⟩ = E_1 |2⟩$ and $H |3⟩ = E_2 |3⟩$. Given a linear combination of these three states, I want ...
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71 views

Use of operators in a time-dependent Hamiltonian quantum system

I am given the following Hamiltonian, $$H=H_1=\frac{p^2}{2m}+\frac{1}{2}m\omega_1^2x^2$$ for $t<0$ and $$H=H_2=\frac{p^2}{2m}+\frac{1}{2}m\omega_2^2x^2$$ for $t\geq0$. For some time $t_1(<0)$, ...
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111 views

Definition of state of a quantum system

In QM, we solve for the eigen kets of the Hamiltonian operator $\hat{H}$ and say that the state of my system lies in a linear superposition of these eigenstates $\{|n\rangle\}$ as the relation implies ...
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1answer
85 views

Reasoning behind the solution to this Hamiltonian

I am confused about Griffith's solution (Example 10.1 pg 374) for the Hamiltonian: $$H = \dfrac{\hbar\omega_1}{2}\begin{bmatrix} \cos\alpha &e^{-i\omega t}\sin\alpha \\ e^{i\omega t}\sin\alpha &...
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42 views

State of a system at previous time

If I am given the state of a quantum system at $t=0$ as $| \psi \rangle$ and I know the Hamiltonian $H$ of the system for time $t<0$, how can I write the state of the system at some time $t<0$?
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84 views

Simple, $3 \times 3$ Hamiltonian with negative eigenvalues and $\langle H \rangle=0$

I have the following exercise: Consider a three-dimensional system whose Hamiltonian is described by the following matrix: $$\begin{bmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 &...
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162 views

One Hamilton Operator for two independent harmonic oscillators

If we consider two independent harmonic oscillators (identical too a two dimensional harmonic oscillator), the hamilton operator is $$ H = \frac{p_1^2}{2m_1}+\frac{1}{2}m_1\omega_1^2x_1^2 + \frac{...
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1answer
44 views

Inverse of canonical quantization: Classical kinetic energy corresponding to a Laplacian on a sphere

In canonical quantization, we replace the canonical conjugate of position($x$), $p$ by $-i \hbar\frac{d}{dx}$. In general we replace $\mathbf{p}$ by $-i\hbar \nabla$. My question is, what if I wanted ...
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1answer
433 views

Harmonic oscillator in microcanonical ensemble

Consider a hamiltonian of a simple classical pendulum $$H=p^2+\omega q^2.$$ How can quantities such as $\langle p^2 \rangle$ or $\langle q^2\rangle$ can be calculated using the microcanonical ...
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1answer
145 views

Time Evolution of Asymptotic Free States in QFT

In equation (4.70) of Peskin, he states that $$_{out}\langle \mathbf{p_1, p_2, \cdots} | \mathbf{k_A,k_B}\rangle_{in} = \lim_{T\rightarrow \infty}\langle \mathbf{p_1, p_2, \cdots} | e^{-iH(2T)} |\...
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311 views

Expectation Value $\langle \frac{1}{r^2} \rangle$ using Hellmann–Feynman theorem

Suppose we have the hydrogen atom$$ H ~=~ \frac{p_r ^2}{2m} + \frac{L^2}{2mr^2} -\frac{e^2}{r} \,.$$And have solved the Schrodinger equation finding $$E_n = - \frac{me^4}{2 \hbar ^2 n^2} $$ and $$ Ψ_{...
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1answer
247 views

Hamilton-Jacobi theory: Differentiating wrt. the constant $E$?

Let's say we have a 1D harmonic oscillator, its Hamiltonian is given by $$H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2$$ we wish to solve it via the Hamilton-Jacobi equation so we have $$\frac{1}{2m}\...
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189 views

Difference between the energy eigenstates and the eigenstates of other physical variables

I am having difficulties understanding the difference between the energy eigenstates and the eigenstates of other physical variables. I was told that if a system is an energy eigenstate at $t = 0$ ...
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action angle variable problem with limits of integration

Good evening everyone, I'm having trouble solving this problem: A particle moves in a vertical XZ plane as: $$x=a(\sin t+t)\, ,\qquad z=a(1-\sin t)\, .$$ It starts at $t=\pi/2$. I calculated the ...
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1answer
81 views

What Hamiltonian stands for?

I read the derivation of Schrödinger equation.It's like below. Let's think about the operator U(t+Δt,t) which transform state vecter at time t to time t+Δt. So ,$|Φ(t+Δt)〉=U(t+Δt,t)|Φ(t)〉$ $〈i|Φ(t+...
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1answer
107 views

Are there quantum systems for which the Hamiltonian has no eigenstates?

I can see there can be multiple interpretations of this question. I will be interested in an answer to any of them. "quantum systems" could include Actual systems observed in experiment with quantum ...
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1answer
41 views

Commonly Used Interaction Hamiltonian and Its Hermiticity

Please see reference ("Meaning of the Density Matrix" by Anandan and Aharonov, Foundations of Physics Letters, vol. 12, No. 6, 1999) p, 573, eq. 2. I am not asking about the overall content or ...
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1answer
58 views

QFT - Show if hamiltonian is unbound

I have a QFT homework to do and there I should show for a given lagrangian $\mathscr{L} = C_1 (\partial_{\nu} A_{\mu}) (\partial^{\nu} A^{\mu}) + C_2 (\partial_{\nu} A_\mu) (\partial^{\mu} A^\nu) + ...
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Why do the operators on either side of Schrodinger's equation show different hermitivity? [duplicate]

Schrödinger's Equation states that: $$i\hbar \frac{\partial \psi(x,t)}{\partial t}=H \psi(x,t).$$ Now the Hamiltonian operator, $H$ is hermitian as long as the potential $V(x)$ is a real valued ...
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40 views

Is there any feature which distinguishes the Hamiltonian in the Poincare algebra?

The Poincare algebra is defined as \begin{align*} i[J^{\mu\nu},J^{\rho\sigma}]&=\eta^{\nu\rho}J^{\mu\sigma}-\eta^{\mu\rho}J^{\nu\sigma}+\eta^{\sigma\nu}J^{\rho\mu}-\eta^{\sigma\mu}J^{\rho\nu}\\ ...
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1answer
107 views

Is perturbation/interaction hamiltonian in interaction theory time-dependent

Reading quantum field theory text, I am confused on regard to whether perturbation (or interaction equivalently) hamiltonian added to free-field hamiltonian is time-dependent or not. In Heisenberg ...
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1answer
128 views

Symmetric part of hamilton matrix and eigenvalues

I'm working with a Density Functional Theory package and I am not sure with a procedure in the code. At some point, after the Hamilton matrix $H_{ij}=\langle\phi_i|\hat{H}|\phi_j\rangle\in\mathbb{R}$ (...
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2answers
205 views

Hamiltonian equation in Cartesian coordinates

My Lagrangian equation is $$L = \dfrac{1}{2}m\dot{q}^{2} \tag{1},$$ where $q=(x,y)$. Performing the Legendre transformation I get the Hamiltonian equation, \begin{equation} H(p,q) =p\dot{q}-\dfrac{1}{...
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1answer
216 views

How to diagonalize the BCS Hubbard Hamiltonian using the Bogoliubov transformation?

How do I diagonalize the following BCS (Bardeen-Cooper-Schrieffer) Hubbard Hamiltonian: \begin{equation} H= \sum\limits_{k \in [-\frac{π}{2}, +\frac{π}{2}[} \begin{bmatrix}c^\dagger_k & c^\...
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160 views

If Hamiltonian is not the total energy, then what to take as $H$ operator in Schrodinger's equation

A particle is confined to move in a circle, and the circle if rotating about its diameter with a constant $\omega$. In this case, I can find the total energy (say, $H_e$), which is the kinetic energy. ...
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1answer
63 views

Dimensions in the Second Quantization of an Operator

Consider the one-particle operator $\hat A_{1p}$. As given in e.g. (Altland and Simons, 2nd ed, 2010; pg47) the second quantized version of this is given by: $$\hat A=\sum_{\mu,\nu} \left< \mu \...