Questions tagged [hamiltonian]

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

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Question in quantum mechanics [on hold]

We consider the following Hamiltonian: $$H=H_0+\omega_1 S_{1z}+\omega_{2}S_{2z}$$ where $H_0=A\vec{S}_1\cdot\vec{S}_2$ with $A$ is constant, $\omega_1=-\gamma_1B_0$ and $\omega_2=-\gamma_2B_0$ with $\...
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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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Work done on a quantum state

I have a Hamiltonian $H _{\lambda(t)}$, where $\lambda(t)$ characterizes a time-dependent path in parameter space. The parameter is changed in finite time from $\lambda(t_i)$ to $\lambda(t_f)$ . At $t=...
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Finding energy Eigenvalue from two spin Hamiltonian

Let's consider a system of two spins, named spin 1 and spin 2. Let's also consider, in a Hamiltonian, spin part has been defined as $\sigma_1 \cdot \sigma_2$. For example: $$H= E_0 + \sigma_1 \...
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Naming symmetries in quantum systems, e.g. $\mathbb{Z}_2$ or $U(1)$

I'm constantly confused by some of nomenclature that is associated with symmetries in quantum Hamiltonians and was hoping someone could set me straight. Specifically, we often have something like a ...
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Hamiltonian with position-spin coupling

I am solving a Hamiltonian including a term $(x\cdot S)^2$. The Hamiltonian is like this form: \begin{equation} H=L\cdot S+(x\cdot S)^2 \end{equation} where $x$ is the position operator, $L$ is ...
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Interaction picture Sakurai

I’m going through Sakurai and got stuck with the following in the interaction picture subsection $$i \hbar \frac{\partial}{\partial t}\left|\alpha, t_{0} ; t\right\rangle_{I}=i \hbar \frac{\partial}{\...
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Prove conservation law in quantum mechanics

I major in Math, and I am studying Quantum Mechanics (QM). I see the conservation law in QM as a mathematical theorem. Please check if my understanding is right, and help me to prove the theorem? ...
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Solution of Time-dependent Schrodinger Equation for Unitary Operator

While reading Quantum Mechanics Book by Sakurai, I found the time-dependent Schrodinger equation for Unitary Operator. $$i\hbar \frac{\partial}{\partial t}\mathcal{U}(t,t_0)=H\mathcal{U}(t,t_0).$$ ...
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Hamiltonian for a mode-shift operator

I have a discrete multi-level degree of freedom in my quantum system (for photons, for example this), which I write as $|l\rangle$. The degree of freedom is unbounded, i.e. $l$ can take ever positive ...
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Symmetry of the hamiltonian $H = \frac{1}{2m}p^2 + V(r) + a \, \vec{s} \cdot \vec{l} $

Consider the hamiltonian \begin{align} H& = H_0 + a\, \vec{s} \cdot \vec{l} \\& = \frac{1}{2m}p^2+ V(r) + a\, \vec{s} \cdot \vec{l}, \end{align} where $V(r)$ denotes an arbitrary central ...
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Does all measured quantities by Hamiltonian operator should be real valued? [duplicate]

am not familiar with QM , and i have checked web many times to know wether all measured Quantities of any arbitray system should be real since it is Hermitian ?
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Commutation with unspecified potential function

Instead of a potential given like $V(r) = k r^2$ or $V(r) = y^2$ , if the potential is given like in the form a function but not clearly specified, can we tell that if that commutes with the ...
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Fourier Transform from lattice site into $k$-space in Hubbard-Holstein model

Say I have a one dimensional lattice with lattice constant $a$. With next nearest neighbor hopping (NNN) included, the hopping term that describe such system would be $$H_{hop} = -t\sum_j(\hat c_{j+1}...
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Is there a physical interpretation to invariant random matrix ensembles?

Disclaimer. I am a graduate student in pure mathematics, so my knowledge of physics more advanced than basic 1st/2nd year undergraduate physics is very limited. I welcome corrections on any ...
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Do systems with level crossings have unstable eigenbases?

It's folklore dating back to von Neumann and Wigner that time-dependent Hamiltonian systems tend not to have level crossings of their energy eigenvalues. However, we can of course consider smoothly ...
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167 views

Equivalent two-levels systems Hamiltonians

I've once read somewhere that the following two Hamiltonians each describing a two-level system (TLS) are equivalent to each other and can thus be used in describing the same TLS. $$\mathcal{H}_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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1answer
90 views

How can we prove that the Hamiltonian for any quantum system is Hermitian? [closed]

By applying partial time derivative to $$\psi_t \rightarrow U \psi_{t_0}$$ we end up with an expression for the Hamiltonian $$H = i\hbar\frac{\partial U_{t}}{\partial t}U^{\dagger}_{t}$$ where $U$ is ...
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In the Schrödinger equation, can I have a Hamiltonian without a kinetic term?

To find out the stationary states of Hamiltonian, we will be finding the eigenvalues and eigenstates. Is there any condition that form of the Hamiltonian should be like, $$\hat{H}=\hat{T}(\hat{p})+\...
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Why is there a need to add the complex conjugate in the tight binding hamiltonian?

So we start with the following hamiltonian describing non-interacting free fermions: $$ \hat{H}_{\text{free}} = \sum_{i,j,\sigma}\tilde{t}_{ij} \hat{c}_{i\sigma}^\dagger\hat{c}_{j\sigma}.$$ Then ...
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Understanding completeness relation and writing Hamiltonian in matrix form

A three level system hamiltonian I found where it is written as: $$\frac{H}{\hbar}= \Omega_1|e\rangle \langle g_1| + \Omega_1^* |g_1\rangle \langle e| + \Omega_2|e\rangle \langle g_2|+ \Omega_2^* |...
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Differentiation of a ket vector with respect to a spatial dimension

Consider a state $|\psi\rangle$. While discussing the Schroedinger equation, we say $$\hat{H}|\Psi(t)\rangle=i\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle$$ We also define the hamiltonian operator ...
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Simple way to modify the diagonal elements of the hamiltonian after adding a strong interaction

Let's say we have a hamiltonian of a non-interactive system of two particles. We have correctly worked out the matrix form of the hamiltonian. Now, if we add a very strong attractive interaction ...
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Dyson Series Iteration - Gives Exact Solution?

When we derive the Dyson series for usage as the time evolution operator in the case of a time dependent Hamiltonian, we start with the equation: \begin{align}\hat{U}_I(t,t_i) = 1 - \frac{i}{\hbar}\...
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Tight Binding Hamiltonian for graphene

The TB Hamiltonian for the tetragonal lattice is $ \hat H_0 = -J\sum_{m,n} (\hat a_{m+1,n}^\dagger \hat a_{m,n}+\hat a_{m,n}^\dagger \hat a_{m,n+1}+h.c.) $ How can this be derived for the hexagonal ...
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Why are transformations $(q,p)\to (Q,P)$ that are canonical, more useful than any $(q,p)\to(Q,P)$?

I am facing a difficulty in understanding canonical transformations. It can be defined as a transformation $$(q_i,p_i)\to \big( Q_i(q_i,p_i,t),P_i(q_i,p_i,t)\big) \tag{1}$$ under which the Hamilton'...
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What is the easiest system to take the matrix representation of a Hamiltonian?

To understand how the unitary operator, preserve the inner products, I wanted to explore the unitary operator as a matrix. Now the equation for the unitary operator (time evolution operator) has a ...
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Unitary Transformation of an Interfering Beam Splitter

I was reading this research paper Quantum interference enables constant time quantum information processing and was confused by one particular expression involving the Hamiltonian of a beam splitter. ...
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Time-dependence of free and interacting Hamiltonians

Consider an interacting field theory with Hamiltonian $$H=H_0+V$$ where $H_0$ is the Hamiltonian of the free theory and $V$ is the added interaction. Now, I know the full Hamiltonian $H$ should be ...
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Why does the Hamiltonian of a Lagrangian that consists of only a coupled term become zero?

Let's say our Lagrangian looks something like this: $$L = \int dz\, Q\cdot \dot{A},\tag{1}$$ where $Q$ and $A$ are two generalized coordinates and $\dot{Q}$ and $\dot{A}$ would be the respective ...
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Good quantum numbers from a given hamiltonian

The primary reason asking this question to understand good quantum number from a giver Hamiltonian. Is there any good approach that we can identify them? For example: We have a square and in that ...
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Is energy $E$ in Schrödinger equation an observable/ Can $E$ be measured?

Take this quantum approach to estimate mean energy of a molecule: $$\langle\psi|H|\psi\rangle=\overline E$$ Question: Is $E$ an observable? How we can compare it to an experimental value? i.e how ...
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PDE from Hamiltonian density

For the wave equation Hamiltonian density is $2H=\phi_t^2+\phi_x^2$ while the Lagrangian density is $2L=\phi_t^2-\phi_x^2$. I can easily compute the pde from the Lagrangian density but how does one do ...
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Help with understanding Pauli matrices in specific Hamiltonian

I am trying to explicitly write out using matrices a Hamiltonian given in this condensed matter paper. In eq (3) of the paper, we have: $$ \hat{H} = a t (\tau k_x \hat{\sigma_x} + k_y \hat{\sigma_y} ) ...
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Hamiltonians and Hilbert spaces in QFT

Suppose we start from a theory with a given $\mathcal{L}$ and correspondigly a Hamiltonian $\mathcal{H}$. Now a state of this $\mathcal{H}$ is (say) $|p,q,r>$. Now suppose that we do a set of ...
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Can the wave-function of any particle in any basis be written as a matrix?

Can the wave-function of any particle in any basis be written as a matrix? If no, how can we explain this, where the Hamiltonian $H$ in U is a QM operator that can be written as a linear ...
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Energy contributions of Hamiltonian density

In Lancaster and Blundell, Quantum Field Theory for the Gifted Amateur, p.99, Hamiltonian density is \begin{equation} \mathcal{H}=\frac{1}{2}[\partial_0\phi(x)]^2+\frac{1}{2}[\nabla\phi(x)]^2+\frac{...
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1answer
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Heisenberg Hamiltonian 2-Spin Terms in Matrix Representation

I am stuck on the interpretation/derivation of the 2-spin terms of the quantum Heisenberg model Hamiltonian. In this model, our electrons, with spin up or down, are confined to sites on a lattice. ...
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1answer
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Scattering, 4 point correlator, #distinct Feynman diagrams

In order to compute the scattering probability that two particles of type 1 (associated to $\phi_1(x)$) which come from the far past with the momenta $p_1$ and $p_2$, to scatter and evolve into two ...
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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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Different formula to find $2\times 2$ Hamiltonian's eigenvalues [closed]

Consider the Hamiltonian $$ \left[ \begin{matrix} E_1 & -A\\ -A& E_2\\ \end{matrix} \right] $$ where $A$, $E_1,E_2$ are real numbers. I have seen a different formula to ...
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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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Trouble getting the matrix representation of a 4-state Hamiltonian

$\newcommand{\bra}[1]{\left\langle #1 \right|} \newcommand{\ket}[1]{\left| #1 \right\rangle}$I have a simple 4-state Hamiltonian and am trying to find the matrix representation (in order to determine ...
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Semiclassical limit $S \to\infty$ in spin model

In many literature, the limit $S \to \infty$ is considered as a semiclassical limit. My question is that when this approximation is valid? Since paticles, say electrons, have the fixed spin number $S=...
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Why does the wave function of a non relativistic particle flatten out over time?

The Hamiltonian I used is the classical one with no potential energy: H=p^2/2m $$i \hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} $$ I want to gain ...
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1answer
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Is Hamiltonian a scalar or tensor in Quantum Mechanics?

According to Wikipeida, a scalar operator is invariant under rotations, and the Hamiltonian satisfies this definition. But at the same time, a Hamiltonian can be written as a matrix, which means it is ...
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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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Why is the generalized momentum replaced by the momentum operator but not the ordinary momentum?

I was trying to understand the derivation of the Hamiltonian for a charged particle in an electromagnetic field. https://en.wikipedia.org/wiki/Hamiltonian_mechanics#...
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1answer
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Significance of energy in a time dependent quantum box

The Hamiltonian for a particle in a finite box is $$H = \frac{p^2}{2m} + V(x)$$ which will give time evolution as $$ i\hbar d/dt|{\psi(t)}\rangle = H|{\psi(t)}\rangle \, .$$ However, if I do a ...