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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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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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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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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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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1answer
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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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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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1answer
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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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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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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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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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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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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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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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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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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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Direct Derivation of Kraus Operator from Interaction Hamiltonian

For the dynamics of open quantum systems, the Kraus operators $K_\kappa$ can be derived from the unitary orbit $U(t)\rho U(t)^\dagger$ for $\rho=\rho_S\otimes\rho_E$ of the composite system given by ...
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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 ...
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(Altland-Simon) Deriving ferromagnetic interaction term from interacting tight-binding Hamiltonian

Below is a part of the book "Condensed Matter Field Theory" by Altland and Simon. My question is about deriving the equation with red arrow. This is outlined in the exercise in the figure, but I don'...
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Completeness condition involving continuum states

Consider a potential $V(x)$ in 1d. Suppose that $V(|x| > a )= 0$ for some positive $a$. We then know that the hamiltonian $H = - \frac{\partial^2}{\partial x^2 } + V(x)$ has non-normalizable or ...
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In quantum search algorithm, how to interpret the effect of $U(t)$ as a rotation on the Bloch sphere?

In Nielsen's QCQI, in page 259, it reads, $$U \left ( \Delta t \right ) = \left ( \cos^2 \left ( \frac {\Delta t} 2 \right ) - \sin ^2 \left ( \frac {\Delta t} 2 \right ) \vec \psi \cdot \hat z \...
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What is a Hamiltonian of a System?

What is a Hamiltonian of a System? When learning about Hamiltonian for first time it is an object introduced as Legendre Dual Transform of Lagrangian of the same system. And we learn further that it ...
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How to construct a Bogoliubov-de Gennes (BdG) matrix?

Recently,I am learning BdG method in superconductor system,I have some question about particle-hole symmetry during construct Hamiltonian matrix for this system. In Hamiltonian if spin orbital term ...
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What is the correct Bogoliubov transformation for a SDW (spin density wave) Hamiltonian in the FBZ (First Brillouin Zone)?

The Hamiltonian for a Antiferromagement with Spin Density Wave (SDW) (in the Reduced Brillouin Zone (RBZ))is written as- \begin{eqnarray} H_{\mathbf{k}\in RBZ}=\sum\limits_{\mathbf{k} \in RBZ}\sum\...
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Why doesn't Wigner's friend interact with the system?

So I was recently modelling something that turned out to be basically Wigner's friend. I saw there were some differences (in the Wiki page) in how it was modelled: Namely, that Wigner's friend ...
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Why are there no potential operators that are non-diagonal in position basis?

The potential energy operator in the Hamilton operator can be expressed in the following way $$ \begin{aligned} \hat V &=\int dx\int dx'|x \rangle\langle x|\hat V|x'\rangle \langle x'| \\ &=\...
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1answer
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Calculating exact energy levels of perturbed Hamiltonian

I wish to find the exact energy levels of the following perturbed hamiltonian. $$\hat{H}=\frac{p^2}{2m}+\frac{m\omega^2}{2}x^2+\alpha x+\beta p^2.$$ I believe that it can be solved by using the ...
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Expection values of the hamiltonian of Klein-Gordon field

The hamiltonian of the quantized Klein-Gordon field $\phi(\textbf{x},t)$ can be writting using the creation and annihilation operators: $$\hat{H} = \frac{1}{2} \int d^{3}\textbf{p} \ \omega_{p} (\hat{...
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Simultaneous eigenstates of Hamiltonian and momentum operator

Given the potential barrier, \begin{align} V(x, y) = \left\{ \begin{array}{cc} V_{0} & \hspace{5mm} \text{if $0 \leq x \leq D$} \\ 0 & \hspace{5mm} \text{...
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1answer
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Kinetic energy always time independent?! Where is my mistake? [closed]

I have some problems understanding the Lagrangian and the Hamiltonian formalism. Those can be condensed in the following "derivation" of $\frac{\partial T}{\partial t} = 0$ from the equation $\frac{\...
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1answer
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Is there always a canonical transformation such that the new Hamiltonian only depends on the new momenta?

Given the Hamiltonian $H(x,p)$ of a system. Is there always a coordinate transformation such that the new Hamiltonian is $K(x',p')=K(p')$?