Questions tagged [hamiltonian]
The central term in the hamiltonian formalism. Can be interpreted as an energy input, or "true" energy.
813
questions
-4
votes
1answer
50 views
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 $\...
0
votes
3answers
64 views
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:
$$H= E_0 + \sigma_1 \...
5
votes
2answers
91 views
Naming symmetries in quantum systems, e.g. $\mathbb{Z}_2$ or $U(1)$
I'm constantly confused by some of nomenclature that is associated with symmetries in quantum Hamiltonians and was hoping someone could set me straight.
Specifically, we often have something like a ...
1
vote
0answers
88 views
+500
Work done on a quantum state
I have a Hamiltonian $H _{\lambda(t)}$, where $\lambda(t)$ characterizes a time-dependent path in parameter space. The parameter is changed in finite time from $\lambda(t_i)$ to $\lambda(t_f)$ . At $t=...
0
votes
1answer
26 views
Interaction picture Sakurai
I’m going through Sakurai and got stuck with the following in the interaction picture subsection
$$i \hbar \frac{\partial}{\partial t}\left|\alpha, t_{0} ; t\right\rangle_{I}=i \hbar \frac{\partial}{\...
0
votes
2answers
52 views
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.
$$i\hbar \frac{\partial}{\partial t}\mathcal{U}(t,t_0)=H\mathcal{U}(t,t_0).$$
...
0
votes
0answers
24 views
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 ...
3
votes
1answer
65 views
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 ...
0
votes
0answers
33 views
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 ?
0
votes
2answers
57 views
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 ...
1
vote
1answer
19 views
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
$$H_{hop} = -t\sum_j(\hat c_{j+1}...
1
vote
1answer
90 views
How can we prove that the Hamiltonian for any quantum system is Hermitian? [closed]
By applying partial time derivative to
$$\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}$$
where $U$ is ...
10
votes
3answers
1k views
In the Schrödinger equation, can I have a Hamiltonian without a kinetic term?
To find out the stationary states of Hamiltonian, we will be finding the eigenvalues and eigenstates. Is there any condition that form of the Hamiltonian should be like, $$\hat{H}=\hat{T}(\hat{p})+\...
0
votes
1answer
44 views
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 ...
0
votes
1answer
41 views
Understanding completeness relation and writing Hamiltonian in matrix form
A three level system hamiltonian I found where it is written as:
$$\frac{H}{\hbar}= \Omega_1|e\rangle \langle g_1| + \Omega_1^* |g_1\rangle \langle e| + \Omega_2|e\rangle \langle g_2|+ \Omega_2^* |...
3
votes
2answers
208 views
Differentiation of a ket vector with respect to a spatial dimension
Consider a state $|\psi\rangle$. While discussing the Schroedinger equation, we say $$\hat{H}|\Psi(t)\rangle=i\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle$$
We also define the hamiltonian operator ...
0
votes
0answers
15 views
Simple way to modify the diagonal elements of the hamiltonian after adding a strong interaction
Let's say we have a hamiltonian of a non-interactive system of two particles. We have correctly worked out the matrix form of the hamiltonian. Now, if we add a very strong attractive interaction ...
0
votes
0answers
52 views
Why are transformations $(q,p)\to (Q,P)$ that are canonical, more useful than any $(q,p)\to(Q,P)$?
I am facing a difficulty in understanding canonical transformations.
It can be defined as a transformation $$(q_i,p_i)\to \big( Q_i(q_i,p_i,t),P_i(q_i,p_i,t)\big) \tag{1}$$ under which the Hamilton'...
-1
votes
0answers
28 views
What is the easiest system to take the matrix representation of a Hamiltonian?
To understand how the unitary operator, preserve the inner products, I wanted to explore the unitary operator as a matrix. Now the equation for the unitary operator (time evolution operator) has a ...
0
votes
1answer
47 views
Unitary Transformation of an Interfering Beam Splitter
I was reading this research paper Quantum interference enables constant time quantum information processing and was confused by one particular expression involving the Hamiltonian of a beam splitter.
...
1
vote
1answer
46 views
Why does the Hamiltonian of a Lagrangian that consists of only a coupled term become zero?
Let's say our Lagrangian looks something like this:
$$L = \int dz\, Q\cdot \dot{A},\tag{1}$$
where $Q$ and $A$ are two generalized coordinates and $\dot{Q}$ and $\dot{A}$ would be the respective ...
1
vote
0answers
57 views
Good quantum numbers from a given hamiltonian
The primary reason asking this question to understand good quantum number from a giver Hamiltonian. Is there any good approach that we can identify them?
For example:
We have a square and in that ...
0
votes
1answer
37 views
PDE from Hamiltonian density
For the wave equation Hamiltonian density is $2H=\phi_t^2+\phi_x^2$ while the Lagrangian density is
$2L=\phi_t^2-\phi_x^2$. I can easily compute the pde from the Lagrangian density but how does one do ...
2
votes
1answer
110 views
Dyson Series Iteration - Gives Exact Solution?
When we derive the Dyson series for usage as the time evolution operator in the case of a time dependent Hamiltonian, we start with the equation:
\begin{align}\hat{U}_I(t,t_i) = 1 - \frac{i}{\hbar}\...
0
votes
0answers
37 views
Help with understanding Pauli matrices in specific Hamiltonian
I am trying to explicitly write out using matrices a Hamiltonian given in this condensed matter paper. In eq (3) of the paper, we have:
$$
\hat{H} = a t (\tau k_x \hat{\sigma_x} + k_y \hat{\sigma_y} ) ...
0
votes
0answers
43 views
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 ...
0
votes
2answers
51 views
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 ...
0
votes
1answer
46 views
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 ...
1
vote
1answer
34 views
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. ...
1
vote
1answer
65 views
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 >\...
0
votes
1answer
80 views
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 ...
1
vote
0answers
21 views
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=...
2
votes
1answer
69 views
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 ...
1
vote
1answer
84 views
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 ...
3
votes
0answers
46 views
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#...
0
votes
0answers
45 views
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 ...
3
votes
1answer
77 views
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 ...
0
votes
0answers
20 views
(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'...
2
votes
2answers
80 views
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 ...
1
vote
0answers
20 views
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,
$$U \left ( \Delta t \right ) = \left ( \cos^2 \left ( \frac {\Delta t} 2 \right ) - \sin ^2 \left ( \frac {\Delta t} 2 \right ) \vec \psi \cdot \hat z \...
0
votes
2answers
91 views
What is a Hamiltonian of a System?
What is a Hamiltonian of a System? When learning about Hamiltonian for first time it is an object introduced as Legendre Dual Transform of Lagrangian of the same system. And we learn further that it ...
0
votes
0answers
32 views
How to construct a Bogoliubov-de Gennes (BdG) matrix?
Recently,I am learning BdG method in superconductor system,I have some question about particle-hole symmetry during construct Hamiltonian matrix for this system.
In Hamiltonian if spin orbital term ...
1
vote
0answers
56 views
What is the correct Bogoliubov transformation for a SDW (spin density wave) Hamiltonian in the FBZ (First Brillouin Zone)?
The Hamiltonian for a Antiferromagement with Spin Density Wave (SDW) (in the Reduced Brillouin Zone (RBZ))is written as-
\begin{eqnarray}
H_{\mathbf{k}\in RBZ}=\sum\limits_{\mathbf{k} \in RBZ}\sum\...
2
votes
0answers
97 views
Why doesn't Wigner's friend interact with the system?
So I was recently modelling something that turned out to be basically Wigner's friend.
I saw there were some differences (in the Wiki page) in how it was modelled: Namely, that Wigner's friend ...
0
votes
0answers
34 views
Why are there no potential operators that are non-diagonal in position basis?
The potential energy operator in the Hamilton operator can be expressed in the following way
$$
\begin{aligned}
\hat V &=\int dx\int dx'|x \rangle\langle x|\hat V|x'\rangle \langle x'| \\
&=\...
0
votes
1answer
48 views
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 ...
0
votes
0answers
40 views
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{...
1
vote
4answers
108 views
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{...
2
votes
1answer
93 views
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{\...
2
votes
1answer
73 views
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 $K(x',p')=K(p')$?
