Questions tagged [hamiltonian]
The central term in the hamiltonian formalism. Can be interpreted as an energy input, or "true" energy.
4
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3answers
120 views
What is meant by unitary time evolution?
According to the time evolution the system changes its state the with the passage of time. Is there any difference between time evolution and unitary time evolution?
2
votes
1answer
29 views
Total angular momentum in QM
Dos the total angular momentum, $J=S+L$, commute with the hamiltonian of a general sistem, with no particularities?
2
votes
1answer
50 views
Perturbation theory: justifying expansion in terms of eigenstates of the basis Hamiltonian
I have been wondering why anyone ever thought that we could find an expansion for eigenstates of some perturbed Hamiltonian in terms fo those for the basis Hamiltonian. My lecturer insisted that this ...
1
vote
1answer
36 views
How to diagonalise a hamiltonian which posesses symmetry?
I have a large hamiltonian but I know that it posseses some symmetries. How do you reduce the hamiltonian in order to find the eigenenergies?
2
votes
2answers
106 views
Is there a unitary transformation such that the Hamiltonian in the time-dependent Schrödinger equation becomes real symmetric?
The time-dependent Schroödinger equation is given as (with $\hbar=1$):
$$i\dfrac{d}{dt}\psi(t)=H(t)\psi(t)\ ,$$
where $\psi$ is some normalized column vector and $H(t)$ is a Hermitian matrix with time-...
0
votes
2answers
50 views
QM: Time evolution with $H = H(t)$
In order to calculate time evolution in QM we use Schrödinger equation
\begin{align*}
i \partial_t |\psi\rangle_t = H(t) | \psi\rangle_t.
\end{align*}
If $H\neq H(t)$ then
\begin{align*}
i \partial_t ...
2
votes
1answer
75 views
Why does the Hamiltonian represent something different after plugging in the solution?
so I am beginning to learn Hamiltonian mechanics. We have learned that the Hamiltonian is a function of q, p, and t. Once we have a Hamiltonian, we can use the Hamiltonian equations to derive the ...
-1
votes
1answer
60 views
How to expand this equation? $H_{1}=\frac{e^{2}}{R}+\frac{e^{2}}{R+x_{1}+x_{2}}-\frac{e^{2}}{R+x_{1}}-\frac{e^{2}}{R+x_{2}}$ [closed]
$$H_{1}=\frac{e^{2}}{R}+\frac{e^{2}}{R+x_{1}+x_{2}}-\frac{e^{2}}{R+x_{1}}-\frac{e^{2}}{R+x_{2}}$$
in the approximation $ \left |x_{1}\right |,\left |x_{2}\right |\ll R $ we expand to obtain in lowest ...
0
votes
0answers
87 views
Definition of Hamiltonian in Quantum Mechanics [duplicate]
Is there any particular reason that the Hamiltonian operator was defined in quantum mechanics to be $$\hat H := \frac{\hat p^2}{2m} + V$$ as opposed to $$\hat H := i\hbar \frac{\partial}{\partial t}?$$...
1
vote
3answers
51 views
Deriving or building a Hamiltonian from a Density Matrix
Is it possible to create a Hamiltonian if given a Density Matrix.
If you already the the Density Matrix, then is the Partition
Function (Z) even needed?
This Q is not about physics. Its about an ...
1
vote
0answers
78 views
How to prove time reversal symmetry in a system (given a hamiltonian)
The generic hamiltonian for a particle that interacts with an electromagnetic field can be written as:
$$H=\frac{1}{2M}\sum_{i}\left(P_i-\frac{q}{c}A_{i}(X_j)\right)^2+V(X_j)+q\phi (X_j)$$
Where $(\...
1
vote
0answers
29 views
Utility of the Magnus expansion (preserving symplectic form?)
There are (at least) two ways to perturbatively solve a matrix initial value problem: the Dyson expansion and the Magnus expansion
To be explicit, suppose you're solving for a density matrix $\rho(t)$...
2
votes
2answers
229 views
How to find the coordinate representation of the kinetic operator?
From my professor's notes on statistical mechanics. $\left|\bf{k}\right\rangle$ is eigenstate of the hamiltonian of the free particle with periodic boundary conditions:
$$
\left\langle{\bf r}|{\bf k}\...
1
vote
0answers
47 views
Transforming the Hamiltonian of a free quantum field
I have been trying to explicitly Lorentz transform the Hamiltonian of a free quantum field between two inertial observers, instead of reading it off the manifestly Lorentz invariant action. My ...
2
votes
1answer
52 views
Hamiltonian for a variable length pendulum
This question is taken from the book "Classical Dynamics of Particles and Systems" - Marion, problem 7.24. The problem is about a pendulum that is set into motion, it's length varies at a constant ...
2
votes
0answers
27 views
Looknig for resources on finding periodic orbit and stability on multidimensional Hamiltonian systems
I am looking for resources (books, papers, algorithms, codes) that explicitly explain the computation and analysis (using the monodromy matrix) of periodic orbits of multidimensional Hamiltonian ...
1
vote
1answer
55 views
When is the Hamiltonian a function of momenta alone?
In my statistical mechanics course, I'm deriving the entropy for an ideal gas and I've come across a statement in the book by Pathria where it states that in the case of an ideal gas, the Hamiltonian ...
0
votes
1answer
82 views
Difference between the energy and the Hamiltonian in a specific example
The problem is the following:
Consider a particle of mass $m$ confined in a long and thin hollow pipe, which rotates in the $xy$ plane with constant angular velocity $\omega$. The rotation axis ...
0
votes
0answers
19 views
How to construct a minimal model based $\vec{k} \cdot \vec{p}$ method and symmetry arguments?
Currently, I am repeating the results of this famous paper written by Di Xiao. In this paper, the authors construct a minimal band model based symmetry arguments and $\vec{k}\cdot\vec{p}$ method. The ...
2
votes
4answers
112 views
How come the energy $E$ appears in the Time-Independent Schrödinger Equation, if only energy differences $\Delta E$ are actually physical?
Consider the time-independent Schrödinger equation:
$$\operatorname{\hat H}\vert\Psi\rangle=E\vert\Psi\rangle$$
Is it not true that $E$ doesn't factor into any physically meaningful relation, and ...
2
votes
4answers
91 views
The definition of the hamiltonian in lagrangian mechanics
So going through the "Analytical Mechanics by Hand and Finch". In section 1.10 of the book, the Hamiltonian $H$ is defined as: $$H = \sum_k{\dot{q_k}\frac{\partial L}{\partial \dot{q_k}} -L}.\tag{1.65}...
0
votes
1answer
49 views
$4\times4$ Dirac Hamiltonian in Graphene
When linearizing the Hamiltonian of Graphene in reciprocal space around $\vec{q} = \vec{k}-\vec{K}_\pm = \vec{0}$, where $\vec{K}_\pm$ are two independent Dirac points, one can get two Hamiltonians, ...
3
votes
1answer
83 views
How can I say whether a Hamiltonian is integrable or not?
The transverse field Ising Hamiltonian $$ H = J\sum_{i=0}^{N}\sigma_{i}^{z}\sigma_{i+1}^{z}+h_{x}\sum_{i=0}^{N}\sigma_{i}^{x} $$ is integrable because it can be exactly solved using Jordan Wigner ...
1
vote
1answer
41 views
Hamiltonian Matrix for XXZ Model
Given the XXZ model Hamiltonian,
$H = -\frac{1}{2}\sum^{N}_{i}(\sigma_{i}^{x}\sigma_{i+1}^{x}+\sigma_{i}^{y}\sigma_{i+1}^{y}+\Delta\sigma_{i}^{z}\sigma_{i+1}^{z})$
The two-site Hamiltonian reads
$H ...
0
votes
0answers
29 views
understanding electron-photon interaction hamiltonian
My apologies for a somewhat out-of-context question in advance.
In equation 26 of the paper here , they have the following hamiltonian
$\hat{H_i} = \sum_{p',p, \mathbf{k}} = (g_{p,p',\mathbf{k}})(u^...
2
votes
2answers
85 views
$q\mathbf{A}\cdot\mathbf{v}$ term in potential energy
In the famous Goldstein mechanics book, there is an example about a single (non-relativistic) particle of mass m and charge q moving in an E&M field.
It says the force on the charge can be ...
1
vote
0answers
35 views
Diagonalizing the free Hamiltonian $\hat{H}$ in terms of smeared wave packet operators
Consider the free real scalar field in (1+1)-dimensions. If the (normal-ordered) Hamiltonian is
$$
\hat{H}=\int_{-\infty}^{\infty} dk\ \sqrt{ k^2 + m^2 } \hat{a}_{k}^{\dagger} \hat{a}_{k}
$$
where $\...
2
votes
1answer
173 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 ...
0
votes
3answers
129 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 ...
1
vote
1answer
52 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 ...
4
votes
1answer
76 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 ...
0
votes
0answers
44 views
How are energy eigenvalues of a wave-function related to the eigenvalue of matrices?
Schrodinger equation is a second order DE. Given a potential, I wish to perform standard -find the ground state energy, plot the wavefunction, find the reduced mass exercises using finite difference ...
0
votes
1answer
78 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 ...
1
vote
0answers
45 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 ...
1
vote
2answers
98 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 ...
1
vote
1answer
60 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 ...
-1
votes
2answers
78 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_{...
1
vote
1answer
63 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 ...
1
vote
1answer
35 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 ...
0
votes
0answers
47 views
Expectation value of time dependent Hamiltonian
Let us assume that we have a time dependent Hamiltonian, precisely a Hamiltonian of a harmonic oscillator with time dependent frequency. \begin{equation} \hat{H}=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\...
1
vote
0answers
219 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 ...
-2
votes
1answer
73 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 ...
1
vote
1answer
249 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 ...
0
votes
1answer
53 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 ...
5
votes
2answers
373 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 ...
1
vote
0answers
40 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 ...
0
votes
0answers
45 views
Diagonalization of Quadratic Fermionic/Bosonic Hamiltonians
I'm currently reading Quantum Theory of Finite Systems by Blaziot and Ripka, and I have a question regarding the first few pages of chapter 3.
In particular, the chapter takes on the task of ...
0
votes
1answer
59 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 ...
0
votes
1answer
56 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
$$
\hat{H} = \left(\hat{a}^{\dagger}\...
7
votes
2answers
1k views
Multi-electron Atom
I'm reading the following text on multi-electron atoms:
for a system of $n$ electrons the Hamiltonian is
$$ \hat H = -\frac 1 2 \sum_{i=1}^n \nabla_i^2 - \sum_{i=1}^n \frac{Z}{R_i} + \frac{1}{2} \...
