Questions tagged [hamiltonian]
The central term in the hamiltonian formalism. Can be interpreted as an energy input, or "true" energy.
1,568
questions
1
vote
0
answers
24
views
Quasi-periodic motion of $N$-particle systems [closed]
My question is about the time evolution of multi-particle systems in QFT. There are such systems evolving a-periodically. I struggle with the treatment of them, always obtaining periodic or quasi-...
- 11
2
votes
0
answers
56
views
Do different Hamiltonians result in different ground states?
I'm learning density functional theory. In the proof of Hohenberg–Kohn theorem I, we assume that different Hamiltonians result in different ground states. Is it true?
In general, for example, we can ...
- 31
1
vote
0
answers
49
views
Using variation principle on quantum oscillator with general potential
Consider a general bounding potential $V(x)$. The hamiltonian is $$H = \frac{p^2}{2m} + V(x).$$ We want to apply the variation principle in equation $$F\leq F_0+\langle H-H_0\rangle_0.$$ $\langle\...
- 289
1
vote
1
answer
47
views
Lindblad equation from microscopic principles for free particles with momentum interactions
I'm rather familiar with the formalism of quantum master equations, but I'm struggling with deriving from microscopic principles the collapse operators for a particular case I need.
I consider two ...
- 855
1
vote
0
answers
24
views
Chemical potential terms in Hamiltonian
In the derivation of grand canonical ensemble, which assumes that the physical system (with Hamiltonian $H$) has an average energy $E$ and an average number of particles $\bar{N}$, the density matrix ...
- 452
2
votes
1
answer
158
views
Non-abelian magnetic field interaction
Consider the magnetic part of a non-Abelian field. I want to know if we can define Hamiltonian for the interaction of this sector with the spin (something such as Hamiltonian of interaction of ...
- 143
1
vote
0
answers
31
views
Why does the Eigen-Energy-Shift for the AC-Stark-Shift, calculated in the "rotating frame", actually matter?
As is neatly described in this answer, the AC-Stark Shift is the observation that the energy-eigenvalues of stationary states in the "rotating frame" of a two-level-system behave as (Adam ...
- 6,336
1
vote
0
answers
16
views
Hamiltonian in a semiconductor quantum well
I am currently studying for a photonics exam, and in particular the topic of this lecture is the energy dispersion of a quantum well in a semiconductor heterojunction. My professor wrote a particular ...
1
vote
0
answers
27
views
Magnon spectrum with in-plane Dzyaloshinski-Moria interraction
I'm attempting to derive a magnon spectra using first principles. In-plane DMI is considered in my system. $z$ axis is out of plane. First of all I wrote a Hamiltonian for interacting spins with two ...
1
vote
0
answers
16
views
Hamiltonian of the BEC in 2nd quantization [closed]
If I have $N$ non-interacting particle (bosons) forming a BEC that is trapped to
$x = 0 $ (assume the system to be 1D) by an applied harmonic potential $V=\frac{1}{2}m\omega^{2}x^{2}$
How can I write ...
- 11
1
vote
0
answers
30
views
When are two quantum descriptions / models equivalent?
I am occupied with different types of quantization methods for constrained systems. I start with a constrained phase space and then follow two different paths to get rid of the constraints. In the end,...
- 115
1
vote
0
answers
55
views
Is the QFT Hamiltonian on an eternal Schwarzschild black hole background unbounded below?
Consider the $t=0$ Cauchy slice of the maximally extended Schwarzschild black hole. Let the parts of the slice to the left and right of the bifurcation surface have Hilbert spaces $\mathcal{H}_L $ and ...
- 992
2
votes
2
answers
79
views
The relation of time-evolution operators from Schwartz's textbook
In the section 7.2.2 of Schwartz's QFT textbook, it says:
define the generation definition of time-evolution operators:
$$U_{21}\equiv U(t_2,t_1)=T{\exp[-i\int^{t_2}_{t_1} dt'V_I(t')]}\tag{7.46}$$
...
- 23
1
vote
0
answers
58
views
Trace of a Hamiltonian and zero-point energy
In finite-dimensional quantum mechanics, we are free to assume that our Hamiltonian is traceless. We can define $H' = H - \mathrm{tr}(H)\cdot I$, and since
\begin{equation}
\exp(-iH't) = \exp(-iHt)\...
- 49
0
votes
1
answer
42
views
How does a general operator in a tensor product space act on operators explicitly written as the tensor product of operators in each subspace?
Suppose I model an atom as a two-level system with states $|g \rangle$ and $|e\rangle$, with eigenvalue equations $\hat{H_1}|g\rangle = g|g \rangle$ and $\hat{H_1}|e\rangle = e|e \rangle$, and an ...
- 51
1
vote
0
answers
62
views
Derivative of parameterized time-evolution operator
Suppose we have a parameterized time-dependent Hamiltonian $$H(t; \theta) = H_0 + f(t; \theta)V_0$$, is there any way to obtain the derivative of time-evolution operator $\frac{d}{d\theta} U(t, t_0; \...
- 63
-2
votes
1
answer
171
views
Why the first-order derivative is missing when composing a Hamiltonian of simple harmonic oscillator by the lowering and the raising operators? [closed]
Given the lowering operator ($a$) and the raising operator ($a^\dagger$)
$$\begin{align*}
a &= \frac{1}{\sqrt{2m \hbar \omega}}\left(-i \hbar \frac{\partial}{\partial x} - i m \omega x\right) \\
a^...
- 239
0
votes
0
answers
21
views
How to conceptually understand the symmetries of the Heisenberg Antiferromagnet
Consider the Heisenberg antiferromagnet's Hamiltonian:
$$
\hat{H} = -\frac{1}{2}\sum_{n = 1}^L\bigg(J_x \hat{\sigma}^x_n \hat{\sigma}^x_{n+1} + J_y \hat{\sigma}^y_n \hat{\sigma}^y_{n+1} + J_z \hat{\...
- 129
0
votes
1
answer
37
views
Derivative term in chain of LC coupled oscillators Hamiltonian
I am taking quantum superconducting circuits course and I cannot recover a formula provided by the lecturer.
I want to calculate the Hamiltonian of the following distributed element model of coplanar ...
- 848
0
votes
2
answers
85
views
How to derive the formal solution of Heisenberg's equation? [closed]
In the book Introductory to Quantum Optics https://ostad.hormozgan.ac.ir/ostad/UploadedFiles/386042/386042-1758823246346514.pdf, we have that for an arbitrary operator $\hat{O}$ having no explicit ...
- 37
3
votes
1
answer
100
views
When solving the Schrodinger Equation, where do we add the condition that $E$ is real?
I'm reading through Griffiths, and I noticed two seemingly contradictory facts:
In Chapter 1, it is proved that for any square-integrable function solving the Schrodinger Equation, $$\frac{d}{dt}\...
0
votes
2
answers
115
views
How do we know that the operator $U(t) = e^{-itH/\hbar}$ corresponds to time?
By Stone's theorem on one-parameter unitary groups we know that there is a one-to-one correspondence between strongly continuous one-parameter unitary groups and self-adjoint operators. Hence, if $H$ (...
- 1,350
0
votes
1
answer
25
views
Justification for Dropping Cross Terms in Transmon Hamiltonian
I am working through the "Introduction to Transmon Physics" page on Qiskit, and there is a part of the transmon Hamiltonian derivation that I am stuck on.
The initial form of the transmon ...
- 99
0
votes
1
answer
28
views
Commutation $[H_0, \phi_0(\vec{x},t)]$ in the Heisenberg picture [closed]
Studying from Schwartz "Quantum Field Theory and the Standard Model" p. 23, I got to the part where he discusses time dependence of the field operator $\phi$ and the annihilation/creation ...
- 109
1
vote
0
answers
42
views
Why is the sign negative in time evolution operator in Quantum Mechanics? [duplicate]
Disclaimer: The question has been answered in related posts, example this one which addresses LSZ reduction formula without explaining the minus sign issue.
We consider a unitary operator $U(t+dt,t):\...
- 491
1
vote
0
answers
29
views
Adiabatic theorem for a 3-level system
If I have a 2-level system, with the energy splitting between the 2 levels $\omega_{12}$ and an external perturbation characterized by a frequency $\omega_P$, if $\omega_{12}>>\omega_P$ I can ...
- 471
0
votes
0
answers
52
views
Understanding (non-degenerate and degenerate) Perturbation theory
I was learning the basics of perturbation theory (PT) in QM, and I had quite a bit of trouble with it, especially (time independent) degenerate PT. My reference books are primarily Sakurai (became too ...
- 23
0
votes
0
answers
50
views
Discretized derivation of Majorana path integral
Shankar's QFT book gives an overview for deriving a path integral representation for Majorana fermions. In the derivation, he works directly in continuous imaginary time, sweeping issues of ...
- 2,498
0
votes
0
answers
54
views
Confusion about the first-order correction term of an eigenstate (Degenerate Perturbation Theory)
I'll try my best to put into words what I want to know. But I have the feeling that it is hard to properly explain and correctly articulate the different notations and formulas for this particular ...
- 1,068
-1
votes
1
answer
55
views
Hamiltonian as a matrix and its elements [closed]
Let us consider an electron in an infinitely deep one-dimensional potential well of thickness L with zero potential energy at the bottom of the well. The normalised eigenfunction solutions to this can ...
- 121
1
vote
0
answers
27
views
Status of Approach of constructing Hamiltonians from Transfer Matrix
I am studying this old paper from J.B.Kogut on lattice gauge theories and spin systems [Rev. Mod. Phys. 51, 659(1979)].
This paper discusses about the way of constructing a quantum Hamiltonian using ...
0
votes
1
answer
45
views
General form of a fermionic Hamiltonian in second quantization
I am a new to quantum chemistry. I've been reading this paper and it seems like the equation (1) in the paper which is,
$H = \sum_{pq} T_{pq} a_p^\dagger a_q + \sum_p U_pn_p + \sum_{pq} V_{pq} n_p n_q$...
- 127
0
votes
0
answers
36
views
Interpretation of the Hamiltonian of a real scalar field in FLRW background
In spatially flat FLRW spacetime, using conformal time, the action of a real free massless scalar field is
$$S=\int d \eta d^3 x \frac{a^2}{2}\left[\left(\phi^{\prime}\right)^2-(\nabla \phi)^2\right]$$...
- 11
0
votes
1
answer
57
views
Numerically approximating eigenstates and energies of a particle subject to some potential. Question about dirac orthonormality and infinite terms
Context
I am attempting to approximate the eigenstates and energies of a particle over an interval $[-a,a]$ subject to some potential.
The goal is then to approximate the wavefunction $\Psi$ with ...
- 146
6
votes
3
answers
1k
views
Boundedness of a Hamiltonian and when does a Hamiltonian have a spectrum?
In the context of Quantum Field Theory we put restrictions on the potentials we can use. One argument is boundedness. If the potential is unbounded, for example $V(\phi) = \phi^3$, then `the field can ...
6
votes
2
answers
110
views
Hamiltonian density is time-independent, so how does energy get transported in QFT?
In a QFT on $d+1$-dimensional Minkowski spacetime, the Hamiltonian is:
$$\tag{1} \hat{H} = \int d^{d}\vec{x} \hat{\mathcal{H}}(\vec{x}), $$
where $\hat{\mathcal{H}}(\vec{x})$ is the Hamiltonian ...
- 992
0
votes
0
answers
58
views
Gauge Invariance and Observables [duplicate]
We know that observables are gauge invariant. We know that energy is an observable. Now, the Hamiltonian of a charged particle in an electromagnetic field is not gauge invariant as:
$$
H=\frac{1}{2 m}\...
0
votes
1
answer
60
views
On time-evolution of a quantum state
Suppose I have a quantum system governed by a time-independent Hamiltonian $H$. Its eigenvectors $\{|\varphi_n\rangle\}_{n\in\mathbb{N}}$ form a complete orthonormal set (or basis) for the Hilbert ...
- 1,374
0
votes
0
answers
56
views
Time evolution operator of a two-level system with a completely general Hamiltonian
Using the identity ($\sigma_{0}$) and the Pauli operators ($\sigma_{i}$) as a basis, the Hamiltonian of any two-level system can be expressed as follows
$$
H =\alpha \cdot \sigma _{0}+\textbf{r}\cdot\...
0
votes
0
answers
36
views
Kinetic term of the Hamiltonian constructed from the action of perturbative string motion is not positive definite
I am trying to reproduce the results from this paper. On page 10 of the paper, they have an equation: $$\frac{S}{T}=\int dt\sum _{n=0,1} (\dot{c_n}{}^2-c_n^2 \omega _n^2)+11.3 c_0^3+21.5 c_0 c_1^2+10....
- 121
0
votes
1
answer
66
views
Condition for stationary density matrix
I have a question about section 5 in 'Statistical mechanics' (Pathria).
According this book, the density matrix (operator) should satisfy the following identity, which describes the time evolution of ...
- 33
0
votes
2
answers
47
views
Why is the full Hamiltonian used instead of the approximate Hamiltonian for determining the effective nuclear charge using the variational principle?
My question is in regards to the variational principle in approximating the wavefunction of Helium.
Some Background:
$$\hat{H}=-\frac{\hbar^2}{2m_{e}}\nabla_{1}^{2}-\frac{\hbar^2}{2m_{e}}\nabla_{2}^{2}...
- 71
5
votes
1
answer
236
views
Why do we ever use the Dyson series?
In dealing with interacting Hamiltonians, it's common to do expansions of some sort to make the problem tractable. Two common methods are the Dyson and Magnus expansion. Of the two, the Magnus ...
- 6,393
-2
votes
3
answers
87
views
Observables commuting with the Hamiltonian [duplicate]
Why is it that observables which commute with the Hamiltonian are constants of the motion? I can't understand why that should be true.
4
votes
1
answer
152
views
Meaning of the transition amplitudes in time dependent perturbation theory
Consider a time dependent Hamiltonian
$$\hat{H}(t)=\hat{H}_0+\hat{V}\tag{1}\label{1}(t)$$
where $\hat{H}_0$ is the unperturbed Hamiltonian, for which the eigenvalue problem has been solved
$$\hat{H_0}\...
- 1,346
3
votes
0
answers
62
views
Factorization of 1d Ising model partition function
If I'm studying a 1-dimensional Ising model such that $\mathcal H = \sum_k J_k\sigma_k\sigma_{k+1}$, where $$J_k=\begin{cases}J&k \in2\mathbb N\\2J&k\in2\mathbb N+1 \end{cases}$$ can I ...
- 137
2
votes
0
answers
35
views
Propagator for radial force field?
The propagator $K(x,y;t)$ is well known for the (1D) harmonic oscillator:
$$H = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + \frac{m}{2}\omega^2 x^2$$
is there a simple closed form solution ...
- 6,510
1
vote
1
answer
47
views
Hamiltonian and Thermal Energy
I was reviewing my lessons when I read this definition on Thermal Energy:
The sum of potential energy and kinetic energy is equal to Thermal Energy
But isn't this the same as the Hamiltonian? So, ...
- 157
3
votes
1
answer
189
views
How does $[H,H] = 0$ imply $dH/dt = 0$?
I have a conceptual question about how $[H,H] = 0$ implies $dH/dt = 0$. Does this relation work for both time-dependent and -independent Hamiltonians, or a general observable $H$?
- 919
1
vote
1
answer
54
views
Time Independent Schrödinger Equation meaning
Working on some QM and we realised we don't understand the simple equation is for the wavefunction.
$H \psi(x) = E \psi(x)$
We know $H$ is the hamiltonian, the sum of the kinetic and potential ...
