Questions tagged [hamiltonian]

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

Filter by
Sorted by
Tagged with
1
vote
1answer
65 views

Classic analog of quantum mechanics when dealing with Hamiltonian operator

I am reading The Principles of Quantum Mechanics by Dirac, in chapter 28 Heisenberg's form for the equations of motion, there is ...
1
vote
0answers
19 views

Quantum Mechanic: Sperically symmetric finite well [duplicate]

Consider the following 3D potential: $$V(r)=\begin{cases}-V_0 \ \ \ \ \ \ & r \leq a\\ 0 &r > a\end{cases}$$ we want to find the eigenfunctions for $\ell=0$, in particular we are interested ...
3
votes
2answers
75 views

Scalar field Hamiltonian $H = 0$ from parameterization independence

This question is related (but not similar) to this old one of mine: How to derive the two Friedmann-Lemaître equations from a Lagrangian? Consider the Lagrangian of an isotropic-homogeneous ...
9
votes
2answers
558 views

Time evolution in quantum mechanics of states not contained in the Hilbert space

Eigenstates of, for example, $\hat p$, are not elements of the standard quantum mechanical Hilbert space, i.e. $\psi(x)=e^{ipx}\notin\mathcal L^2(\Bbb R)$. This prompts the question of - given that ...
2
votes
0answers
24 views

Obtaining exponential form of Bogoliubov Operator

Dicke Model Hamilonian after Holstein- Primakoff Transformation in the large spin limit can be represented as - \begin{equation} \begin{split} \hat H = \omega \hat a^{\dagger}a + \varepsilon \hat b^{\...
1
vote
0answers
42 views

Markov approximation at low temperatures

I'm interested in a two-level system in contact with a reservoir of harmonic oscilators, described by the following Hamiltonian: $$ \hat{H}=\hat{H}_S+\hat{H}_R+\hat{H}_{RS}=\hbar\omega_0\hat{a}^\...
0
votes
1answer
95 views

First relativistic correction to the heat capacity of free relativistic gas

I would like to compute the first relativistic correction to the heat capacity of free relativistic gas. At first I would chose the following Hamiltonian: $$H(p,q)=\sum_{a=0}^N \sqrt{m^2c^4+c^2\vec{p}...
1
vote
1answer
46 views

If the time-ordering symbol does not respect commutation relations, why is the expansion of the Dyson series valid?

A standard result in QFT is the expression of the interacting theory correlation functions in terms of field operators in the interaction picture and the free theory vacuum: $$\langle\Omega|\mathcal T\...
0
votes
0answers
22 views

Eigenstates of Rashba Spin-Orbit Hamiltonian

I have been studying on Rashba Spin-Orbit Hamiltonian and trying to find its eigenstates. I am kinda stuck and need some advice. Hamiltonian: H = P^2/(2*m) + a * (Px * sigma-y - Py * sigma-x) + b * ...
0
votes
0answers
52 views

Expanding field operators at a fixed time $t_0$ (from Peskin/Schroeder)

This relates to the bottom of page 83 in Peskin and Schroeder. The following claim is made: At any fixed time $t_0$ we can of course expand $\phi$ in terms of ladder operators $$\phi(\textbf{x},t_0)=\...
0
votes
1answer
79 views

What does the Hermiticity have to do with the conservation of energy?

Naomichi Hatano's Non-Hermitian quantum mechanics (link here) published in PTEP 12, 2020 says that Hermitian operators are used when the energy is conserved but when it isn't then non-Hermitian ...
0
votes
1answer
44 views

How to reconcile two different derivations of the time-independent Schrödinger equation?

On one hand, using the Spectral decomposition of the Hamiltonian operator $H$, assumed to be an Hermitian operator, it is relatively simple to derive the equation $U(t) = \sum |v_j\rangle\langle v_j| ...
2
votes
1answer
92 views

States with defined energy and time evolution

Consider the following simple problem: We have a step potential: $$V=V_0\Theta (x)$$ so the Hamiltonian is: $$H=\frac{p^2}{2m}+V_0\Theta(x)$$ and we want to find the eigenfunctions of the Hamiltonian $...
0
votes
1answer
73 views

Canonical transformation and generating function

I am stuck on an exercise of which I do not understand the solution. The exercise is the following: Consider the Hamiltonian system $$H(\theta, p, t) = \dfrac{(p-\omega t)^2}{2} - k\cos(\theta)$$ ...
0
votes
0answers
57 views

What kind of system would have both discrete as well as continuous eigenvalues?

I have been given an example of a hydrogen atom, but i don't really understand how that works out, maybe someone can provide a better example.
0
votes
0answers
23 views

Energy of a moving system in quantum mechanics

I have a question concerning the Hamiltonian operator. As I understand it, the eigenvectors of the Hamiltonian are "steady states," and their corresponding eigenvalues are the observed ...
0
votes
0answers
34 views

Heisenberg's equation of motion without reference to Schrödinger's picture

Well I'm reading Mukhanov & Winitzki's Introduction to quantum effects in gravity, and I got to the exercise 2.8 that ask to derive Heisenberg's equation of motion \begin{equation} \frac{d\hat{A}}{...
-1
votes
1answer
63 views

Dealing with $\delta^{(3)}(0)$ using normal ordering

On page 109 of David Tong's lecture notes on QFT, equations (5.11) and (5.12) read: $$ H = \int \frac{d^{3}p}{(2\pi)^{3}} E_{\vec{p}}[(b_{\vec{p}}^{s})^{\dagger}b_{\vec{p}}^{s}-c_{\vec{p}}^{s}(c_{\vec{...
0
votes
0answers
33 views

Peskin Schroeder 2.44 application [duplicate]

After 2.44, the book calculates the commutators of $\phi$ and $\pi$ with the Hamiltonian. But in the first expression the Hamiltonian has a different form than in the second. The first one matches ...
2
votes
1answer
110 views

Low-energy Hamiltonian from Dirac Lagrangian

The low-energy QED Hamiltonian that I would like to derive is ($c=\hbar=1$): $$ H = \frac{(p-eA)^2}{2m} = \frac{p^2}{2m} - \frac{e}{m} \vec{p}\cdot \vec{A} + \frac{e^2}{2m}A^2$$ where $A$ is the ...
1
vote
0answers
47 views

Time dependency of interacting Hamiltonian in QFT

Suppose I have a Lagrangian of the form $$\mathscr{L} = \frac{1}{2} \left( \partial_{\mu} \phi \partial^{\mu} \phi - m^2\right) - \frac{\lambda}{4!} \phi^4, $$ which is just an interacting real scalar ...
3
votes
2answers
94 views

$k$-dependence of the energy in solid state physics

In a crystal, the electrons are subject to a periodic potential due to the fact that the atoms form a periodic lattice. From this periodicity we can obtain the Bloch theorem, and get a general formula ...
0
votes
1answer
70 views

Diagonalizing a Hamiltonian

I am currently self studying many-body physics from Introduction to Many Body Physics and am trying to derive (3.153) from (3.152). The Bogoliubov transformation for fermions $a_1$ and $a_2$ are given ...
0
votes
1answer
60 views

Unitary Equivalence of Parity Operator

I recently read a statement that 'parity operator is defined only up to unitary equivalence' in a paper about PT symmetric quantum mechanics. But I didn't understand the meaning of it. It was ...
4
votes
0answers
57 views

Mechanical systems with their configuration space being a Lie group

In Marsden, Ratiu - Introduction To Mechanics And Symmetry there is a certain focus on reducing cotangent bundles of Lie groups. More precisely, if $G$ is a Lie group, then $T^*G$ is naturally a ...
0
votes
0answers
44 views

How to obtain the wave function of time-dependent coupled two harmonic oscillators?

The form of the Hamiltonian of this system is \begin{equation} H = \frac{p_1^2}{2} + \frac{p_2^2}{2} + \frac{x_1^2}{2} + \frac{1}{2}\omega^2(t)x_2^2 + \frac{q}{d}x_1x_2 \end{equation} where $p_1$ and $...
1
vote
1answer
83 views

Question about Klein-Gordon field Lagrangian

I was studying Klein-Gordon field with Peskin QFT. I know that the Hamiltonian of the scalar field can be written as $$H=\int d^3x\left[\frac{1}{2}\pi^2+\frac{1}{2}(\nabla\phi)^2+\frac{1}{2}m^2\phi^2\...
0
votes
0answers
31 views

How to move around an orbit?

Suppose I have a function $f(x)$ (for drawing purposes, I will image it to be $2D$). Is there some sort of dynamics/set of differential equations etc that allows me to move around the orbit defined by ...
1
vote
0answers
69 views

Why does the minimization of the Hartree-Fock Hamiltonian consider only the complex conjugate wave function?

We have the Hartree wave function of N particles: $ \psi_{H} ( 1, ...., N) := \phi_{1}(1) \cdot \phi_{2}(2) ... \phi_{N}(N) $ where $\phi_{j}(I)$ is the one particle wave function of the I-th particle ...
4
votes
1answer
191 views

What is the difference between the time dependent and time independent Schrödinger equation?

I've already gone through a couple of questions regarding the Schrödinger equation and none of them seem to solve my doubt. Some say that the Time Independent Schrödinger Equation (TISE) is just a ...
0
votes
1answer
43 views

Second consequence of invariance under regular canonical transformation in Shankar's QM book

Near the end of chapter 2 of R. Shankar's book Principles of Quantum Mechanics, he talks about two consequences of invariance of the Hamiltonian under a regular canonical transformation. My problem is ...
2
votes
5answers
382 views

Eigenstates/vectors of the sum of non-commuting Hamiltonians

Suppose I have two Hamiltonians $H_1$ and $H_2$, and they're both two-level systems (they do not commute, such as pauli $X$ and $Z$). $H_1$ has eigenstates $|\psi_{11}\rangle$ and $|\psi_{12}\rangle$, ...
1
vote
1answer
61 views

How to transform free field Hamiltonian from position to momentum space?

I'm reading Srednicki's Quantum Field Theory. The equation (3.1) says $$ H=\int\mathrm{d}^3xa^\dagger(\boldsymbol{x})\left(-\frac{1}{2m}\nabla^2\right)a(\boldsymbol{x}) $$ will be transformed $$ H=\...
0
votes
1answer
64 views

Sakurai: Time evolution operator

How does Sakurai derive the infinitesimal time-evolution operator from scratch without Hamiltonian? $$\mathcal{U}(t_0+dt,t_0) = 1 - i\Omega dt.$$ It is definitely from Taylor's expansion. But complex $...
0
votes
1answer
60 views

A relationship between Lagrangian formalism and Hamiltonian formalism

In the Lagrangian formalism, The Lagrangian L = T(kinetic energy) - V(potential energy). The equations of motion for a given system is given by minimizing the action functional which is a integration ...
0
votes
1answer
44 views

How to diagonalize a coupled Hamiltonian

Let $H(r, R)$ be a hamiltonian of a system in center-of-mass and relative coordinates, being $r = r_1 - r_2$ and $$R = \frac{m_1r_1 + m_2 r_2}{m_1 + m_2}$$ Consider the Hamiltonian to contain coupling ...
1
vote
2answers
48 views

Confusion regarding second quantised notation

Suppose Hamiltonian of the system in 2nd quantised notation is $H= t\sum_{x=1}^{N}d_{x}^{\dagger}d_{x}$. Does this mean that eigenstates of the Hamiltonian is $N$-fold degenerate with energy $t$?
1
vote
1answer
15 views

Analytical expression for density of random matrix level ratios

Consider a hermitian matrix $H$ with eigenvalues $E_{i-1}<E_i$. The level spacings are defined as $s_i=E_i-E_{i-1}$ and the level ratios as $r_i = s_i/s_{i-1}$. To make the support of an underlying ...
1
vote
1answer
40 views

How to write lattice $\phi^4$ hamiltonian in terms of Pauli matrices?

I want to decompose lattice~$\phi^4$ hamiltonian in terms of Pauli matrices. Particularly, how can I decompose $$ H_\text{Lattice}=a^d\sum_{{n}\in{Z}}\left[\frac{1}{2}\Pi_{n}^2+\frac{1}{2}\left(\...
1
vote
0answers
54 views

$\mathcal{PT}$ symmetry is a weaker condtion for a Hamiltonian than Hermicity?

In this paper by Bender (end of page 3), he says: And, because PT-symmetry is a weaker condition than Hermiticity, there are infinitely many Hamiltonians that are PT-symmetric but non-Hermitian; we ...
-2
votes
1answer
45 views

Energy Eigenstate of a Hamiltonian

What exactly are energy eigenstates? For something like H = $h\omega \left( \begin{matrix}1 & 2i \\-2i & 4 \end{matrix}\right)$ , what the eigenstates be like the eigenvectors, so $(i\,\,\,2)$ ...
0
votes
0answers
24 views

How is the Hamiltonian of propagating modes transformed with a boundary condition $\phi(0,t)=0$?

I am confused by the following. Imagine that I have wave propagating in a waveguide. I can have left and right moving waves. The total Hamiltonian reads: $$H=\sum_{\omega>0} \hbar \omega (a^{\...
0
votes
1answer
74 views

Statistical weight for $N$ harmonic oscillators in microcanonical ensemble

I would like to compute the statistical weight for the microcanonical ensemble for $N$ harmonic oscillators. To do that i use the hamiltonian of the harmonic oszillator: $$H(q,p)=\sum\limits_{i=1}^N \...
0
votes
0answers
12 views

Transmon: why do we need unharmonic hamiltonian to isolate energy levels? [duplicate]

With a quantum harmonic oscillator, we cannot isolate energy levels, e.g. to create a qubit. We need to embed anharmonicity, to get unevenly-spaced energy levels and so making them distinguishable. ...
0
votes
0answers
26 views

Sketching dynamics of Bloch vector for ground state with dynamics induced by Hamiltonian

How can I sketch a Bloch vector for a system that is in the ground state |g> that is induced by Hamiltonian $ H_x = \frac{ω_x}{2}σ_z$? Is there a general method that I can follow for drawing Bloch ...
0
votes
0answers
22 views

Why should the energy (of an eigenstate) be greater than the lowest point of the potential? [duplicate]

In this lecture (on Scattering states and the step potential) by Dr. Barton Zwiebach, at $2:28$, the professor says: "The energy of any energy eigenstate here has to be bigger than the lowest ...
1
vote
1answer
59 views

Strong equality in Quantization of Gauge Systems by Henneaux and Teitelboim

I am new to the concept of weak and strong equalities, and I have a doubt trying to derive an expression. In section $1.2.1$ of Henneaux and Teitelboim's Quantization of Gauge Systems, there is a ...
0
votes
0answers
45 views

Proca Field Hamiltonian density

Having the most general Lagrangian of the Proca Field given by $$\mathcal{L}=C_1(\partial_\nu A_\mu)(\partial^\nu A^\mu)+C_2(\partial_\nu A_\mu)(\partial^\mu A^\nu)+C_3 A_\mu A^\mu$$ the canonical ...
4
votes
0answers
140 views

Conditions for the Hamiltonian's spectrum to be discrete

I came across this article [1], in which the author studies some Hamiltonian that have a discrete spectrum even though they do not go to infinity at infinity. In there, the author makes several claims,...
1
vote
1answer
65 views

Ground state energy

I am trying to get familiar with the ground state energy of an operator. In my lecture we defined the ground state energy of a self-adjoint operator $H$ that is bounded from below as $$ E_0= \inf_{\...

1
2
3 4 5
26