Questions tagged [hamiltonian]
The central term in the hamiltonian formalism. Can be interpreted as an energy input, or "true" energy.
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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 ...
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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 ...
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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 ...
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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 ...
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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^{\...
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Proof of statement involving hamiltonian operator and projection operator [closed]
$$
\text{Prove: }\,\,i h \frac{\partial{P_j}}{\partial t} = \hat{H}P_j - P_j \hat{H} \\
\text{Where: } \hat{H}\psi(x, t) = ih\frac{\partial{\psi}}{\partial t} \\
P_j = \text{projection operator onto } ...
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Markov approximation at low temperatures
I'm interested in a two-level system in contact with a reservoir of harmonic oscilators, described by the following Hamiltonian:
$$
\hat{H}=\hat{H}_S+\hat{H}_R+\hat{H}_{RS}=\hbar\omega_0\hat{a}^\...
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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:
$$H(p,q)=\sum_{a=0}^N \sqrt{m^2c^4+c^2\vec{p}...
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1answer
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If the time-ordering symbol does not respect commutation relations, why is the expansion of the Dyson series valid?
A standard result in QFT is the expression of the interacting theory correlation functions in terms of field operators in the interaction picture and the free theory vacuum:
$$\langle\Omega|\mathcal T\...
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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 * ...
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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
$$\phi(\textbf{x},t_0)=\...
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What does the Hermiticity have to do with the conservation of energy?
Naomichi Hatano's Non-Hermitian quantum mechanics (link here) published in PTEP 12, 2020 says that Hermitian operators are used when the energy is conserved but when it isn't then non-Hermitian ...
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How to reconcile two different derivations of the time-independent Schrödinger equation?
On one hand, using the Spectral decomposition of the Hamiltonian operator $H$, assumed to be an Hermitian operator, it is relatively simple to derive the equation $U(t) = \sum |v_j\rangle\langle v_j| ...
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States with defined energy and time evolution
Consider the following simple problem:
We have a step potential:
$$V=V_0\Theta (x)$$
so the Hamiltonian is:
$$H=\frac{p^2}{2m}+V_0\Theta(x)$$
and we want to find the eigenfunctions of the Hamiltonian $...
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Canonical transformation and generating function
I am stuck on an exercise of which I do not understand the solution. The exercise is the following:
Consider the Hamiltonian system
$$H(\theta, p, t) = \dfrac{(p-\omega t)^2}{2} - k\cos(\theta)$$
...
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Eigenvalues of $H=aX+bY+cZ+dI$ [migrated]
Suppose I have the hamiltonian $H=aX+bY+cZ+dI$, where $a,b,c,d$ are some real constants, and $X,Y,Z,I$ are Pauli matrices. I'm trying to figure out the range of possible energy eigenvalues. If I limit ...
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What kind of system would have both discrete as well as continuous eigenvalues?
I have been given an example of a hydrogen atom, but i don't really understand how that works out, maybe someone can provide a better example.
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Energy of a moving system in quantum mechanics
I have a question concerning the Hamiltonian operator. As I understand it, the eigenvectors of the Hamiltonian are "steady states," and their corresponding eigenvalues are the observed ...
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Proof that the relativistic Hamiltonian for a free particle corresponds to its total energy
I was trying to prove the usual mass-energy relationship in special relativity using Lagrangian mechanics. The idea is to form the relativistic Lagrangian using spacetime intervals like so:
$$S_\text{...
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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}}{...
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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:
$$ H = \int \frac{d^{3}p}{(2\pi)^{3}} E_{\vec{p}}[(b_{\vec{p}}^{s})^{\dagger}b_{\vec{p}}^{s}-c_{\vec{p}}^{s}(c_{\vec{...
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Peskin Schroeder 2.44 application [duplicate]
After 2.44, the book calculates the commutators of $\phi$ and $\pi$ with the Hamiltonian. But in the first expression the Hamiltonian has a different form than in the second. The first one matches ...
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Low-energy Hamiltonian from Dirac Lagrangian
The low-energy QED Hamiltonian that I would like to derive is ($c=\hbar=1$):
$$ H = \frac{(p-eA)^2}{2m} = \frac{p^2}{2m} - \frac{e}{m} \vec{p}\cdot \vec{A} + \frac{e^2}{2m}A^2$$
where $A$ is the ...
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Time dependency of interacting Hamiltonian in QFT
Suppose I have a Lagrangian of the form
$$\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 ...
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2answers
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$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 ...
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1answer
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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 ...
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1answer
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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 ...
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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 ...
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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 $...
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1answer
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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\...
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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 ...
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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 ...
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1answer
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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 ...
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1answer
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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 ...
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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 $|\psi_{12}\rangle$, ...
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1answer
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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
$$
H=\...
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1answer
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Sakurai: Time evolution operator
How does Sakurai derive the infinitesimal time-evolution operator from scratch without Hamiltonian?
$$\mathcal{U}(t_0+dt,t_0) = 1 - i\Omega dt.$$
It is definitely from Taylor's expansion. But complex $...
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1answer
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A relationship between Lagrangian formalism and Hamiltonian formalism
In the Lagrangian formalism, The Lagrangian L = T(kinetic energy) - V(potential energy). The equations of motion for a given system is given by minimizing the action functional which is a integration ...
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1answer
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How to diagonalize a coupled Hamiltonian
Let $H(r, R)$ be a hamiltonian of a system in center-of-mass and relative coordinates, being $r = r_1 - r_2$ and $$R = \frac{m_1r_1 + m_2 r_2}{m_1 + m_2}$$ Consider the Hamiltonian to contain coupling ...
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Confusion regarding second quantised notation
Suppose Hamiltonian of the system in 2nd quantised notation is
$H= t\sum_{x=1}^{N}d_{x}^{\dagger}d_{x}$.
Does this mean that eigenstates of the Hamiltonian is $N$-fold degenerate with energy $t$?
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Analytical expression for density of random matrix level ratios
Consider a hermitian matrix $H$ with eigenvalues $E_{i-1}<E_i$. The level spacings are defined as $s_i=E_i-E_{i-1}$ and the level ratios as $r_i = s_i/s_{i-1}$. To make the support of an underlying ...
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How to write lattice $\phi^4$ hamiltonian in terms of Pauli matrices?
I want to decompose lattice~$\phi^4$ hamiltonian in terms of Pauli matrices. Particularly, how can I decompose
$$ H_\text{Lattice}=a^d\sum_{{n}\in{Z}}\left[\frac{1}{2}\Pi_{n}^2+\frac{1}{2}\left(\...
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$\mathcal{PT}$ 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 ...
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1answer
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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 $(i\,\,\,2)$ ...
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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:
$$H=\sum_{\omega>0} \hbar \omega (a^{\...
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1answer
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Statistical weight for $N$ harmonic oscillators in microcanonical ensemble
I would like to compute the statistical weight for the microcanonical ensemble for $N$ harmonic oscillators.
To do that i use the hamiltonian of the harmonic oszillator:
$$H(q,p)=\sum\limits_{i=1}^N \...
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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.
...
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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 ...
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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 ...
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1answer
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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 ...
