# 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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### 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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### 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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### 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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### 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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### 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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### $\mathcal{PT}$ 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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### 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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### 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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### 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 ...