All Questions
Tagged with hamiltonian variational-principle
9
questions
3
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How to get a lower bound of the ground state energy?
The variational principle gives an upper bound of the ground state energy. Thus it is quite easy to get an upper bound for the ground state energy. Every variational wave function will provide one.
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Why energy eigenstates are extrema of the energy functional? [duplicate]
We have the energy functional of a system:
$$E[\psi] = \frac{\langle \psi | \hat{H} | \psi \rangle}{\langle \psi | \psi \rangle}$$
and over numerous textbooks it is said that the eigenstates of the ...
1
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3
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Why does the variational method simplify in this way when $H$ Hermitian?
Ballentine (Quantum Mechanics: A Modern Development 2nd edition, page 290) writes the attached in his introduction of the variational method. My question is about his very last line: why does $H$ ...
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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:
$$\hat{H}=-\frac{\hbar^2}{2m_{e}}\nabla_{1}^{2}-\frac{\hbar^2}{2m_{e}}\nabla_{2}^{2}...
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9
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Proof of the Variational Theorem
I have trouble understanding the proof of the Variational Theorem. I'll recall quickly the proof to show my problems (see also this post and the answer by Mateus Sampaio for a detailed proof).
Let $H$ ...
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1
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Can Jacobi's formulation of Maupertuis' principle be derived in Riemannian geometry?
I want to arrive to Hamilton-Jacobi equation using the Riemannian geometry.
So let $\textbf{X}\in \mathfrak{X}(M)$, where $M$ is Riemannian manifold whose metric is $g:\textbf{T}M \times \textbf{T}M \...
1
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1
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Variational principle, functional gradient
Given the energy functional
$$E[\Psi] = \frac{\langle \Psi \vert H \vert \Psi \rangle}{\langle \Psi \vert \Psi \rangle},$$ its functional gradient is
$$\frac{\delta E[\Psi]}{\delta \langle \Psi \vert}...
- 153
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2
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Variational Theorem proof
I have been trying to prove variational theorem in quantum mechanics for a couple of days but I can't understand the logic behind certain steps. Here is what I have so far:
\begin{equation}
E=\frac{\...
user58536
9
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Hamilton's characteristic and principal functions and separability
Just hoping for some clarity regarding Hamilton's characteristic function $W$. When we take a time independent Hamiltonian we can separate the Principal function $S$ up into the characteristic ...
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