# Tagged Questions

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

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I am looking for a proof of the next theorem: "If the higher order time derivative Lagrangian is non-degenerate, there is at least one linear instability in the Hamiltonian of this system." Where ...
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### Where does this commutator relation come from?

What is the origin of this relation: $$[H,a_n^\dagger] = \epsilon_n a_n^\dagger$$ for Hamiltonian $H$, creation operator $a_n^\dagger$, and eigenvalue $\epsilon_n$. The square brackets denote ...
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### How can I write the Anderson hamiltonian as a matrix? [closed]

How can I write this Hamiltonian: $$H = \sum E_d \hat{n}_d + \sum_k \epsilon_k\hat{n}_k + \sum_k V_{kd} (\hat{a}^\dagger_k \hat{a}_d + \hat{a}^\dagger_d \hat{a}_k)$$ in matrix form using its ...
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### What does the relation between mass and energy of a free particle mean?

What does the Hamiltonian for a free particle mean? Does it mean that the kinetic energy of the particle is in reverse relation with mass? $H$ or $E=\hbar^{2}k^{2}/2m$. Or better to ask: what's the ...
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### About the double orthogonality of the eigenfunctions of the Hamiltonian

Consider the usual Hamiltonian describing the motion of a particle, $$\hat H = \frac {\hat P^2}{2m} + V(r),$$ where for simplicity the problem can be considered in 1D on the semiaxis $[a, \infty)$ , ...
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### Contradiction in classical analysis of the hamiltonian $\mathcal{H}=xp$?

I am writing an essay on the Berry Keating article proposing to use the $\mathcal{H}=xp$ hamiltonian to get a correspondence between the nontrivial riemann zeros and the eigenvalues of an Hermitian ...
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### Commutation relation of a operator with Hamiltonian [duplicate]

Given that the eigenvalues of a Hamiltonian operator $H$ are bounded below, will a Hermitian operator $T$ exist such that $[T, H] = i\hbar{\bf 1}$ identity operator?
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### What really generates time evolution?

A fundamental principle of quantum mechanics, as far as I can tell, states that the Hamiltonian generates time evolution. A common result about generators are the following: let $\mathrm T$ be the ...
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### Is a gapless system always conducting and a gapped system insulating?

In an answer to this question, @user566 mentioned that there is a qualitative difference between gapped and gapless systems; that gapless systems are conducting and gapped system are insulating. Is ...
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### Two-state Hamiltonian matrix in basis

I have a homework problem as following: Write the two-state Hamiltonian matrix in a certain basis |1>, |2> in a general form as \begin{array}{ccc} H_{11} & H_{12} \\ H_{21} & H_{22} \end{...
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### Hamiltonian split into Mass term and Decay Width

I have encountered the following procedure several times now, and none of the sources ever explain the physical reason behind it: The Hamiltonian $H$ is split into $M$ and $\Gamma$. WHY? Where ...
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### About the derivation of the Hamilton-Jacobi equation

It is an old question for me. In Goldstein's book, the H-J equation is derived in this way. We want to find a generating function $F(q,P,t)$ such that the transformed Hamiltonian vanishes identically, ...
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Suppose we have a particle in an infinite potential well, with $V(x) = 0,\space 0< x < a$ and infinity everywhere else. Now suppose we have a perturbation on the LHS of the well: $V_1(x) = v, ... 1answer 104 views ### Hamiltonian Operator for Harmonic Oscillator I have been solving the harmonic oscillator problem in quantum mechanics using Algebraic Method and since then I am consulting the books of Tannoudji and Griffiths for that matter. While studying both ... 2answers 342 views ### Feynman's$i \epsilon$prescription in loop expansion I have some questions about the$i\epsilon$factor in Feynman diagrams. First, what is the physical meaning of$i\epsilon$in loop amplitudes. Second, how does it ensures unitarity? And third, Dyson ... 1answer 105 views ###$\gamma^5$factor in Quantum Field Theory I have a problem with interpretation of$\gamma^5$factor in the interaction Hamiltonian. I know that$\frac{1\pm\gamma^5}{2}$is a helicity projection and it requires helicity conservation in ... 1answer 123 views ### On use of Hamiltonians for Helium The Hamiltonian of helium can be expressed as the sum of two hydrogen Hamiltonians and that of the Coulomb interaction of two electrons. $$\hat H = \hat H_1 + \hat H_2 + \hat H_{1,2}.$$ The wave ... 1answer 135 views ### Expectation value of Hamiltonian in different pictures of quantum mechanics We start with the familiar Schrodinger equation: $$i\hbar \frac{\partial \left|\psi_S\right\rangle}{\partial t} = \hat{H}_S \left|\psi_S\right\rangle$$ As we switch to a different picture than ... 2answers 194 views ### Showing that Hamiltonian expectation value is time independent I want to check that I am getting the concept right here, and my question is: if the expectation value of a Hamiltonian is the same whether you use the time dependent version or not. I thought I had ... 1answer 5k views ### Calculating the expectation value of a Hamiltonian I want to calculate the expectation value of a Hamiltonian. I have a wave function that is $$\psi = \frac{1}{\sqrt{5}}(1\phi_1 + 2\phi_2).$$ I want to know if I set this up properly. The Hamiltonian ... 1answer 93 views ### Expectation value of Hamiltonian on number state [closed] Hamiltonian is defined by$H_I = \hbar \omega (\hat{a}^+ \hat{a} + 1/2)$What is the expectation value of the energy on the number state $$\vert \psi \rangle = \frac{1}{\sqrt{2}} ( \vert 1 \rangle +... 1answer 139 views ### Apply the Heisenberg Equation to the Hamiltonian [closed] \frac{d}{dt}$$\hat{H}$ = \$\frac{i}{\hbar}$$[\hat{H},\hat{H}] +\frac{\partial{\hat{H}}}{\partial{t}} That's as far as I've got. I do not know much about the Heisenberg equation or even what it ... 1answer 124 views ### How would Hamiltonian for several fermions with spin look? All discussions of Pauli exclusion principle I read usually talked about antisymmetric wavefunctions, from which the princinple appears. But I would like to see a Hamiltonian for multiple fermions, ... 3answers 1k views ### Correct way to write the eigenvector of a diagonalized hamiltonian in second quantization I am studying diagonalization of a quadratic bosonic Hamiltonian of the type:$$ H = \displaystyle\sum_{<i,j>} A_{ij} a_i^\dagger a_j + \frac{1}{2}\displaystyle\sum_{<i,j>} [B_{ij} a_i^\...
In quantum mechanics, we usually write the Hamiltonian as: $$\hat{H}=\hat{T}+\hat{V}$$ But in classical mechanics, there are several reasons why it would not have this form: We've chosen some ...