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You can think of Hartree-Fock as a self-consistent mean field method. The idea is that you start with each of the particles in their initial orbits. These particles generate a mean field, and you can solve for single-particle eigenfunctions of this mean field. This is done by solving the time-independent Schrodinger equation $$ ...


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It depends on the specifics what the notation means, but in general one might say: we pretend, mathematically, that we can label each particle $1, 2, \dots n$, and look at its individual distribution in space, which is (for pure states) some wavefunction $a_k(\vec r, t)$. The product state simply means "particle #1 is in state $a_1$, particle #2 is in state ...


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Normal ordering is an ordering of a product of field operators, in which all the annihilation operators are placed in to the right of all creation operators. This mean that the expectation value of a normal ordering in relation to the vacuum state is zero. If $c_a^\dagger c_b = \rho_{ba} + :c_a^\dagger c_b:$ and $|\Phi\rangle$ is the vacuum state, then ...


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You need to perform tensor product of the matrices. By doing this you will get the matrix with the dimension 4x2=8. The similar tensor product you should perform with the wavefunctions, so they span 8 dimensional space. Your new wavefunction will be a vector in this new basis.


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According to this lecture from the University of Edinburgh, numerical simulations of N-body systems suggest a half-mass relaxation time: $$ t_\text{rh} = 0.138\frac{N^{1/2}r_\text{h}^{3/2}}{m^{1/2}G^{1/2}\ln(\gamma N)} $$ where $r_\text{h}$ is the radius that initially contains half the mass of the system, $G$ is the gravitational constant, $m$ is the ...


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The Hamiltonians of the two particle in the composite system are $H_0\otimes I_2$ and $I_2\otimes H_0$ respectively, so that the total Hamiltonian is $$H = H_0\otimes I_2 + I_2\otimes H_0,$$ and this leads, as expected, to the doubling of the energy if both particles are in the same state (provided this is not excluded by symmetry properties of the system). ...



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