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1answer
95 views

Can a general many-body Hamiltonian with quadratic and biquadratic terms be diagonalized?

Can an arbitrary many-body hamiltonian in second quantization form with quadratic and biquadratic terms $$H=\sum_{v_1,v_2} \alpha_{v_1 v_2}\ c_{v_1}^{\dagger}c_{v_2}+ \sum_{v_1,v_2,v_3,v_4}\beta_{v_1 ...
2
votes
1answer
112 views

A naive question on the eigenvalues of fermionic operators?

Let $A$ be a fermionic operator which is a product of odd number of fermion operators or a summation of them, say $A=C_{i_1}^{\dagger}\cdot \cdot\cdot C_{i_m}^{\dagger}C_{j_1}\cdot \cdot\cdot C_{j_n}...
1
vote
1answer
398 views

Single particle operator in second quantization

I want to understand why we write in the formalism of second quantization for a single particle operator \begin{equation} \hat H=\sum_i \varepsilon_i \hat a_i^{\dagger} \hat a_i \end{equation} where ...
1
vote
1answer
221 views

Eigenvalues of fermionic field operators

Consider the fermionic field operators $\psi_a(x), \psi^{\dagger}_b(y)$ with the canonical anti-commutation relations $$\{\psi_a(x),\psi_b(y)\} = 0 $$ and $$\{\psi^{\dagger}_b(t,\vec{x}),\psi_a(t,\vec{...
14
votes
1answer
2k views

Schrödinger wavefunctional quantum-field eigenstates

The reason that I have this problem is that I'm trying to solve problem 14.4 of Schwartz's QFT book, which've confused me for a long time. The problem is to construct the eigenstates of a quantum ...
2
votes
0answers
411 views

Tight-binding model of diatomic solid

I am working on a problem based on Problem 1.6 in Gerald D. Mahan - Many Particle Physics, 3rd edition. The problem: Consider a tight-binding solid which has alternating atoms of type A and B. The ...
1
vote
0answers
80 views

Eigenvalues of Hamiltonian with on-diagonal coordinate

A bit abstract, but if I take the standard graphene Hamiltonian (around the Dirac point) and introduce an on-diagonal term proportional to the coordinate $\hat{y}$, how would I find the eigenstates ...
1
vote
1answer
813 views

numerical diagonalization of tight-binding hamiltonian

I would like to find the exact eigenvalues of the following tight-binding Hamiltonian, written here in second quatization: \begin{eqnarray} \hspace{-0.25in}{\mathcal{H}} &=& \mathcal{H}_0+ \...