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thanks to the answers I received to my previous questions, I could derive correctly an elegant partition function for my problem which resembles a second quantized model taking the particles to be Bosons. The related Hamiltonian for every $i$th ensemble is there: $$H_i=\sum_{q=1}^\infty \gamma_{q,i} \; \lambda_q$$ while $q$ and $i$ positive integers.

What we find is that $\gamma_{q,i}$ is an infinite matrix, and we have all elements of this infinite matrix from observations. $\gamma_{q,i}$ contains only positive integers (and zero), it is a lower triangular matrix with all diagonal elements equal zero except of $\gamma_{11}=\gamma_{22}=1$. The matrix is not random.

I wonder if someone in this community may help me to understand where and how such matrices are dealt with in physics? Is there a specific name/notation for them? Is there a certain intuition behind of such Hamiltonian and such tzpe of matrix?

I want to analyse $\gamma_{q,i}$ further from a physicist's point of view.

Thanks

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  • $\begingroup$ if you're asking for a physics point of view, you'll have to give more info on what the physical system is you're studying; what do you mean by "ensemble", what do the $\lambda_q$ represent, ... $\endgroup$ – Carlo Beenakker Jun 14 '13 at 13:19
  • $\begingroup$ @CarloBeenakker indeed my question is not from physics point of view, I was asking for a mathematical intuition independent of the physical system - this is why raising this question in this forum. However, I can surely tell you more on the physical background w.r. to your questions. $\lambda_q=E_q/E_0$, where $E_q$ are the energies corresponding to the eigenstates. We applied here the term ensemble for a multiparticle state. $\endgroup$ – al-Hwarizmi Jun 26 '13 at 21:24
  • $\begingroup$ if your Hamiltonian $H$ is a sum of lower triangular matrices $\gamma$, then how can $H$ be Hermitian? $\endgroup$ – Carlo Beenakker Jun 27 '13 at 7:26

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