Switching to a Matrix Hamiltonian - Conceptual Issues

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Discussion Overview

The discussion revolves around the conceptual issues related to switching to a matrix Hamiltonian in quantum mechanics, particularly in the context of numerical simulations involving discretized Hamiltonians. Participants explore the transition from analytical to matrix representations of Hamiltonians, addressing both theoretical and practical aspects of discretization.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant discusses their routine use of a discretized single particle, one-band effective mass Hamiltonian and raises questions about transitioning to an atomistic Hamiltonian derived from first principles.
  • Another participant suggests that the matrix elements of the Hamiltonian can be expressed in terms of a basis, specifically mentioning the formula h_{ij} = ⟨ψ_i|H|ψ_j⟩, although they express uncertainty about whether this fully addresses the original question.
  • A third participant inquires about the methods of discretizing the Hamiltonian and questions the distinction between exact and numerical representations.
  • Another participant provides specific methods for discretizing the Hamiltonian on a discrete lattice, mentioning first and second order differentiation techniques, and notes that the time step, represented by tau, is a critical factor.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and clarity regarding the original question, with some seeking clarification while others provide technical insights. There is no consensus on the best approach to transition from an analytical to a matrix representation of the Hamiltonian.

Contextual Notes

Participants highlight the complexity of the transition from analytical to matrix representations, noting potential ambiguities in definitions and the need for a clear understanding of discretization methods. The discussion reflects a mix of theoretical and practical considerations without resolving the underlying conceptual issues.

sokrates
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Switching to a Matrix Hamiltonian -- Conceptual Issues

It's probably very clear and well-established for those who rigorously studied Quantum Mechanics but I don't think what I am going to ask is easily 'google'-able but if it is so - please send me to the correct source before spending time.

But don't recommend the whole Landau-Lifgarbagez QM text please.

The thing is, I am routinely performing numerical simulations that involve a discretized single particle, one-band effective mass Hamiltonian almost everyday. I discretize free space (note that I am using an effective mass, so that's okay even for an electron moving in a solid
which is okay.

Which is not okay is that if I don't use an effective mass approach and decide to go to an ATOMISTIC hamiltonian which could be derived from first principles the lattice I am going to work on will be discrete by itself! The point is, everything is made up of atoms and I will have to work on a discrete lattice anyway.

And this Hamiltonian will be exact if I am not mistaken. Now the question:

How do you start from an analytical Hamiltonian and obtain an exact matrix representation??

I kind of know everything (numerical discretization, real lattice discretization, eigenspace discretization,) boils down to the concept of basis functions but I just can't connect the dots.

I hope there'll be some interest in this,
 
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For me the grammar in some parts of your post is hard to understand, you may want to rephrase your question.

If [itex]|\psi_i\rangle[/itex] form a basis then the matrix elements [itex]h_{i j}[/itex] of the Hamiltonian in that basis are given by [itex]h_{i j} = \langle\psi_i|H|\psi_j\rangle[/itex]. I doubt this answers your question, but it seems related.
 


How do you discretize the Hamiltonian?

Is it exact?

What is the real space representation?

What is analytical and what is purely numerical?
 


sokrates said:
How do you discretize the Hamiltonian?

Working on a discrete lattice you might typically use:

[tex]\frac\tau2~\Big\{~-1~,~~0~~,~+1~\Big\}[/tex] for first order differentiation and

[tex]\tau^2\Big\{~~~1~,~-2~,~~~1~~~\Big\}[/tex] for second order differentiation.

The rest should stay the same. Tau determines the time steps.Regards, Hans
 

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