I am writing an undergrad (Bachelor) thesis covering the harmonic and anharmonic oscillator using path integrals on discrete time lattice. With the material from [CF80], I have created a program that creates quantum trajectories on the lattice and calculates [itex]|\psi(x)|^2[/itex] and [itex]E_0[/itex] from that.(adsbygoogle = window.adsbygoogle || []).push({});

Now I have been given [Blo+09] as well to calculate higher energy eigenvalues. From what I understand so far, the energies are given by

[tex] E_n^\text{eff}(t, t_0) = - \frac1a \left( \log \lambda_n(t+a, t_0) - \log \lambda_n(t, t_0) \right),[/tex]

where the eigenvalues [itex]\lambda_n[/itex] are determined by the GEVP (generalized eigenvalue problem)

[tex] C(t) v_n(t, t_0) = \lambda_n(t, t_0) C(t_0) v_n(t, t_0).[/tex]

The matrices [itex]C[/itex] are “matrices of Euclidean space correlation function”, see (2.1) in the paper. Those contain operators/fields [itex]O_i[/itex] that are “some interpolating fields”.

I see that the energy values are what I need, but I am not sure where I can put my data to use. Since I have just one harmonic oscillator, I am not sure what my “fields” are supposed to be.

Do you see how I can extend my Metropolis Monte Carlo algorithm to compute higher energies with the contents of that paper? If so, could you please point me into the right direction?

- [CF80] M.[/PLAIN] [Broken] Creutz und B. Freedman. „A Statistical Approach to Quantum Mechanics“. In: Annals of

Physics 132.1981 (1980), S. 427–462.

- [Blo+09] Benoit Blossier et al. On the generalized eigenvalue method for energies and matrix elements in lattice field theory. Feb. 2009. arXiv: 0902.1265 [hep-lat]

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# Apply GEVP methods to find energies of oscillators in lattice theory

Can you offer guidance or do you also need help?

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