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The Variational Methods of Quantum Systems in Holomorphic Representation.

M. W. AlMasri, Mohamed Ridza Wahiddin

Published Nov 28, 2021• 0 citations
OpenAlex

Abstract

We show that variational method for harmonic oscillator in Bargmann representation breaks down since we always end up with one of the exact eigenvalues of Hamiltonian using any normalized monomials in the complex plane. Using the holomorphic representation of pure spin systems , we also encounter the same result found earlier for harmonic oscillators in Bargmann representation. We conclude that the exact energy eigenvalues of these systems are unavoidable starting from any normalized monomials of the complex phase-space coordinate. Since $\{\frac{z^{n}}{\sqrt{n!}}\}$ form an orthonormal basis, one could in principle expand any complex function $f(z)$ in powers of $z$ and use this expansion as energy eigenstates. Such expansion is valid since the magnitude of $z$ carry no physical information only their non-negative exponents $n$ are physically important. These results may lead to new selection principle regarding the quantized energy spectrum of physical systems and enhance our understanding of quantum world.

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We show that variational method for harmonic oscillator in Bargmann representation breaks down since we always end up with one of the exact eigenvalues of Hamiltonian using any normalized monomials in the complex plane. Using the holomorphic representation of pure spin systems , we also encounter the same result found earlier for harmonic oscillators in Bargmann representation. We conclude that the exact energy eigenvalues of these systems are unavoidable starting from any normalized monomials of the complex phase-space coordinate. Since {znn!}\{\frac{z^{n}}{\sqrt{n!}}\}{n!​zn​} form an orthonormal basis, one could in principle expand any complex function f(z)f(z)f(z) in powers of zzz and use this expansion as energy eigenstates. Such expansion is valid since the magnitude of zzz carry no physical information only their non-negative exponents nnn are physically important. These results may lead to new selection principle regarding the quantized energy spectrum of physical systems and enhance our understanding of quantum world.
We show that the variational method for harmonic oscillator in Bargmann representation blows up since we always end up with one of the exact eigenvalues of the Hamiltonian using any normalized monomials in the complex plane. Using the holomorphic representation of pure spin systems , we also encounter the same result found earlier for harmonic oscillators in Bargmann representation. We conclude that the exact energy outcomes of these systems are unavoidable starting from any normalized monomials of the complex phase-space coordinate. These results may lead to new selection principle regarding the quantized energy spectrum of physical systems. Starting from any normalized complex-variable monomial always give the corresponding exact energy eigenvalue. This fact opens the road for extending the possible energy configuration solutions and enable us to use the whole series expansion techniques in quantum theory.
We present a systematic study of the Rayleigh--Ritz variational method for quantum oscillators in the Segal--Bargmann space. We rigorously derive the normalizability condition ∣α∣<12|α| < \tfrac{1}{2}∣α∣<21​ for generalized Gaussian trial functions ψ(z)=eαz2+βzψ(z) = e^{αz^2 + βz}ψ(z)=eαz2+βz through convergence analysis of Gaussian integrals in the complex plane. Applications to the harmonic oscillator demonstrate exact recovery of the ground state in Segal--Bargmann space when the trial family contains the true solution. For the quartic anharmonic oscillator (H^=−12∂x2+12x2+λx4\hat{H} = -\tfrac{1}{2}\partial_x^2 + \tfrac{1}{2}x^2 + λx^4H^=−21​∂x2​+21​x2+λx4), adaptive Gaussian ansätze in position space yield a cubic stationarity equation and perturbative energy expansions beyond first order, capturing anharmonic wavefunction narrowing. In contrast, monomial trial functions (ψn(z)=znψ_n(z) = z^nψn​(z)=zn) in the Segal--Bargmann space -- while providing rigorous upper bounds En=n+12+3λ4(2n2+2n+1)E_n = n + \tfrac{1}{2} + \tfrac{3λ}{4}(2n^2 + 2n + 1)En​=n+21​+43λ​(2n2+2n+1) for excited states -- lack width adaptability and are limited to first-order accuracy for ground-state calculations. We further analyze displaced Gaussians and displaced monomials for asymmetric potentials (e.g., x3+x4x^3 + x^4x3+x4), showing that displacement parameters are essential to capture parity breaking and stabilization effects (E0≈12+3μ4−9λ24+⋯E_0 \approx \tfrac{1}{2} + \tfrac{3μ}{4} - \tfrac{9λ^2}{4} + \cdotsE0​≈21​+43μ​−49λ2​+⋯).
Instead of representing quantum states by vectors in the Hilbert space one could represent them as a convergent monomial in the space of holomorphic functions with Gaussian measure using the Bargmann representation of the lowering and raising operators. This strategy simplifies the calculations since for a given quantum system the coordinate representation of any quantum state is always more complicated than its corresponding expression in the Bargmann space. We solve the quantum absorption refrigerator analytically in term of the phase space coordinates ziz_{i}zi​ and their partial derivatives. We finally discuss the computational complexity of the holomorphic representation and compare it with the computational complexity of the standard operator method and prove the efficiency of the holomorphic representation in computing some tasks. Our treatment is applicable in all quantum heat engines and refrigerators.
We demonstrate that the Bargmann representation of quantum mechanics is ideally suited for semiclassical analysis, using as an example the WKB method applied to the bound-state problem in a single well of one degree of freedom. While the WKB expansion formulas are basically the usual ones, in this representation they describe approximations that are uniform and nonsingular in the classically allowed region of phase space because no turning points appear there. The quantization of energy levels relies on a complex contour integral that tests the eigenfunction for analyticity. For the harmonic oscillator, this WKB method trivially gives the exact eigenfunctions in addition to the exact eigenvalues. For an anharmonic well, a self-consistent variational choice of the representation greatly improves the accuracy of the semiclassical ground state. Also, a simple change of scale illuminates the relationship of semiclassical versus linear perturbative expansions, allowing a variety of multidimensional extensions. All in all, the Bargmann representation appears to combine the advantages of a linear description and of a phase-space representation of the quantum state vectors.
Spin chain Hamiltonians can be written in terms of complex differential operators using the Bargmann representation of the Jordan-Schwinger map. In this case, the eigenfunctions are expressed as the product of orthonormal monomials of the phase-space coordinates zi=Qi+iPiz_{i}=Q_{i}+i P_{i}zi​=Qi​+iPi​ in the complex plane. Furthermore, the series constructed from each phase-space coordinate converges uniformly in any compact domain of the complex plane. Formulating spin chains with respect to the phase-space coordinates helps in discussing their classical limit and in the calculations of quasi-probability distributions.
This chapter introduces the holomorphic representation of quantum mechanics, because it allows a study of the properties of boson systems both from the point of view of evolution and of quantum statistical physics (in the so-called second quantization formalism). Path integrals then take the form of integrals over trajectories in phase space in a complex parametrization. A parallel formalism, based on integration over anti-commuting type or Grassmann variables, makes it possible to handle fermions in a way quite analogous to bosons. The corresponding path integral representation of the statistical operator is then derived. The holomorphic formalism is specially well adapted to a study of the harmonic oscillator and, more generally, of perturbed harmonic oscillators. As an illustration, the formalism is applied to the Bose-Einstein condensation. This chapter also considers complex integrals and Wick's theorem, kernel of operators, general Hamiltonians, partition function, and the quantum Bose gas.