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arXiv:0907.1380

On Oscillation Theorem for Two-Component Schrodinger Equation

Vladimir I. Pupyshev, E. A. Pazyuk, A. V. Stolyarov, M. Tamanis, R. Ferber

Published Jul 8, 2009• 1 citation
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Abstract

Conventional one-dimensional oscillation theorem is found to be violated for multi-component Schrödinger equations in a general case while for two-component eigenstates coupled by the sign-constant potential operator the following statements are valid: (1) the ground state ($v = 0$) is not degenerate; and (2) the arithmetic mean of nodes $n_1$, $n_2$ for the two-component wavefunction never exceeds the ordering number $v$ of eigenstate: $(n_1 + n_2)/2\leq v$.

DOI
10.48550/arxiv.0907.1380

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Conventional one-dimensional oscillation theorem is found to be violated for multi-component Schrödinger equations in a general case while for two-component eigenstates coupled by the sign-constant potential operator the following statements are valid: (1) the ground state (v=0v = 0v=0) is not degenerate; and (2) the arithmetic mean of nodes n1n_1n1​, n2n_2n2​ for the two-component wavefunction never exceeds the ordering number vvv of eigenstate: (n1+n2)/2≤v(n_1 + n_2)/2\leq v(n1​+n2​)/2≤v.
Considering energy eigenvalues E(m1{m}_{1}m1​,m2{m}_{2}m2​) with respect to two particles whose masses are m1{m}_{1}m1​ and m2{m}_{2}m2​, respectively, one would expect that if M>m then E(M,m2{m}_{2}m2​)>E(m,m2{m}_{2}m2​), but actually the opposite can happen if one solves the Schr\"odinger equation. In other words, the derivative \ensuremath{\partial}E/\ensuremath{\partial}m1{m}_{1}m1​ can be negative. In this paper a general condition of this problem is given.
Parwani and Tabia established in 2006 that in information-theoretic nonlinear extensions of the Schrödinger equation the energy shift of an excited state is carried predominantly by the nodes of the unperturbed wavefunction, while nodeless ground states receive only subleading shifts. This paper shows that for the rotation-invariant branch of that family, regulated by the small components of the Dirac spinor as proposed by Liew and Parwani, the node contribution closes in a single expression: each node of the density contributes exactly −4πC²/ε to the shift integral, where C is the slope of the unnormalised wavefunction at the node and ε is the ratio of the reduced Compton wavelength to the characteristic length of the system. The rule contains no free parameter and covers every bound state of every smooth potential. Two applications follow. For hydrogen it reproduces in three lines the closed coefficient obtained previously by radial integration. For the harmonic oscillator it gives a finite sum over the roots of the Hermite polynomials; the first two evaluate in closed form. The oscillator spectrum ceases to be equidistant, with two parameter-free level-spacing ratios, 1.4261 and 1.7514, which separate this model from the commutator branch, where the corresponding ratios are integers. The relative shift scales as the square root of the trap frequency. Two questions left open in the companion paper are settled: the momentum entering the deformed uncertainty relation is the one a time-of-flight measurement records, and the leading constant of that relation is 4, so that the Gaussian is its minimum-uncertainty state to first order. The same constant governs the entropy-power form, which is uniformly stronger for every non-Gaussian state; the corresponding statement for the product of entropy powers is formulated as a conjecture, with the obstruction to proving it identified. Throughout, other deformations are discussed under an explicit rule: parameters of different frameworks are never compared, only predictions for one and the same measured observable. The paper claims neither the family of equations nor the node mechanism as new, and reports no evidence for a nonzero deformation. Companion paper: https://doi.org/10.5281/zenodo.22072059A working log documenting all calculations and twenty-one self-corrections is included in the files.
We consider a one-dimensional mean-field-hydrodynamic model of a two-component degenerate Fermi gas in an external trap, each component representing a spin state of the same atom. We demonstrate that the interconversion between them (linear coupling), imposed by a resonant electromagnetic wave, transforms the immiscible binary gas into a miscible state, if the coupling constant, $\ensuremath{\kappa}$, exceeds a critical value, ${\ensuremath{\kappa}}_{\mathrm{cr}}$. The effect is predicted in a variational approximation, and confirmed by numerical solutions. Unlike the recently studied model of a binary Bose-Einsten condensate with the linear coupling, the components in the immiscible phase of the binary fermion mixture never fill two separated domains with a wall between them, but rather form antilocked ($\ensuremath{\pi}$-phase-shifted) density waves. Another difference from the bosonic mixture is spontaneous breaking of symmetry between the two components in terms of the numbers of atoms in them, N1{N}_{1}N1​ and N2{N}_{2}N2​. The latter effect is characterized by the parameter $\ensuremath{\nu}\ensuremath{\equiv}({N}_{1}\ensuremath{-}{N}_{2})∕({N}_{1}+{N}_{2})$ (only N1+N2{N}_{1}+{N}_{2}N1​+N2​ is a conserved quantity), the onset of miscibility at $\ensuremath{\kappa}\ensuremath{\geqslant}{\ensuremath{\kappa}}_{\mathrm{cr}}$ meaning a transition to $\ensuremath{\nu}\ensuremath{\equiv}0$. At $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{\mathrm{cr}}$, $\ensuremath{\nu}$ features damped oscillations as a function of $\ensuremath{\kappa}$. We also briefly consider an asymmetric model, with a chemical-potential difference between the two components. The relation between the imbalance in the spin population, induced by the linear coupling, and the developing spatial structure resembles the known Larkin-Ovchinnikov-Fulde-Ferrell states in the Fermi mixture. Dynamical states, when $\ensuremath{\kappa}$ is suddenly switched from zero to a value exceeding ${\ensuremath{\kappa}}_{\mathrm{cr}}$, are considered too. In the latter case, the system features oscillatory relaxation to the mixed state.
In (1), it is argued that if the classical energy of an oscillator E = pp/2 + wwxx/2 (m=1) is written as E = (p+iwx)(p-iwx) and this in turn written using the Moyal star product { d/dp left d/dx right - d/dx left d/dp right } in terms of a parameter ( i hbar /2), one obtains: E = pp/2 + wwxx/2 + hbar w/2 i.e. a semiclassical expression for the quantum result with the ground state energy hbar w/2. In Part I, we showed that this result follows from combining a time-independent Schrodinger equation in x-space with one in p (momentum) - space. The two second order equations yield second order derivatives composed of two parts. One part returns the form pp/2m and k/2 xx and the other returns a constant which is the ground state energy if one considers the ground state p and x wavefunctions which are of the form exp(- (pp/2 + wwxx/2) /b) where b is the ground state energy.. The first part cancels with kinetic energy from the p-space equation and .5kxx from the x-space equation. Thus, this combination of two Schrodinger equations really involves three pieces: ((A)) an energy term in p,x, ((B)) a cancellation term of this energy and ((C)) a constant ground state energy term. In (2) it is stated that the Moyal star product ** has the property: Integral dx dp f(p,x) g(p,x) = Integral dx dp f(p,x) ** g(p,x) This is then used to argue that: H(p,x) ** f(p,x) = E f(p,x) ((1)). For the ground state oscillator case, only the first and second order operator terms contribute. The first order operator term is zero, while the second order term behaves in exactly the same manner as the combined x-space and p-space time-independent Schrodinger equations used in Part I. Thus the combined p and x space time-independent Schrodinger equations for the oscillator Hamiltonian pp/2 +wwxx/2 is identical to the Moyal approach.
In most textbooks on quantum mechanics, it is asserted that the bound states of a localized potential have negative energies. To stress that this view should be taken with caution, we give several examples of scalar potentials that tend to zero at infinite distance and yet possess one or several discrete positive bound states for the Schrödinger single-particle wave equation. The consequence that usual tunneling may not occur, though it should according to a superficial view, is of interesting pedagogical value.