Ground State of a Quantum Particle in a Potential Field
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D MATTER
Ground State of a Quantum Particle in a Potential Field A. M. Dyugaeva and P. D. Grigorieva, b, c, * a
Landau Institute for Theoretical Physics, Russian Academy of Sciences, Chernogolovka, Moscow region, 142432 Russia
b National c
University of Science and Technology MISIS, Moscow, 119049 Russia
Lebedev Physical Institute, Russian Academy of Sciences, Moscow, 119991 Russia *e-mail: [email protected] Received April 20, 2020; revised June 1, 2020; accepted June 1, 2020
A solution of the Schrödinger equation for the ground state of a particle in a potential field is analyzed. Since the wavefunctions of the ground state are nodeless, potentials of various kinds can be unambiguously determined. It turns out that the ground state corresponds to zero energy for a wide class of model potentials. Moreover, the zero level can be a single one at the boundary of the continuous spectrum. Crater-like potentials monotonically dependent on coordinates in one-, two-, and three-dimensional cases are studied. Instanton-type potentials with two local minima are of interest in the one-dimensional case. For the Coulomb potential, the energy of the ground state is stable with respect to both long- and short-range screening of this potential. Two-soliton solutions of the nonlinear Schrödinger equation are found. It is demonstrated that the proposed version of the inverse scattering transform is efficient in the analysis of solutions of differential equations.
DOI: 10.1134/S002136402014009X
1. In some cases, the form of the scattering potential for a quantum particle can be reconstructed using the experimentally found scattering phases [1, 2]. In this work, we show that, for a particle having at least one bound state in a potential field, it is possible to uniquely determine this potential by specifying a model wavefunction Ψ0 of the ground state. This statement is based on the oscillation theorem [3, 4], according to which the function Ψ0 is nodeless, and this makes it possible to write the Schrödinger equation for Ψ0 in the form
ΔΨ 0 V (r) − E0 = . 2m Ψ 0 2
Here, v (r ) and ε0 are the dimensionless potential and energy of the ground state, respectively; 2 2 (3) V (r ) = 2 v (r ); E0 = 2 ε0; 2ma0 2ma0 and primes denote differentiation with respect to the dimensionless variable r∗ = r /a0 . Everywhere below, we omit asterisk at r. The equation for Ψ 0 represented in form (2) shows that, for finding v (r ) and ε0 , it is sufficient to specify the logarithmic derivative ϕ of the ground state wavefunction rather than this wavefunction itself
(1)
v (r ) − ε0 = ϕ2 + ϕ' +
According to Eq. (1), the model function Ψ 0 determines the model potential V (r) and the energy of the ground state E0 . For the spherically symmetric potential V = V (r ), it is possible to specify the characteristic scale a0 such that Eq. (1) can be represented in the dimensionless form [5]
Ψ' Ψ' Ψ' v (r ) − ε0 = 0 + 0 + 2 0 1 . Ψ0 Ψ0 Ψ0 r
(2)
ϕ=
Ψ 0' . Ψ0
(4)
Two of the functions Ψ 0 having a physical meaning
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