Exact solutions for the inverse problem of the time-independent Schrödinger equation

As opposed to the traditional method, we propose an alternative prescription, the “inverse method.” Here, we limit our studies to only $V(r→)$ that have at least one bound state and furthermore restrict ourselves to only finding the ground state. Asking the inverse question makes it elementary for instructors or students to construct their own solvable $V(r→)$ for the TISE. Namely, given a pdf $P(r→)$⁠, find which potential $V(r→)$ and energy E in the TISE is needed with $ψ(r→)=P(r→)$ to give the desired pdf. Our inverse method works for a wavefunction that can be chosen to be a non-negative function. Mathematical requirements for $P(r→)$ are given in Sec. II.

For beginning students, instructors can describe sufficient mathematical constraints. First, $P(r→)$ should be continuous as well as twice continuously differentiable. Since $P(r→)$ is a pdf, it must be normalized. It is permissible for $P(r→)$ to be zero on a subset of measure zero, including approaching zero at infinity, in which case at those points $f(r→)$ can be undefined.

For more advanced students, less restrictive mathematical constraints than in the previous paragraph may be described. Our inverse method works only when the ground state wavefunction $ψ(r→)$ can be chosen to be real, which is possible if the TDSE (time dependent Schrödinger equation) has time reversal invariance. For example, the inverse method will not work for a charged particle in a magnetic field. The method works if $f(r→)$ is twice piecewise differentiable, in which case the first and second derivatives need not be continuous when approaching a boundary between adjacent regions. For the piecewise differentiable case, note the energy E in Eq. (3) must be the same in all regions, which may require a constant shift added to the potential in some regions.

Exact solutions for example 1D, 2D, and 3D pdfs in real space are obtained and explained in this section. We implemented our inverse method to obtain solutions of the TISE both for some well-known pdfs and some more exotic pdfs. Additional analysis of some of these pdfs is in Ref. 3.

We also solved the inverse problem for five additional 1D distributions: Gaussian, Gumbel, Lorentzian (Cauchy), logistic, and student's t pdfs. The results are displayed in Table I. These five pdfs have the domain $x∈[−∞,+∞]$⁠. Additional analysis for each 1D pdf is in Ref. 3.

The first row in Table I is the well-known Gaussian pdf in 1D with standard deviation σ. The corresponding potential is the simple harmonic oscillator (SHO or quadratic) potential with the potential minimum at x = x₀ with $V(x0)=0$⁠. The pdf P(x) is the ground state probability distribution. The ground state energy $E0=ℏ2/4mσ2$⁠, which is more commonly written as $E0=(1/2)ℏω$⁠. The angular frequency is $ω=k/m$⁠, with $V(x)=(1/2) k(x−x0)2$ with k the spring constant of the SHO. Note the SHO relationship $σ2=⟨(x−x0)2⟩=ℏ/2mω$⁠.

Table I lists four additional common 1D pdfs. A Gumbel pdf is an extreme value type I distribution and is different for maximum or minimum extreme values.⁴ We analyze only the Gumbel pdf for maximum extreme values, leaving the Gumbel minimum extreme value pdf as an exercise for the reader. Implementing our approach for the Gumbel (maximum extreme value) pdf in 1D, we found the corresponding potential $VGbl(x)$ has a minimum at $xmin=x0−β ln(2)$⁠. Here, we have chosen $VGbl(xmin)=0$⁠, and the energy associated with the ground state is found to be $E0=3ℏ2/8mβ2$⁠. The Lorentzian (or Cauchy) pdf has the corresponding potential with $VCauchy(x→±∞)→ℏ2/2mγ2$⁠. $VCauchy(x)$ has a minimum at $xmin=x0$ with $VCauchy(xmin)=0$⁠, and it has maxima located at $xmax=x0 ± γ2$ with the value $VCauchy(xmax)=2ℏ2/3mγ2$⁠. Nevertheless, we have found the single bound state for the $VCauchy(x)$ potential, with the ground state energy $E0=ℏ2/2mγ2$⁠. The potential associated with the logistic pdf is zero at its minimum x = x₀ and achieves its maximum $ℏ2/4 ms2$ as $x→±∞$⁠. We found the ground state energies (in units where $ℏ2/4m=1$⁠) to be $1/σ2$⁠, $2/γ2$⁠, $3/2β2$⁠, and $1/2 s2$ for the Gaussian, Lorentzian, Gumbel, and logistic distributions, respectively. We also analyzed the student's t distribution, which for $ν=1$ is the Cauchy pdf with $γ=1$⁠. As $ν→∞$ the student's t pdf approaches the Gaussian pdf with $σ=1$⁠. The exact potentials V(x) and ground state energy E₀ for each of these 1D pdfs are provided in Table I where column 3 contains the associated potentials for the pdfs in column 2.

Solutions to Eq. (3) in higher dimensions are also easily found using the inverse prescription. We briefly present two pdfs in 2D and one in 3D. Additional details are included in Ref. 3.

We could have chosen P(x) and P(y) as any of the pdfs in Table I. Then, the associated $V(x,y)=V(x)+V(y)$ and the ground state energies are the sum of the two associated ground state energies in Table I. Although the pdfs in Table I all are 1D and have a single maximum in P(x), these conditions are not a requirement, as seen in Fig. 1. Very convoluted pdfs, including multimodal $P(r→)$⁠, can be constructed that with the inverse problem prescription gives a very complicated $V(r→)$⁠.

In summary, we have introduced an inverse problem methodology to solve the time-independent Schrödinger equation (TISE). The inverse method for the TISE is simple enough to be taught to students the first time they are introduced to the TISE. The introduced prescription gives a wide class of new solutions for bound states of the TISE and works in any dimension. We have illustrated the method by obtaining solutions for six 1D pdfs with the domain $x∈[−∞,+∞]$⁠, see Table I for five common 1D pdfs. Solutions for the 1D pdfs Rayleigh and Chi with the domain $x∈[0,+∞]$ as well as the Beta and Kumaraswamy pdfs with the domain $x∈[0,1]$ are documented in Ref. 5. In this paper, our inverse method was also illustrated for select 2D and 3 D pdfs.

Two generalizations can be envisioned. The inverse method prescription should work for a pdf that is an excited state of the TISE, using an analogue of the fixed-node approximation for locations where P(x) = 0. One can also generalize the inverse problem to the TDSE, which corresponds to two coupled nonlinear differential equations^6,7 as derived, for example, using the Fisher information theory.^8–11 The inverse problem then uses an Ansatz for $P(r→,t)$ as well as the action. Note the two coupled nonlinear equations derived using Fisher information are the analogue mathematically of Bohm's formulation of quantum mechanics.^12,13 Some implementation of the inverse method for the TDSE is in Ref. 5.

The project reported here was motivated by the ability to change the potential $V(r→)$ for an atomic gas Bose–Einstein condensate (BEC). Painted potentials for a BEC have been reported.¹⁴ A cloud-based BEC device with a programmable ‘painted’ $V(r→)$ was recently introduced by the company Infleqtion.¹⁵ The BEC state measured is the ground state associated with $V(r→)$⁠, motivating concentrating on the inverse method for only the ground state of the TISE. For the BEC, the underlying equation is the non-linear Schrödinger equation, namely, the Gross–Pitaevskii equation (GPeq), which takes into account atomic interactions.¹⁶ In fact, the inverse method was introduced to provide a solution for the GPeq.^5,17–19 Exact solutions for the GPeq obtained with the inverse method can also be used to validate numerical codes, such as the GPeq multigrid renormalization method.²⁰ The inverse method will also work in other cases where the GPeq may be valid, for example for including ψ-dark-matter²¹ in large-scale structure investigations of the universe.

The authors thank Ms. A. Tingle, Dr. D. Dahl, and Profs. H. de Raedt, G. Rupak, and Y. Koshka for useful discussions. M.A.N. gratefully acknowledges partial support from Grant No. DOE DE-SC0024286.

The authors have no conflicts to disclose.