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The Schrödinger equation has been solved in closed form for about 20 quantum-mechanical problems. This Demonstration describes one such example published some time ago. A particle moves in a potential that is zero everywhere except on a spherical bubble of radius ... , drawn as a red circle in the contour plots. This result has been applied to model the buckminsterfullerene molecule ... and also to approximate the interatomic potential in the helium van der Waals dimer ... . The relevant Schrödinger equation is given by ... , in units with ... , ... and ... in bohrs, and ... in hartrees. For ... , the equation has separable continuum solutions ... , where the ... are spherical harmonics. The radial function has the form ... for ... and ... ... for ... . Here ... and ... are spherical Bessel functions and the ... are phase shifts. For each value of ... , a single bound state will exist, provided that ... . The bound-state radial function is ... , where ... and ... are the greater and lesser of ... and ... , and ... is a Hankel function. The energy is given by ... , with ... determined by the transcendental equation ... . Both the bound and continuum wavefunctions are continuous at ... but have discontinuous first derivatives. The produces a deltafunction in the second derivative. This Demonstration shows plots of the radial functions ... ... and a cross section of the density plots of ... for ... . The wavefunction is positive in the blue regions and negative in the white regions. Be cautioned that the density plots might take some time to complete.

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      EUN,LOM,LRE4,work-cmr-id:398421,http://demonstrations.wolfram.com:http://demonstrations.wolfram.com/SchroedingerEquationForADiracBubblePotential/,ilox,learning resource exchange,LRE metadata application profile,LRE

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