TAILIEUCHUNG - Application of a generalised function method to the infinitely deep square well problem

The Schrodinger equation for the eigenvalues of the infinitely deep square well potential is solved within the class of generalised functions. It is found that the ground state consists of a step function like eigenfunction with the eigenvalue zero. | Turk J Math (2017) 41: 605 – 610 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Application of a generalised function method to the infinitely deep square well problem ∗ ¨ Basri UNAL Department of Physics Engineering, Faculty of Engineering, Ankara University, Ankara, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: The Schr¨ odinger equation for the eigenvalues of the infinitely deep square well potential is solved within the class of generalised functions. It is found that the ground state consists of a step function like eigenfunction with the eigenvalue zero. Key words: Eigenvalue, eigenfunction, complete set, generalised function 1. Introduction The basic problem of quantum mechanics is the solution of the eigenvalue equation L1 uλ (x) = λuλ (x) (1) 2 d 2 Here L1 is a self-adjoint operator in one dimension of the type L1 = − dx is the eigenvalue 2 +V (x) , λ = 2mE/ℏ with E the energy parameter, and m and ℏ are particle mass and Planck constant, respectively. The function V (x) is related to the potential Vp (x) by V (x) = 2mVp (x)/ℏ2 . Since equation (1) is of second order, there must be some data about the sought eigenfunction uλ (x) itself and about its derivative u′λ (x) on the system boundaries. Although the shape of V (x) for the infinitely deep square well (IDSW) is very simple, satisfaction of boundary conditions (BCs) proved to be difficult [2]. The presently known solution, what I call the old one, uses ordinary functions, satisfies the BC for uλ (x), but does not satisfy the BC for u′λ (x). The old solution of IDSW problem is given in almost every physics book [5,6] on quantum mechanics and every mathematics book [1] on partial differential equations and boundary value problems. They, in fact, repeat a wrong result about the ground state energy that was found long time ago [7,8]. The earliest solution [8]

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