It is shown that it is possible to establish sum rules that must be satisfied at the nodes and extrema of the eigenstates of confining potentials which are functions of a single variable. At any boundstate energy the Schrödinger equation has two linearly independent solutions one of which is normalizable while the other is not. In the domain after the last node of a boundstate eigenfunction the unnormalizable linearly independent solution has a simple form which enables the construction of functions analogous to Green's functions that lead to certain sum rules. One set of sum rules gives conditions that must be satisfied at the nodes and extrema of the boundstate eigenfunctions of confining potentials. Another sum rule establishes a relation between an integral involving an eigenfunction in the domain after the last node and a sum involving all the eigenvalues and eigenstates. Such sum rules may be useful in the study of properties of confining potentials. The exactly solvable cases of the particle in a box and the simple harmonic oscillator are used to illustrate the procedure. The relations between one of the sum rules and two-particle densities and a construction based on supersymmetric quantum mechanics are discussed.
We study the effect of a
δ distribution potential placed at
x0≥0 and multiplied by a parameter
α on a quantum mechanical particle in an infinite square well over the segment
[−2L,2L]. We obtain the limit of the eigenfunctions of the time independent Schrödinger equation as
α↗+∞ and as
α↘−∞. We see how each solution of the Schrödinger equation corresponding to
α=0 changes as
α runs through the real line. When
x0 is a rational multiple of
L, there exist solutions of the Schrödinger equation which vanish at
x0 and are unaffected by the value of
α. We show that each one of these has an energy that coincides with the energy of a certain limiting eigenfunction obtained by taking
∣α∣→∞. The expectation value of the position of a particle with wave function equal to the limiting eigenfunction is
x0.
This project enumerates methods utilizing discretized centered-difference approximations on the second order differential equation for quantum particles known as Schrodinger’s Equation. An eigenvalue-eigenfunction scheme is developed to sieve for valid solutions to The Time Independent Schrodinger Equation. Additionally the Crank-Nicolson method is applied to the Time Dependent Schrodinger Equation to describe wavefunction (eigenfunction) time evolution. The validity of these methods is discussed with applications to several fundamental pedagogical introductory quantum mechanic systems.