Question

Schrödinger equation for a particle of mass m is given by,d²Ψ(x)/dx² + 2m/ℏ² (E−U)Ψ(x)=0 Examine the equation to derive the expression of Eigen function Ψ(x) and Eigen energy E(x) for infinite potential well. Interpret why energy quantization concept is contradict with classical physics.

Answer 1

The eigenfunctions and eigenenergies for a particle in an infinite potential well are derived from the Schrödinger equation, resulting in quantized energy levels which illustrate a key difference between quantum and classical physics.

Step-by-step explanation:

The Schrödinger equation for a particle in an infinite potential well can be solved to find the eigenfunctions Ψ(x) and eigenenergies E. In an infinite potential well, the particle is confined to a region with zero potential energy inside, and infinite potential energy outside. The boundary conditions require that Ψ(x) be zero at the walls of the well. Consequently, only sinusoidal wave functions that have nodes at the walls are solutions. These wave functions are given by Ψ(x) = A sin(nπx/a), where A is a normalization constant, n is a positive integer, and a is the width of the well. The corresponding energy eigenvalues are quantized and given by E_n = (n^2π^2ĩ^2)/(2ma^2), with n = 1, 2, 3, ... This is in contrast to classical physics where a particle can have any energy, highlighting one of the fundamental differences between classical and quantum mechanics. The quantization of energy arises from the boundary conditions imposed by the infinite potential well which allow only certain standing wave solutions.