Question

An electron is confined by some potential energy well centered about the origin, and is represented by the wave function ψ(x) = Axe−x2/L2, where L = 4.48 nm. The electron's total energy is zero. (a) What is the potential energy (in eV) of the electron at x = 0? eV (b) What is the smallest value of x (in nm) for which the potential energy is zero? nm

Answer 1

An electron is confined by some potential energy well centered about the origin.

It is represented by a wave function which given as :

(x) = A x exp (x² / L²)

Time independent Schrodinger equation for the wavefunction (x) of a particle of mass 'm' in potential energy U(x) which is given by -

- [(hbar)² / 2m] [d²(x) / dx²] + U(x) .(x) = E (x)

When a particle with zero energy has wavefunction (x), then the potential energy which will be given as -

U(x) = [2 (hbar)² / m L⁴] [x² - (3 L² / 2)]

(a) The potential energy of an electron at x = 0 which will be given as -

U(0) = [2 (hbar)² / m L⁴] [(0)² - (3 L² / 2)]

U = - 3 (hbar)² / m L²

where, m = mass of an electron = 9.11 x 10^-31 kg

L = length = 4.48 x 10^-9 m

hbar = constant value = 1.05 x 10^-34 J.s

then, we get

U = - [3 (1.05 x 10^-34 J.s)²] / [(9.11 x 10^-31 kg) (4.48 x 10^-9 m)²]

U = - [(3.3075 x 10^-68 J².s²) / (1.82 x 10^-47 kg.m²)]

U = - 1.81 x 10^-21 J

converting J into eV :

U = - [(1.81 x 10^-21 J)[(6.241509 x 10¹⁸ eV/J)]

U = - 0.0113 eV