Here we explore further the idea introduced in Impact I8.1 that quantum mechanical effects need to be invoked in the description of the electronic properties of metallic nanocrystals, here modelled as three-dimensional boxes. 

(a) Set up the Schrödinger equation for a particle of mass m in a three-dimensional rectangular box with sides L₁, L₂, and L₃. Show that the Schrödinger equation is separable. 

(b) Show that the wavefunction and the energy are defined by three quantum numbers.

(c) Specialize the result from part (b) to an electron moving in a cubic box of side L=5nm and draw an energy diagram resembling Fig. 8A.2 and showing the first 15 energy levels. Note that each energy level may consist of degenerate energy states.

(d) Compare the energy level diagram from part (c) with the energy level diagram for an electron in a one-dimensional box of length L=5nm. Are the energy levels more or less sparsely distributed in the cubic box than in the one-dimensional box?

Data in Fig. 8A.2

Transcribed Image Text:

Energy, E/E₁; E₁ = h²/8mL² 100 81 64 49 36 25 16 9 0 41 n -10 Classically allowed energies 9 8 7 CO 6 5 NWA