An electron in a quantum dot is modelled as an isotropic three-dimensional harmonic…

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Central potentials and canonical three-dimensional systems · Central potentials and canonical three-dimensional systems

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch The Schrödinger equation in three dimensions and central potentials

Problem

An electron in a quantum dot is modelled as an isotropic three-dimensional harmonic oscillator, with potential energy V(x,y,z)=12meω2(x2+y2+z2)V(x,y,z) = \tfrac12 m_e\omega^2(x^2+y^2+z^2) and ℏω=0.500\hbar\omega = 0.500 eV. The one-dimensional oscillator of Chapter 7 has levels (n+12)ℏω(n + \tfrac12)\hbar\omega with n=0,1,2,…n = 0, 1, 2, \dots, and the zero of energy is at the bottom of the potential. ((a)) By writing ψ=X(x)Y(y)Z(z)\psi = X(x)Y(y)Z(z) in the three-dimensional time-independent Schrödinger equation, show that the equation separates into three one-dimensional oscillators and hence that E=(N+32)ℏωE = \left(N + \tfrac32\right)\hbar\omega with N=nx+ny+nzN = n_x+n_y+n_z. ((b)) How many states does the shell N=4N = 4 contain? Obtain the number by counting the triples (nx,ny,nz)(n_x, n_y, n_z). ((c)) Give the energy of that shell in eV.

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