Using the method outlined in Section 3.4, write a computer program to solve the Schrödinger wave equation for the first four eigenvalues and eigenstates of an electron with effective mass m*e = 0.07 × m0 confined to a parabolic potential well in such a way that V(x) = ((x − L/2)²/(L/2)²) eV and L =

Question

Using the method outlined in Section 3.4, write a computer program to solve the Schrödinger wave equation for the first four eigenvalues and eigenstates of an electron with effective mass [latex]m_{\mathrm{e}}^{*} =0.07 imes m_{0}[/latex] confined to a parabolic potential well in such a way that [latex]V(x)=((x-L/2)^{2}/(L/2)^{2})\,\mathrm{eV\,and}\ L= 100\,\mathrm{nm}.[/latex]

Accepted Answer

We would like to use the method outlined in Section 3.4 to numerically solve the one dimensional Schrödinger wave equation for the first four eigenvalues and eigenstates of an electron with effective mass [latex] m_{ \mathrm{e}}^{*} =0.07 imes m_{0}[/latex] confined to a parabolic potential well in such a way that [latex]V(x)=((x-L/2)^{2}/(L/2)^{2}) \, \mathrm{eV\,and}\ L= 100\,\mathrm{nm}.[/latex]

Because we use a nontransmitting boundary condition, the value of L must be large enough to ensure that the wave function is approximately zero at the boundaries [latex]x_{0}=0[/latex] and [latex]x_{N}=L.[/latex] The number of sample points N should also have a large enough value to ensure that the wave function does not vary significantly between adjacent discretization points.

The main computer program calls solve_schM, which was used in the solution of Exercise 3.7.

In this exercise, the first four energy eigenvalues are [latex]E_{0} = 0.0147~\mathrm{eV},~E_{1}=[/latex][latex]0.0443~\mathrm{eV} , ~E_{2}=0.0738~\mathrm{eV},~\mathrm{and}~E_{3}=0.1032~\mathrm{eV}.[/latex] As expected for a harmonic oscillator, the separation in energy between adjacent states is independent of eigenvalue, in this case [latex]\hbar\omega=0.0295\;\mathrm{eV}.[/latex] The eigenfunctions generated by the program and plotted in the figure below are not normalized.

Listing of MATLAB program for Exercise 6.10

%simple harmonic oscillator clear; clf; length = 100; %length of well (nm) n=400; %number of sample points x=-length/2:length/n:length/2; %position of sample points in potential mass=0.07; %effective electron mass num sol=4; %number of solutions sought v0=1; %potential scale (eV) for i=1:n+1 %potential (eV) v(i)=v0*(x(i)ˆ2)/((length/2)ˆ2); end [energy,phi]=solve schM(length,n,v,mass,num sol); for i=1:num sol sprintf(['eigenfunction (',num2str(i),') =',num2str(energy(i)),'eV']) %energy eigenvalues end figure(1); plot(x,v,'b');xlabel('Distance (nm)'),ylabel('Potential energy, (eV)'); title(['m* =',num2str(mass),'m0, Length =',num2str(length),'nm']); s=char('y','k','r','g','b','m','c'); %plot curves in different colors figure(2); for i=1:num sol j=1+mod(i,7); plot(x,phi(:,i),s(j)); %plot eigenfunctions hold on; end xlabel('Distance (nm)'),ylabel('Wave function'); title(['m* =',num2str(mass),'m0, Length =',num2str(length),'nm']); hold off;

Question 6.10: Using the method outlined in Section 3.4, write a computer p......

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