display2d:false$

Pmat:matrix([cos(q*a),sin(q*a)/q],[-q*sin(q*a),cos(q*a)])$
Dmat:matrix([1,0],[2*m*H/hbar^2,1])$
Mmat:Dmat.Pmat$
Pdim:m*H*a/hbar^2$
residual_transfer_trace:trigreduce(ratsimp((Mmat[1,1]+Mmat[2,2])/2-(cos(q*a)+Pdim*sin(q*a)/(q*a))))$
print("residual_kronig_penney_trace =",residual_transfer_trace)$

detgap:(E0-E)^2-U^2$
residual_gap_plus:ratsimp(ev(detgap,E=E0+U))$
residual_gap_minus:ratsimp(ev(detgap,E=E0-U))$
print("residual_gap_upper_root =",residual_gap_plus)$
print("residual_gap_lower_root =",residual_gap_minus)$

Eband:Eedge+hbar^2*(k-k0)^2/(2*mstar)$
residual_effective_mass:ratsimp(diff(Eband,k,2)/hbar^2-1/mstar)$
print("residual_effective_mass =",residual_effective_mass)$