Question

8. Consider one electron in a 1D box of side L. Its wavefunction is given by V3 V3 2V3i where ф1(x), фг(x), and фз(x) are the first 3 eigenfunctions of the Hamiltonian, A, of a particle in a 1D box, h2 d2 a) Is Ψ(x) normalized? If it is not normalized it, normalize it! b) Is ų (x) an eigenfunction of A? If it is an eigenfunction, what is the eigenvalue?

Answer 1

a) If the wave function is normalized then the square of the absolute value of the coefficients should add up to 1

$\left(\frac{\sqrt{3}}{4} \right )^2+\left(\frac{\sqrt{3}}{2\sqrt{2}} \right )^2+\left(\frac{2+\sqrt{3}i}{4} \right )\left(\frac{2-\sqrt{3}i}{4} \right )=\frac{3}{16}+\frac{3}{8}+\frac{7}{16}=1$

Hence the wave function is normalized.

b)

Note that

$\hat H\phi_n(x)=E_n\phi_n(x)$

The wave function is not an eigenfunction of the Hamiltonian because

$\hat H\Psi(x)=\left(\frac{\sqrt{3}}{4} \right )E_1\phi_1(x)+\left(\frac{\sqrt{3}}{2\sqrt{2}} \right )E_2\phi_2(x)+\left(\frac{2+\sqrt{3}i}{4} \right )E_3 \phi_3(x)$

This cannot be written as

$\hat H\Psi(x)=E\Psi(x)$