Question

Use the variational principle to obtain an upper limit to ground state energy of a particle in one dimensional box.

Answer 1

$= <H>=<\psi/H/\psi>=\int_{-\infty}^{\infty}\psi*(H\psi)dz$

$H=\frac{eB_x}{m}Sy=\frac{e\bar{h}By}{m}[_1^0 \space _0^{-i}] /\psi>=[^{cos\theta}_{sin\theta}]$

So, $<\psi/=[cos\theta sin\theta]$

$=<H>=<\psi/H/\psi/>$

$=[cos \theta sin \theta][_1^0 \space_0^{-i}][_{sin\theta}^{cos \theta}] \frac{ehBy}{m}$

$=[-icos\theta sin\theta+sin \theta cos \theta]\frac{ehBy}{m}=0$

$\therefore <H>= <\psi/\mu/\psi>=0$