A particle of mass \( m \) is confined to move in the region between two concentric spherical shells with radii \( r = a \) and \( r = b \). The particle experiences no potential within this region, except for the constraints imposed by the spherical shells. The ground state energy and the normalized wave function of the particle is
1
\( E = \frac{\hbar^2 \pi^2}{2m(b - a)^2} \) and \( \psi(r) = \sqrt{\frac{1}{4\pi}} \sqrt{\frac{2}{b - a}} \frac{\sin\left[\frac{\pi(r - a)}{b - a}\right]}{r} \)
2
\( E = \frac{\hbar^2 \pi^2}{4m(b - a)^2} \) and \( \psi(r) = \sqrt{\frac{1}{4\pi}} \sqrt{\frac{2}{b - a}} \frac{\sin\left[\frac{\pi(r - a)}{b - a}\right]}{r} \)
3
\( E = \frac{\hbar^2 \pi^2}{3m(b - a)^2} \) and \( \psi(r) = \sqrt{\frac{1}{4\pi}} \sqrt{\frac{2}{b - a}} \frac{\sin\left[\frac{\pi(r - a)}{b - a}\right]}{r} \)
4
\( E = \frac{\hbar^2 \pi^2}{m(b - a)^2} \) and \( \psi(r) = \sqrt{\frac{1}{4\pi}} \sqrt{\frac{2}{b - a}} \frac{\sin\left[\frac{\pi(r - a)}{b - a}\right]}{r} \)