GATE 2022 PH – Question 48
A mass m is confined to a spherical infinite well of radius a. Which statements about the ground energy E0 and radial wavefunction R(r) are correct?
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Correct answer: (A) $E_0=\hbar^2\pi^2/(2ma^2)$; (B) $-\hbar^2[\partial_r(r^2\partial_rR)]/(2mr^2)=E_0R$; (D) $R(r)=\sin(\pi r/a)/r$ up to normalization
Explanation
A particle in a spherical infinite well of radius $a$. The ground state has $l=0$ (no centrifugal barrier).
**Radial equation for $l=0$.** In spherical coordinates
$$-\frac{\hbar^2}{2m}\frac1{r^2}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)=E_0R .\quad(\text{statement B is correct})$$
**Solution.** Put $R(r)=u(r)/r$. Then $\dfrac1{r^2}\dfrac{d}{dr}\left(r^2R^\prime\right)=\dfrac{u^{\prime\prime}}{r}$, and the equation becomes $-\dfrac{\hbar^2}{2m}u^{\prime\prime}=E_0u$.
With the boundary conditions $u(0)=0$ (so that $R$ is finite) and $u(a)=0$:
$$u=\sin\frac{\pi r}{a},\qquad R(r)=\frac{\sin(\pi r/a)}{r}\quad(\text{statement D is correct}),$$
$$E_0=\frac{\hbar^2\pi^2}{2ma^2}\quad(\text{statement A is correct}).$$
Statement C, $-\dfrac{\hbar^2}{2mr^2}R^{\prime\prime}=E_0R$, is not the correct radial equation (it omits the first-derivative term and the factor $r^2$ in the right place).
Answer **A, B and D**.