This problem deals with a particle of unit mass in a finite spherical potential well of radius 2 a o and well height 2 E h . The trial wave function is given below. \[ \psi (r, \beta ) := ( \frac{2 \beta}{ \pi})^{ \frac{3}{4}} exp(- \beta r^2) \nonumber \] \[ T = - \frac{1}{2r} \frac{d^2}{dr^2} (r \blacksquare ) \nonumber \] \( V(r) := if [(r \leq 2), 0, 2]\) a. Demonstrate that the wave function is normalized. \[ \int_{0}^{ \infty} \psi (r, \beta )^2 4 \pi r^2 dr |_{simplify}^{assume,~ \beta > 0} \rightarrow 1 \nonumber \] b. Evaluate the variational integral. \[ E( \beta ) := \int_{0}^{ \infty} \psi (r, \beta ) [- \frac{1}{2r} \frac{d^2}{dr^2} (r \psi (r, \beta ))] 4 \pi r^2 dr ... |_{simplify}^{assume,~ \beta > 0} + \int_{2}^{ \infty} 2 \psi (r, \beta )^2 4 \pi r^2 dr \nonumber \] \[ E( \beta ) := \frac{1}{2} \frac{3 \pi^{ \frac{1}{2}} \beta + 4 \pi ^{ \frac{1}{2}} + 16 exp(-8 \beta) 2^{ \frac{1}{2}} \beta^{ \frac{1}{2}}-4 \pi ^{\frac{1}{2}} erf((2) 2^{ \frac{1}{2}} \beta^{ \frac{1}{2}}) }{ \pi ^{ \frac{1}{2}}} \nonumber \] c. Minimize the energy with respect to the variational parameter \( \beta\). \( \beta\) := 5 \( \beta\) := Minimize (E, \beta ) \( \beta\) = 0.381 \( E ( \beta ) = 0.786\) d. Calculate the average value of r. \( \int_{0}^{ \infty} r \psi (r, \beta )^2 4 \pi r^2 dr = 1.293\) e. Calculate the kinetic and potential energy. Potential energy: \( \int_{2}^{ \infty} r \psi (r, \beta )^2 4 \pi r^2 dr = 0.215\) Kinetic energy: \( E( \beta ) - 0.215 = 0.571\) f. Calculate the probability that the particle is in the barrier. \( 1 - \int_{0}^{2} \psi (r, \beta )^2 4 \pi r^2 dr = 0.107\) g. Plot the wavefunction on the same graph as the potential energy.