Normalize the wavefunction for the = 0 finite spherical well. In terms of and n, derive an expression for the probability of finding the particle outside the well. Evaluate your result numerically for the case discussed in Example 7.1. EXAMPLE 7.1 How many energy states are available to an alpha-particle trapped in a finite spherical well of depth 50 MeV and radius 10-14 m? Assume l = 0. From Equation (13), or, (2n-1)²< 32 μανο h² On substituting the appropriate numerical values we find (2n-1)² < 32 (6.646 x 10-27 kg) (10-14 m) 2 (8.010 × 10-12) (6.626 x 10-34 sec)2 (2n-1)² <388. From this we find n < 10.4: the alpha-particle has only ten bound states of l=0 available. We can determine the energy of the lowest-energy bound state for this system as follows. From Equation (12) we have 2μανο = 956.6. Using this result to eliminate n in Equation (9) provides a constraint on §: cot+V956.6 - ² = 0. The lowest-valued root of this equation is = 3.04305, which gives, via Equation (10) and the definition of k₁ in Equation (2), E = 0.48 MeV. 2 μαζ