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The particle in a spherical box has V = 0 for r ≤ b and V = ∞ for r > b. For this system:

(a) Explain why ψ = R(r)f(θ, ϕ), where R(r) satisfies (6.17). What is the function f(θ, ϕ)?

(b) Solve (6.17) for R® for the l = 0 states. The substitution R(r) = g(r)/r reduces (6.17) to an easily solved equation. Use the boundary condition that c is finite at r = 0 [see the discussion after Eq. (6.83)] and use a second boundary condition. Show that for the l = 0 states, c = N[sin(nπr/b)4] / r and E = n2h2/8mb2 with n = 1, 2, 3, . . . . (For l ≠ 0, the energy-level formula is more complicated.)

(a) Explain why ψ = R(r)f(θ, ϕ), where R(r) satisfies (6.17). What is the function f(θ, ϕ)?

(b) Solve (6.17) for R® for the l = 0 states. The substitution R(r) = g(r)/r reduces (6.17) to an easily solved equation. Use the boundary condition that c is finite at r = 0 [see the discussion after Eq. (6.83)] and use a second boundary condition. Show that for the l = 0 states, c = N[sin(nπr/b)4] / r and E = n2h2/8mb2 with n = 1, 2, 3, . . . . (For l ≠ 0, the energy-level formula is more complicated.)

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