The wave functions for the three states with the dot plots shown in figure, which have n = 2, ? = 1, and m? = 0, +1, and ? 1, are in which the subscripts on ?(r, ?) give the values of the quantum numbers n, ?, m_? and the angles ? and ? are defined in figure. Note that the flrst wave function is real but the others, which involve the imaginary number i, are complex. Find the radial probability density P(r) for?

(a) ?₂₁₀ and?

(b) ?₂₁₊₁ (same as for ?_21-1).?

(c) Show that each P(r) is consistent with the corresponding dot plot in figure.?

(d) Add the radial probability densities for ?₂₁₀, and ?_21-1 and then show that the sum is spherically symmetric, depending only on r.?

Wzso(r. 0) = (1/4N2))(a 2)(rla)e cos 0. a (r, 0) = (1/8V7)(a (ria)e-a(sin 6)e i, V-1(r. 0) = (18V7)(a-)(r/a)e(sin 0)e-, COS %3D