In Section 10.2, we discussed two-lobed px, py, and pz states and four-lobed hybrid sp3 st...

In Section 10.2, we discussed two-lobed px, py, and pz states and four-lobed hybrid sp3 states. Another kind of hybrid state that sticks out in just one direction is the sp, formed from a single p state and an s state. Consider an arbitrary combination of the 2s state with the 2pz, state. Let us represent this by cos T ψ2,0,0 + sin T ψ2,1,0. (The trig factors ensure normalization, In carrying out the integral, cross terms integrate to 0, leaving

(a) Calculate the probability that an electron in such a state would be in the +z-hemisphere. (Note: Here, the cross terms do not integrate to 0.)

(b) What value of T leads to the maximum probability, what is the value of the maximum probability, and what is the corresponding ratio of ψ2,0,0 to ψ2,1,0?

(c) Using a computer, make a density (shading) plot of the probability density—density versus r and 0- for the T-value found in part (b).