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We solve the Thouless-Anderson-Palmer (TAP) variational principle associatedto the spherical pure $p$-spin mean field spin glass Hamiltonian and present adetailed phase diagram.  In the high temperature phase the maximum of variational principle is theannealed free energy of the model. In the low temperature phase the maximum,for which we give a formula, is strictly smaller.  The high temperature phase consists of three subphases. (1) In the firstphase $m=0$ is the unique relevant TAP maximizer. (2) In the second phase thereare exponentially many TAP maximizers, but $m=0$ remains dominant. (3) In thethird phase, after the so called dynamic phase transition, $m=0$ is no longer arelevant TAP maximizer, and exponentially many non-zero relevant TAP solutionsadd up to give the annealed free energy.  Finally in the low temperature phase a subexponential number of TAPmaximizers of near-maximal TAP energy dominate.

Phase diagram for the tap energy of the $p$-spin spherical mean field spin glass model