The (time-independent) momentum space wave function in three dimensions is defined by the natural generalization of Equation 3.54: φ(p) ≡ 1 (2π)3/2 e−i(p·r)/ψ (r) d3r. (4.223) (a) Find the momentum space wave function for the ground state of hydrogen (Equation 4.80). Hint: Use spherical coordinates, setting the polar axis along the direction of p. Do the θ integral first. φ(p) = 1 π 2a 3/2 1 + (ap 1 /)22 . (4.224)
(b) Check that φ(p) is normalized.
(c) Use φ(p) to calculate p2 , in the ground state of hydrogen.
(d) What is the expectation value of the kinetic energy in this state? Express your answer as a multiple of E1, and check that it is consistent with the virial theorem