Texts: Hydrogen atom problem, only part e.
Problem 1: The Hydrogen Atom. 20 points
a) For the 1s orbital of the hydrogen atom, the quantum mechanical solution is:
ψ1s = (1/πa0^3)^(1/2) * e^(-r/a0)
where r = √(x^2 + y^2 + z^2) and a0 is the Bohr radius. Write down the expression for the probability of the electron in the range [x, x+dx], [y, y+dy], and [z, z+dz]. In other words, P = ?drdydz. (3 points)
b) Perform a change of variables and convert the probability to the spherical coordinate system. In other words, P = drdθdϕ. Hint: dV = dxdydz and dV = r^2sinθdrdθdϕ. (3 points)
c) Prove that the total probability, i.e., the integration over space, is normalized to 1. (3 points) (Hint: the integration in spherical coordinates is ∫∫∫ r^2sinθdrdθdϕ)
d) Based on the expression in (b), find the radius r that corresponds to the highest probability, i.e., the most likely r where the electron is located. How is it related to the Bohr radius a0? (3 points)
e) Prove that ψ1s and ψ2s are orthogonal, and ψ2s and ψ2p are orthogonal. (8 points)
4(2T)1/
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