The solution to the t-i schr¨odinger equation for hydrogen-like atoms is of the form ψnℓmℓ (r, θ, φ) = cnℓmℓrnℓ(r)y mℓ ℓ (θ, φ) consider the first excited state, n = 2, with ℓ = mℓ = 0, so the wave function takes the form ψ200 = c200 2 − zr a0 e −zr/2a0 if the value of the constant c200 is given by c200 = 1 √ 32π z a0 3/2 for this state (n = 2, ℓ = 0, mℓ = 0) in hydrogen, find the values and evaluate at r = a0 for each of the following: (a) (b) |ψ| 2 , (c) the radial probability density p(r). give your answers in terms of a0

An alpha particle with kinetic energy 12.5 mev makes a collision with lead nucleus, but it is not "aimed" at the center of the lead nucleus, and has an initial nonzero angular momentum (with respect to the stationary lead nucleus) of magnitude l = p0 b, where p0 is the magnitude of the initial momentum of the alpha particle and b=1.40×10−12 m. (assume that the lead nucleus remains stationary and that it may be treated as a point charge. the atomic number of lead is 82. the alpha particle is a helium nucleus, with atomic number 2.) what is the distance of closest approach?