Question

Let’s say that at t=0 the hydrogen atom is in the state that is described by the following wave function:

Ψ(r,0) = N{ψ211(r,θ,φ) + 1/4ψ32-1(r,θ,φ)+1/4ψ320(r,θ,φ)}

where ψ are the the normalized eigenfunctions of the hydrogen atom.

For a random time t calculate

1. The normalized wave function Ψ(r,t)

2. The expected value of the z component of the angular momentum, <Lz>

3. The expected value of the square of the angular momentum, <L^2

GIVE ANSWER ONLY IN WORD FORMAT TYPED ANSWER.

Answer 1

1)

---------------1

The normalising condition is,

------------------2 This the normalise wavefunction

Now time evolution of the wavefunction is given by,

  --------------------------3 Energy depends only on n.

2) The angular momentum Lz eigenvalue is given as,

and   

3) Now