Question

Question 4: (6 points) How many quantum numbers do you have for the particle in a box model we derived in class? How many quantum numbers do you have for a Hydrogen atom? If the two are different, explain why this is so. Note: You do not need to account for spin in this problem.

Answer 1

1)Particle in a Box. A total of four quantum numbers are used to describe completely the movement and trajectories of each electron within an atom

2) The hydrogen atom solution to the Schrodinger equation produces three quantum numbers which can be seen to arise naturally from geometrical constraints on the wavefunction.

3)the wavefunction takes the form

Y(r,0,0) = R(r)P(0)F(0)

which gives three equations. The equation for each of the three variables gives rise to a quantum number and the quantized energy states of the atom can be specified in terms of these quantum numbers

3)In the solution to the Schrodinger equation for the hydrogen atom, three quantum numbers arise from the space geometry of the solution and a fourth arises from electron spin. No two electrons can have an identical set of quantum numbers according to the Pauli exclusion principle , so the quantum numbers set limits on the number of electrons which can occupy a given state and therefore give insight into the building up of the periodic table of the elements.

N -- The three spherical coordinates are associated with the three spatial quantum numbers.

R(r)   Principal quantum number n=1,2,3.... P(θ) Orbital quantum number (= 0,1,2,...n-1 F(φ) Magnetic quantum number m, =-1,-1+1,...0...1-1,1 or 21 +1 values Spin quantum number ב- += m