8.1 The Hydrogen Atom

Can the magnitude of $L_{z}$ ever be equal to L?

Identify the physical significance of each of the quantum numbers of the hydrogen atom.

Distinguish between Bohr’s and Schrödinger’s model of the hydrogen atom. In particular, compare the energy and orbital angular momentum of the ground states.

The wave function is evaluated at rectangular coordinates ($x , y , z$) $=$ (2, 1, 1) in arbitrary units. What are the spherical coordinates of this position?

What are the possible values of m for an electron in the $n = 4$ state?

How many possible states are there for the $l = 4$ state?

The force on an electron is “negative the gradient of the potential energy function.” Use this knowledge and the equation to show that the force on the electron in a hydrogen atom is given by Coulomb’s force law.

The wave function is evaluated at spherical coordinates $\left(r , \theta , \phi\right) = \left(\sqrt{3} , 45 ° , 45 °\right) ,$ where the value of the radial coordinate is given in arbitrary units. What are the rectangular coordinates of this position?

Write an expression for the total number of states with orbital angular momentum l.

What is the probability that the 1s electron of a hydrogen atom is found outside the Bohr radius?

What is the maximum number of orbital angular momentum electron states in the $n = 2$ shell of a hydrogen atom? (Ignore electron spin.)