Prove that the radial probability density peaks at for the state of hydrogen.
has its maximum peak at
Radial probability densityFor the state where role="math" localid="1648542807295" is
When is maximum, then
At has either a maximum or a minimum. To determine whether it is a maximum or minimum. we find
has its maximum peak at .
II In general, an atom can have both orbital angular momentum and spin angular momentum. The total angular momentum is defined to be . The total angular momentum is quantized in the same way as and . That is, , where is the total angular momentum quantum number. The z- component of is , where goes in integer steps from to . Consider a hydrogen atom in a state, with .
a. role="math" localid="1648549390842" has three possible values and has two. List all possible combinations of and . For each, compute and determine the quantum number . Put your results in a table.
b. The number of values of that you found in part a is too many to go with a single value of . But you should be able to divide the values of into two groups that correspond to two values of . What are the allowed values of ? Explain. In a classical atom, there would be no restrictions on how the two angular momenta and can combine. Quantum mechanics is different. You've now shown that there are only two allowed ways to add these two angular momenta.
94% of StudySmarter users get better grades.Sign up for free