a. Calculate and graph the hydrogen radial wave function over the interval .
b. Determine the value of (in terms of ) for which is a maximum.
c. Example and Figure showed that the radial probability density for the state is a maximum at . Explain why this differs from your answer to part .
The expression determined is and At , the is maximum
We are given
Then we have the following plot
We are given
is maximum when
At , the is maximum
Probability density depends on the total wave function therefore we will not get the maximum value of the probability density at the same point where we get the maximum radial wave function.
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.
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