Chapter 7: Q46E (page 281)

In Table 7.5, the pattern that develops with increasing n suggests that the number of different sets of $(l, m_{l})$ values for a given energy level n is $n^{2}$ . Prove this mathematically by summing the allowed values of $m_{l}$ for a given $l$ over the allowed values of $l$ for a given n.

Short Answer

Expert verified

The number of different sets of $(l, m_{l})$ values for a given energy level n is $n^{2}$ .

Step by step solution

Given data

Energy level = n.

Approach to find the final answer

For each n that is the principal quantum number, l, which is the azimuthal quantum number, can be from 0 to n – 1; for each n, there are 2l+1 allowed values of $m_{l}$ .

To prove the number of different sets of l,ml values for a given energy level n is n2

Summing all the allowed values of $m_{l}$ for a given l over the allowed values of l for a given n as:

$\sum_{0}^{n - 1} 2 l + 1 = 2 (\frac{0 + n - 1}{2}) n + n = (n - 1) n + n = n^{2} - n + n = n^{2}$

Thus, the number of different sets of $(l, m_{l})$ values for a given energy level n is $n^{2}$ .

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In Table 7.5, the pattern that develops with increasing n suggests that the number of different sets of $(l, m_{l})$ values for a given energy level n is $n^{2}$ . Prove this mathematically by summing the allowed values of $m_{l}$ for a given $l$ over the allowed values of $l$ for a given n.

Short Answer

Step by step solution

Given data

Approach to find the final answer

To prove the number of different sets of l,ml values for a given energy level n is n2

One App. One Place for Learning.

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In Table 7.5, the pattern that develops with increasing n suggests that the number of different sets ofl,mlvalues for a given energy level n isn2. Prove this mathematically by summing the allowed values ofmlfor a givenlover the allowed values oflfor a given n.

Short Answer

Step by step solution

Given data

Approach to find the final answer

To prove the number of different sets of l,ml values for a given energy level n is n2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electric Charge Field and Potential

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In Table 7.5, the pattern that develops with increasing n suggests that the number of different sets of $(l, m_{l})$ values for a given energy level n is $n^{2}$ . Prove this mathematically by summing the allowed values of $m_{l}$ for a given $l$ over the allowed values of $l$ for a given n.