How many possible values for pair l and m_{l} are there when n_1 = 3?


How many possible values for pair l and m_{l} are there when n_2 = 5?

Express your answer as an integer.

I understand the following solution:
"we know that for a particular value of n , l ranges from 0 ton-1
and m ranges from -l to + l including 0.
so here in this question
for n = 2 we have l value ranging from 0 to 1
now ml ranges from -l to + l
so for l = 0, we have m l = 0
and for l = 1 , we have ml = -1 , 0 , +1
so there are total 2 values of l and 3values of ml
now in the other case where n = 5
we have l values ranging from 0 to 4
so there are 5 possible values of l
and ml ranges from -l to + l , so total 2l + 1values
so ml possible values = 2*4 + 1
= 9
so there are 5 possible values of l and 9possible values of ml"

I tried to answer 6 for n = 3 and 45 for n = 5 but the answer is incorrect.

I must confess that I have no inkling of what "pair l" means although I understand the example you hve give but don't see the connection since the word "pair" is never used in the example.

For n = 3 there will be 3 possible values for l (0,1,2) and there will be 5 different values for ml. How that fits into "pair l" I don't know.
For n = 5, there will be 5 possible values for l (0,1,2,3,4) and 9 separate values for ml.

To determine the possible values for the pair (l, ml) when n_1 = 3, we follow the given explanation which states that for a particular value of n, l ranges from 0 to n - 1, and ml ranges from -l to +l.

For n_1 = 3, l can take the values of 0, 1, and 2. For each value of l, ml can take the values from -l to +l. So, for l = 0, we have ml = 0, for l = 1, ml can be -1, 0, or +1, and for l = 2, ml can be -2, -1, 0, +1, or +2. Therefore, there are a total of 3 possible values for l and 5 possible values for ml.

Hence, the correct answer is 3 * 5 = 15 possible values for the pair (l, ml) when n_1 = 3.

Similarly, for n_2 = 5, we can apply the same logic. The possible values for l are 0, 1, 2, 3, and 4. For each value of l, ml can take the values from -l to +l. So, for l = 0, ml = 0, for l = 1, ml can be -1, 0, or +1, for l = 2, ml can be -2, -1, 0, +1, or +2, and so on. Hence, there are a total of 5 possible values for l and 2 * 5 + 1 = 11 possible values for ml.

Therefore, the correct answer for the pair (l, ml) when n_2 = 5 is 5 * 11 = 55.

So, the correct answers are 15 for n_1 = 3 and 55 for n_2 = 5.

For a given value of n, the possible values for pair l and m_l are calculated as follows:

For n = 3:
Since n_1 = 3, the value of l ranges from 0 to (n_1 - 1), which is 0 to 2 in this case.
For each value of l, the number of possible values for m_l is 2l + 1.
- For l = 0, m_l has 2(0) + 1 = 1 possible value.
- For l = 1, m_l has 2(1) + 1 = 3 possible values.
- For l = 2, m_l has 2(2) + 1 = 5 possible values.

Therefore, the total number of possible values for pair l and m_l when n_1 = 3 is 1 + 3 + 5 = 9.

For n = 5:
Since n_2 = 5, the value of l ranges from 0 to (n_2 - 1), which is 0 to 4 in this case.
For each value of l, the number of possible values for m_l is 2l + 1.
- For l = 0, m_l has 2(0) + 1 = 1 possible value.
- For l = 1, m_l has 2(1) + 1 = 3 possible values.
- For l = 2, m_l has 2(2) + 1 = 5 possible values.
- For l = 3, m_l has 2(3) + 1 = 7 possible values.
- For l = 4, m_l has 2(4) + 1 = 9 possible values.

Therefore, the total number of possible values for pair l and m_l when n_2 = 5 is 1 + 3 + 5 + 7 + 9 = 25.

Thus, for n_1 = 3, there are 9 possible values for pair l and m_l, and for n_2 = 5, there are 25 possible values for pair l and m_l.