When The Angular Quantum Number Is $l = 3$, How Many Possible Values Are There For The Magnetic Quantum Number $m_l$?A. 2 B. 7 C. 5 D. 3 E. 1

Feb 28, 2025 by ADMIN 152 views

**Understanding the Angular Quantum Number and Magnetic Quantum Number**

In the realm of quantum mechanics, the angular quantum number (l) and the magnetic quantum number (m_l) play crucial roles in determining the properties of atomic orbitals. The angular quantum number (l) represents the orbital angular momentum of an electron, while the magnetic quantum number (m_l) represents the projection of the orbital angular momentum along the z-axis. In this article, we will delve into the relationship between the angular quantum number (l) and the magnetic quantum number (m_l), with a focus on the specific case where l = 3.

The Relationship Between Angular Quantum Number (l) and Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) can take on values ranging from -l to +l, including zero. This means that for a given value of l, there are 2l + 1 possible values of m_l. For example, if l = 1, then m_l can take on values of -1, 0, and +1, resulting in a total of 3 possible values.

Calculating the Number of Possible Values for m_l When l = 3

Given that l = 3, we can use the formula 2l + 1 to calculate the number of possible values for m_l. Substituting l = 3 into the formula, we get:

2(3) + 1 = 6 + 1 = 7

Therefore, when the angular quantum number (l) is 3, there are 7 possible values for the magnetic quantum number (m_l).

Conclusion

In conclusion, the relationship between the angular quantum number (l) and the magnetic quantum number (m_l) is a fundamental concept in quantum mechanics. By understanding this relationship, we can determine the number of possible values for m_l given a specific value of l. In this article, we have shown that when l = 3, there are 7 possible values for m_l.

Key Takeaways

The angular quantum number (l) represents the orbital angular momentum of an electron.
The magnetic quantum number (m_l) represents the projection of the orbital angular momentum along the z-axis.
The number of possible values for m_l is given by the formula 2l + 1.
When l = 3, there are 7 possible values for m_l.

Frequently Asked Questions

Q: What is the angular quantum number (l)?

A: The angular quantum number (l) represents the orbital angular momentum of an electron.

Q: What is the magnetic quantum number (m_l)?

A: The magnetic quantum number (m_l) represents the projection of the orbital angular momentum along the z-axis.

Q: How many possible values are there for m_l when l = 3?

A: There are 7 possible values for m_l when l = 3.

Q: What is the formula for calculating the number of possible values for m_l?

A: The formula is 2l + 1.

References

[1] Quantum Mechanics by Lev Landau and Evgeny Lifshitz
[2] Atomic Physics by Herbert Goldstein
[3] Quantum Chemistry by Linus Pauling
Quantum Mechanics Q&A: Angular Quantum Number and Magnetic Quantum Number ====================================================================

In our previous article, we explored the relationship between the angular quantum number (l) and the magnetic quantum number (m_l) in quantum mechanics. We also calculated the number of possible values for m_l when l = 3. In this article, we will continue to answer more questions related to the angular quantum number and magnetic quantum number.

Q&A Session

Q: What is the significance of the angular quantum number (l)?

A: The angular quantum number (l) represents the orbital angular momentum of an electron. It determines the shape of the atomic orbital and the energy level of the electron.

Q: What are the possible values of the angular quantum number (l)?

A: The possible values of the angular quantum number (l) are 0, 1, 2, 3, and so on. Each value of l corresponds to a specific type of atomic orbital.

Q: What is the relationship between the angular quantum number (l) and the magnetic quantum number (m_l)?

A: The magnetic quantum number (m_l) can take on values ranging from -l to +l, including zero. This means that for a given value of l, there are 2l + 1 possible values of m_l.

Q: How many possible values are there for m_l when l = 0?

A: When l = 0, there is only one possible value for m_l, which is 0.

Q: How many possible values are there for m_l when l = 1?

A: When l = 1, there are 3 possible values for m_l, which are -1, 0, and +1.

Q: How many possible values are there for m_l when l = 2?

A: When l = 2, there are 5 possible values for m_l, which are -2, -1, 0, +1, and +2.

Q: How many possible values are there for m_l when l = 3?

A: When l = 3, there are 7 possible values for m_l, which are -3, -2, -1, 0, +1, +2, and +3.

Q: What is the formula for calculating the number of possible values for m_l?

A: The formula is 2l + 1.

Q: Can you give an example of how to use the formula to calculate the number of possible values for m_l?

A: Yes, for example, if l = 4, then the number of possible values for m_l is 2(4) + 1 = 9.

Q: What is the significance of the magnetic quantum number (m_l)?

A: The magnetic quantum number (m_l) represents the projection of the orbital angular momentum along the z-axis. It determines the orientation of the atomic orbital in space.

Q: Can you explain the concept of orbital angular momentum?

A: Orbital angular momentum is a measure of the tendency of an electron to rotate around the nucleus of an atom. It is a vector quantity that has both magnitude and direction.

Q: How does the orbital angular momentum affect the energy level of an electron?

A: The orbital angular momentum affects the energy level of an electron by determining the shape of the atomic orbital. The shape of the atomic orbital, in turn, affects the energy level of the electron.

Conclusion

In conclusion, the angular quantum number (l) and the magnetic quantum number (m_l) are fundamental concepts in quantum mechanics. Understanding the relationship between these two quantities is essential for determining the properties of atomic orbitals. We hope that this Q&A session has helped to clarify any doubts you may have had about these concepts.