Question

For a given value of l , what are the allowed values of m_l ? What are the allowed values of m_l for a given value of n ? Give an example in each case.

a) m_l can take any integer value from - l to l ; for n , m_l can take any integer value from - n to n .

b) m_l can take any integer value from 0 to l ; for n , m_l can take any integer value from 0 to n .

c) m_l can take any integer value from - l to l ; for n , m_l can take any integer value from - n to n .

d) m_l can take any integer value from 0 to l ; for n , m_l can take any integer value from - n to n .

Answer 1

The magnetic quantum number (
`\(m_l\)`) spans from -l to l, and for a given value of n,
`\(m_l\)` ranges from -n to n, accurately describing the allowed values of
`\(m_l\)`, thus the correct option is c)
`m_l` can take any integer value from - l to l; for n,
`m_l` can take any integer value from - n to n.

Step-by-step explanation:

In quantum mechanics, the quantum numbers n and l represent the principal and azimuthal quantum numbers, respectively. The magnetic quantum number (
`\(m_l\)`) specifies the orientation of an atomic orbital in space.

For a given value of l, the allowed values
`\(m_l\)` range from -l to l. This is because
`\(m_l\)` can take any integer value within this range, representing the different orientations of the orbital.

For a given value of n, the allowed values
`\(m_l\)` similarly range from -n to n. The restriction on
`\(m_l\)` is based on the principal quantum number n, which determines the main energy level of the electron.

For example, if l = 2, the allowed values of
`\(m_l\)` would be -2, -1, 0, 1, 2. If n = 3, the allowed values of
`\(m_l\)` would be -3, -2, -1, 0, 1, 2, 3. Therefore, option c) accurately describes the allowed values of
`\(m_l\)` for a given l and n.

Answer 2

Final Answer:

c)
`(m_l\)` can take any integer value from
`\(-l\)` to
`\(l\);` for (n),
`\(m_l\)` can take any integer value from (-n) to (n).

Step-by-step explanation:

The correct statement is c). For a given value of (l), the magnetic quantum number
`\(m_l\)` can take any integer value from
`\(-l\)` to
`\(l\)`. This is a consequence of the fact that the magnetic quantum number represents the orientation of the orbital angular momentum with respect to a chosen axis. Since there are
`\(2l+1\)` possible values for
`\(m_l\)`, ranging from
`\(-l\)` to
`\(l\),` this aligns with the known properties of angular momentum in quantum mechanics.

For a given value of \(n\), the principal quantum number,
`\(m_l\)` can take any integer value from
`\(-n\)` to (n\). The principal quantum number dictates the energy level of an electron in an atom, and the range of
`\(m_l\)` is determined by the azimuthal quantum number
`(\(l\))`. The range of
`\(m_l\)` is broader when considering a higher energy level
`(\(n\))`, as (n) is directly related to the size of the electron's orbit.

In summary, the magnetic quantum number
`\(m_l\)` varies based on the angular momentum quantum number
`\(l\)` and has a range from
`\(-l\)` to
`\(l\)`. For a given principal quantum number (n),
`\(m_l\)` has a range from
`\(-n\)` to (n)