Question

Consider an electron in the state n=4n=4, l=3l=3, m=2m=2, s=1/2s=1/2.

1)In what shell is this electron located?
2)In what subshell is this electron located?
3)How many other electrons could occupy the same subshell as this electron?
.4)What is the orbital angular momentum LL of this electron? Express your answer in units of ℏ.
5)What is the z component of the orbital angular momentum of this electron, LzLz Express your answer in units of ℏ.
6)
What is the z component of the spin angular momentum of this electron, SzSz Express your answer in units of ℏ.

Answer 1

1. This electron is located in the fourth shell.
2. This electron is located in the l=3 subshell, also known as the p subshell.
3. There could be six other electrons occupying the same subshell as this electron.
4. The orbital angular momentum of this electron is LL = √(l(l+1))ℏ = √(3(3+1))ℏ = √12ℏ = 2ℏ.
5. The z component of the orbital angular momentum of this electron is Lz = mℏ = 2ℏ.
6. The z component of the spin angular momentum of this electron is Sz = m_sℏ = (1/2)ℏ = 1/2ℏ.

Step-by-step explanation:

1. In quantum mechanics, an electron is described by four quantum numbers: n, l, m, and s. The first quantum number, n, is known as the principal quantum number and determines the energy level or shell in which the electron is located. In this case, the electron is in the n=4 shell.
2. The second quantum number, l, is known as the angular momentum quantum number and determines the subshell in which the electron is located. The possible values of l are 0, 1, 2, ..., n-1, where n is the principal quantum number. In this case, the electron is in the l=3 subshell, which is also known as the p subshell.
3. The Pauli exclusion principle states that no two electrons in an atom can have the same set of quantum numbers. Therefore, the maximum number of electrons that can occupy a given subshell is determined by the number of distinct combinations of the remaining quantum numbers, m and s. For a given value of l, there are 2l+1 possible values of m, and two possible values of s (±1/2). Therefore, the maximum number of electrons that can occupy a subshell with l=3 is 2(3+1) = 6.
4. The orbital angular momentum of an electron is given by the formula LL = √(l(l+1))ℏ, where l is the angular momentum quantum number and ℏ is the reduced Planck constant. In this case, the electron has l=3, so its orbital angular momentum is LL = √(3(3+1))ℏ = √12ℏ = 2ℏ.
5. The z component of the orbital angular momentum, Lz, is given by the formula Lz = mℏ, where m is the magnetic quantum number. In this case, the electron has m=2, so its z component of orbital angular momentum is Lz = 2ℏ.
6. The spin angular momentum of an electron is given by the formula Sz = m_sℏ, where m_s is the spin quantum number. In this case, the electron has m_s=1/2, so its spin angular momentum is Sz = (1/2)ℏ = 1/2ℏ.