Quantum Mechanics

Step 1: Understanding the given problem.
We are given that the electron is in the 4f orbital.
The 4f orbital corresponds to (principal quantum number) and (azimuthal quantum number), which indicates it's an -type orbital.
Step 2: Determining valid quantum numbers.
means the electron is in the fourth shell.
For the 4f orbital, . The quantum number corresponds to the type of orbital:
for -orbitals
for -orbitals
for -orbitals
for -orbitals
Step 3: Determining the possible values of .
The magnetic quantum number can take values from to , including zero.
For , the possible values for are: .
Step 4: Determining the possible values of .
The spin quantum number can only have two values: or .
Step 5: Analyzing the options.
Option (1) is incorrect because corresponds to the third shell, not the fourth shell, and it’s a -orbital, not .
Option (2) is correct because it satisfies the conditions for a 4f orbital.
Option (3) is incorrect because must be between and .
Option (4) is incorrect because cannot be 4 for an -orbital. For , the highest value of is 3.
Step 6: Conclusion.
Thus, the correct answer is option (2).

The maximum number of electrons that can occupy an energy level is given by , where is the principal quantum number. This is derived from the formula for the number of orbitals in each energy level.

The de Broglie wavelength is related to momentum by: The kinetic energy is: If kinetic energy is increased to three times, then: Corresponding momentum: Therefore, new wavelength : Given the original wavelength is , so: