Question

The orbital angular momentum is a 3D vector. For an electron at a state associated with quantum numbers (?, ?, ??), the magnitude of the orbital angular momentum is √?(? + 1) ℏ. The orbital angular momentum measured along the z axis, however, will be exactly ??ℏ. For a real orbital that combines two complex orbitals associated with two different ?? values, a measurement along the z axis will result in either of the two ?? values with an equal probability. However, once the measurement is done and either of the two ?? values is determined to describe the state of the electron, the new orbital will be a complex orbital that is associated with the specific ?? measured in the experiment. For instance, ?? orbital combines ?¹₁ and ?^-1₁ with ?? = 1 and −1, respectively. If the orbital angular momentum is measured along the z axis, the outcome would either be ℏ or −ℏ. After the measurement, the orbital is no longer the ?? orbital but it is either the ?¹₁ or the ?^-1₁ orbital, depending on the outcome of the measurement.

a. What is the magnitude of the orbital angular momentum associated with each orbital in Problem 1.

b. What are the possible outcomes of the angular momentum measurements along the z axis for each orbital in Problem 1. For each possible outcome, describe the new orbital associated with the state of the system after the measurement.

Answer 1