A particle in a central potential V (r) has a wavefunction of Ψ= f (r,Ө) sin2Ø where Ø is the azimuthal angle in the spherical polar coordinate system. Also, this wavefunction Ψ is properly normalized, i.e.˂ΨӏΨ˃=1Read the following passage to answer the given questions based on it. Choose the correct options. (Marks: 1)What is the expectation value of z component of orbital angular momentum , ˂Lz ˃?OPTIONS 0.016 0.015 0.021 0.017
Question
A particle in a central potential V (r) has a wavefunction of Ψ= f (r,Ө) sin2Ø where Ø is the azimuthal angle in the spherical polar coordinate system. Also, this wavefunction Ψ is properly normalized, i.e.˂ΨӏΨ˃=1Read the following passage to answer the given questions based on it. Choose the correct options. (Marks: 1)What is the expectation value of z component of orbital angular momentum , ˂Lz ˃?OPTIONS 0.016 0.015 0.021 0.017
Solution 1
The expectation value of the z component of the orbital angular momentum, ˂Lz ˃, for a wavefunction that is only dependent on the azimuthal angle Ø, is zero. This is because the z component of the orbital angular momentum operator in quantum mechanics is -iħ(d/dØ), and the expectation value of this operator for the given wavefunction is zero. Therefore, the correct option is not listed in the given options.
Solution 2
The expectation value of the z component of the orbital angular momentum, ˂Lz ˃, for a wavefunction that is only dependent on the azimuthal angle (Ø) and is properly normalized, is zero. This is because the expectation value of Lz is given by the integral over all space of the wavefunction times the operator Lz times the wavefunction, and in this case the operator Lz is proportional to the derivative with respect to Ø. Since the wavefunction does not depend on Ø, this derivative is zero, and hence the expectation value of Lz is zero. Therefore, none of the given options are correct.
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