Question

For \(\ell<4,\) which values of \(\ell\) and \(m\) correspond to wave functions that have their maximum probability in the \(x y\) -plane?

Short answer

Answer: The pairs \((\ell, m) = (1, 0)\) and \((1, ±1)\) have the maximum probability in the xy-plane.

Step-by-step solution

Review of Spherical Harmonic Functions

Spherical Harmonic functions are given by: \(Y_{\ell}^m(θ, φ) = N_{\ell m} P_{\ell}^m (\cos{θ}) e^{imφ}\), where \(N_{\ell m}\) is the normalization constant, \(P_{\ell}^m(\cos{θ})\) are the associated Legendre functions, and \(θ\) and \(φ\) are the polar and azimuthal angles, respectively. Additionally, the values of \(\ell\) and \(m\) are limited by: \(0 \leq \ell\) and \(-\ell \leq m \leq \ell\). Since the given condition is that \(𝑙<4\), we have four possible sets of values (\(\ell = 0, 1, 2\), and \(3\)).

Maximum probability in the xy-plane

The probability density is proportional to the square of the magnitude of the spherical harmonics. Thus, we need to find the maximum of \(|Y_{\ell}^m(θ, φ)|^2\) along the z-axis, which corresponds to \(θ = \frac{π}{2}\). For each value of \(\ell\) from \(0\) to \(3\), we will check all possible values of \(m\) (ranging from \(-\ell\) to \(\ell\)) and evaluate the square of the magnitude of the spherical harmonics at \(θ = \frac{π}{2}\), while varying the azimuthal angle \(\phi\). Then, we will find the combinations of \(\ell\) and for which the spherical harmonic's square magnitude is maximized.

Evaluate and find the maximum probability

Evaluating the magnitude square of spherical harmonics at \(θ = \frac{π}{2}\): For \(\ell = 0\) and \(m = 0\): \(|Y_{0}^0|^2 \propto N_{00}^2\) For \(\ell = 1\): 1. \(|Y_{1}^0|^2 \propto N_{10}^2P_{1}^0\sin^2{θ}\) has a maxima when \(\sin^2{θ}=1\). 2. \(|Y_{1}^{±1}|^2 \propto N_{1,±1}^2P_{1}^{±1}\sin^2{θ}\) has a maxima when \(\sin^2{θ}=1\). For \(\ell = 2\): 1. \(|Y_{2}^0|^2 \propto N_{20}^2P_{2}^0\sin^2{θ}\) has a maxima when \(\sin^2{θ}=1\). 2. \(|Y_{2}^{±1}|^2 \propto N_{2,±1}^2P_{2}^{±1}\sin^2{θ}\) is less than \(|Y_{1}^{±1}|^2\) magnitude. 3. \(|Y_{2}^{±2}|^2 \propto N_{2,±2}^2P_{2}^{±2}\sin^2{θ}\) equals \(0\) in the xy-plane. For \(\ell = 3\): 1. \(|Y_{3}^0|^2 \propto N_{30}^2P_{3}^0\sin^2{θ}\) is less than \(|Y_{2}^0|^2\) magnitude. 2. \(|Y_{3}^{±1}|^2 \propto N_{3,±1}^2P_{3}^{±1}\sin^2{θ}\) is less than \(|Y_{1}^{±1}|^2\) magnitude. 3. \(|Y_{3}^{±2}|^2 \propto N_{3,±2}^2P_{3}^{±2}\sin^2{θ}\) is less than \(|Y_{1}^{±1}|^2\) magnitude. 4. \(|Y_{3}^{±3}|^2 \propto N_{3,±3}^2P_{3}^{±3}\sin^2{θ}\) equals \(0\) in the xy-plane. After calculating and analyzing the maximum probability for each combination of \(\ell\) and \(m\), we find that the pairs \((\ell, m) = (1, 0)\), \((1, ±1)\) have the maximum probability in the xy-plane.