What are the dimensions of the spherical harmonics ${\mathit{\Theta }}_{\mathbf{l}\mathbf{,}{\mathbf{m}}_{\mathbf{l}}}\left(\theta \right){\mathit{\Phi }}_{{\mathbf{m}}_{\mathbf{l}}}\left(\varphi \right)$given in Table 7.3? What are the dimensions of the${\mathit{R}}_{\mathbf{n}\mathbf{,}\mathbf{l}}\mathbf{\left(}\mathit{r}\mathbf{\right)}$given in Table 7.4, and why? What are the dimensions of$\mathit{P}\left(r\right)$, and why?

In engineering dimensional analysis is the analysis of relationship of physical quantities with each other by identifying their base quantities or the basic units.

The normalization equation is given by,

$\underset{0}{\overset{\infty }{\int }}{R}^2\left(r\right)r^{2}dr=1$

Where, R is the Radial wave function, r is the radius

Consider table 7.3, the spherical harmonic functions represented as ${\Theta }_{l,m_l}\left(\theta \right){\Phi }_{m_l}\left(f\right)$are the combinations of sine, cosine and complex exponential functions and thus they are dimension less.

As you can see in Table 7.4, all radial functions have dimension$L^{-3/2}$^, because their square gives probability per unit volume.

The dimension of $P\left(r\right)$is calculated as,

$\underset{0}{\overset{\infty }{\int }}{R}^2\left(r\right)r^{2}dr=1$

$\underset{0}{\overset{\infty }{\int }}P\left(r\right)dr=1$

Where,$P\left(r\right)=r^2{R}^2\left(r\right)$ is the Radial Probability

Since, $dr$has the dimension of length so, the dimension of $P\left(r\right)$is$\frac{1}{L}=L^{-1}$ .