Question

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?

Short answer

The spherical harmonics ${\Theta }_{l,m_l}\left(\theta \right){\Phi }_{m_l}\left(f\right)$are dimensionless.

All radial functions have dimension$L^{-3/2}$_.

The dimension of $P\left(r\right)$is$\frac{1}{L}=L^{-1}$ .

Step-by-step solution

Dimensional analysis

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

 Formula used

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

 The dimensions of Spherical Harmonics and radial functions

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 dimensions of Pr

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$\frac{1}{L}=L^{-1}$ .