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Expert-verified Found in: Page 278 ### Modern Physics

Book edition 2nd Edition
Author(s) Randy Harris
Pages 633 pages
ISBN 9780805303087 # What are the dimensions of the spherical harmonics ${{\mathbit{\Theta }}}_{\mathbf{l}\mathbf{,}{\mathbf{m}}_{\mathbf{l}}}\left(\theta \right){{\mathbit{\Phi }}}_{{\mathbf{m}}_{\mathbf{l}}}\left(\varphi \right)$ given in Table 7.3? What are the dimensions of the ${{\mathbit{R}}}_{\mathbf{n}\mathbf{,}\mathbf{l}}{\mathbf{\left(}}{\mathbit{r}}{\mathbf{\right)}}$ given in Table 7.4, and why? What are the dimensions of ${\mathbit{P}}\left(r\right)$, and why?

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

See the step by step solution

## Step 1: 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.

## Step 2:  Formula used

The normalization equation is given by,

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

## Step 3:  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.

## Step 4:  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$\frac{1}{L}={L}^{-1}$ .

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