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Q10CQ

Expert-verifiedFound in: Page 278

Book edition
2nd Edition

Author(s)
Randy Harris

Pages
633 pages

ISBN
9780805303087

**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?**

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

**I****n 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$\frac{1}{L}={L}^{-1}$ .

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