The Radial Wavefunction Solutions

Defining the Bohr radius

\begin{displaymath}\bgroup\color{black}a_0={\hbar\over\alpha mc} ,\egroup\end{displaymath}

we can compute the radial wave functions Here is a list of the first several radial wave functions \bgroup\color{black}$R_{n\ell}(r)$\egroup.

R_{10} &=& 2\left({Z\over a_0}\right)^{3\over 2}\; e^{ - Z r/ ...
...0}+{2\left(Zr\right)^2\over 27a_0^2}\right)\;e^{ - Z r/ 3a_0}\\

For a given principle quantum number \bgroup\color{black}$n$\egroup,the largest \bgroup\color{black}$\ell$\egroup radial wavefunction is given by

\begin{displaymath}\bgroup\color{black}R_{n,n-1}\; \propto\; r^{n-1}\; e^{ - Z r/na_0}\egroup\end{displaymath}

The radial wavefunctions should be normalized as below.

\begin{displaymath}\bgroup\color{black}\int\limits_0^\infty r^2 R^*_{n\ell}R_{n\ell}\; dr=1 \egroup\end{displaymath}

* Example: Compute the expected values of $E$, $L^2$, $L_z$, and $L_y$ in the Hydrogen state ${1\over 6}(4\psi_{100}+3\psi_{211}-i\psi_{210}+\sqrt{10}\psi_{21-1})$.*

The pictures below depict the probability distributions in space for the Hydrogen wavefunctions.


The graphs below show the radial wave functions. Again, for a given \bgroup\color{black}$n$\egroup the maximum \bgroup\color{black}$\ell$\egroup state has no radial excitation, and hence no nodes in the radial wavefunction. As \bgroup\color{black}$\ell$\egroup gets smaller for a fixed \bgroup\color{black}$n$\egroup, we see more radial excitation.


A useful integral for Hydrogen atom calculations is.

\begin{displaymath}\bgroup\color{black}\int\limits_{0}^{\infty}\; dx\; x^n\; e^{-ax} ={n!\over a^{n+1}} \egroup\end{displaymath}

* Example: What is the expectation value of ${1\over r}$ in the state $\psi_{100}$?*

* Example: What is the expectation value of $r$ in the state $\psi_{100}$?*

* Example: What is the expectation value of the radial component of velocity in the state $\psi_{100}$?*

Jim Branson 2013-04-22