Question

Write down the wave functions (i) ψ210 and (ii) ψ300 for the hydrogen atom. Obtain the
expectation value of r for the ground state hydrogen atom.

Answer 1

Answer on question #60875 – Physics / Atomic and Nuclear Physics

Write down the wave functions (i) $\psi 210$ and (ii) $\psi 300$ for the hydrogen atom. Obtain the expectation value of $r$ for the ground state hydrogen atom.

Solution

$$\psi_{nlm}(r, \vartheta, \varphi) = \frac{1}{\sqrt{2n(n-l-1)!(n+l)!}} \left(\frac{2}{na_0}\right)^{\frac{3}{2}} \exp\left(-\frac{r}{na_0}\right) \left(\frac{2r}{na_0}\right)^l L_{n-l-1}^{2l+1} \left(\frac{2r}{na_0}\right) Y_{lm}(\vartheta, \varphi),$$

Where

$a_0$ is the Bohr radius,

$L_{n-l-1}^{2l+1}$ is a generalized Laguerre polynomial of degree $n - \ell - 1$,

$Y_{lm}(\vartheta, \varphi)$ is a spherical harmonic function of degree $\ell$ and order $m$.

Then

$$\begin{aligned} \psi_{210}(r, \vartheta, \varphi) &= \frac{1}{\sqrt{24}} \left(\frac{1}{a_0}\right)^{\frac{3}{2}} \exp\left(-\frac{r}{2a_0}\right) \left(\frac{r}{a_0}\right) L_0^3 \left(\frac{r}{a_0}\right) Y_{10}(\vartheta, \varphi) \\ &= \frac{1}{4} \sqrt{\frac{1}{2\pi}} \left(\frac{1}{a_0}\right)^{\frac{3}{2}} \exp\left(-\frac{r}{2a_0}\right) \left(\frac{r}{a_0}\right) \end{aligned}$$ $$\begin{aligned} \psi_{300}(r, \vartheta, \varphi) &= \frac{1}{\sqrt{72}} \left(\frac{2}{3a_0}\right)^{\frac{3}{2}} \exp\left(-\frac{r}{3a_0}\right) L_2^1 \left(\frac{2r}{3a_0}\right) Y_{00}(\vartheta, \varphi) \\ &= \frac{1}{12} \sqrt{\frac{3}{2\pi}} \left(\frac{2}{3a_0}\right)^{\frac{3}{2}} \exp\left(-\frac{r}{3a_0}\right) \left(\frac{1}{2} \left(\frac{2r}{3a_0}\right)^2 - 6 \left(\frac{2r}{3a_0}\right) + 3\right) \cos \vartheta \end{aligned}$$

For ground state we have:

$$\psi_{100}(r, \vartheta, \varphi) = \sqrt{\frac{1}{\pi a_0^3}} \exp\left(-\frac{r}{a_0}\right)$$

The expectation value of $r$ is:

$$\int_V r |\psi_{100}|^2 dV = \frac{4\pi}{\pi a_0^3} \int_0^\infty r^3 \exp\left(-\frac{2r}{a_0}\right) dr = \frac{4}{a_0^3} \left(\frac{a_0}{2}\right)^4 \int_0^\infty x^3 \exp(-x) dx = \frac{3}{2} a_0$$

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