÷ÈÓ°Ö±²¥

Calculate the angles that a spin angular momentum vector for an individual electron can make with the \(z\) axis.

In this problem, you will use the variational method to find the optimal \(1 s\) wave function for the hydrogen atom starting from the trial function \(\Phi(r)=e^{-\alpha r}\) with \(\alpha\) as the variational parameter. You will minimize \\[ E(\alpha)=\frac{\int \Phi^{*} \hat{H} \Phi d \tau}{\int \Phi^{*} \Phi d \tau} \\] with respect to \(\alpha\) a. Show that \\[ \begin{aligned} \hat{H} \Phi &=-\frac{\hbar^{2}}{2 m_{e}} \frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial \Phi(r)}{\partial r}\right)-\frac{e^{2}}{4 \pi \varepsilon_{0} r} \Phi(r) \\ &=\frac{\alpha \hbar^{2}}{2 m_{e} r^{2}}\left(2 r-\alpha r^{2}\right) e^{-\alpha r}-\frac{e^{2}}{4 \pi \varepsilon_{0} r} e^{-\alpha r} \end{aligned} \\] b. Obtain the result \(\int \Phi^{*} \hat{H} \Phi d \tau=4 \pi \int_{0}^{\infty} r^{2} \Phi^{*} \hat{H} \Phi d r=\) \(\pi \hbar^{2} /\left(2 m_{e} \alpha\right)-e^{2} /\left(4 \varepsilon_{0} \alpha^{2}\right)\) using the standard integrals in the Math Supplement. c. Show that \(\int \Phi^{*} \Phi d \tau=4 \pi \int_{0}^{\infty} r^{2} \Phi^{*} \Phi d r=\pi / \alpha^{3}\) using the standard integrals in the Math Supplement. d. You now have the result \(E(\alpha)=\hbar^{2} \alpha^{2} /\left(2 m_{e}\right)-e^{2} \alpha /\left(4 \pi \varepsilon_{0}\right)\) Minimize this function with respect to \(\alpha\) and obtain the optimal value of \(\alpha\) e. Is \(E\left(\alpha_{\text {optimal }}\right)\) equal to or greater than the true energy? Why?

Classify the following functions as symmetric, antisymmetric, or neither in the exchange of electrons 1 and 2 : a. \([1 s(1) 2 s(2)+2 s(1) 1 s(2)] \times[\alpha(1) \beta(2)-\beta(1) \alpha(2)]\) b. \([1 s(1) 2 s(2)+2 s(1) 1 s(2)] \alpha(1) \alpha(2)\) c. \([1 s(1) 2 s(2)+2 s(1) 1 s(2)][\alpha(1) \beta(2)+\beta(1) \alpha(2)]\) d. \([1 s(1) 2 s(2)-2 s(1) 1 s(2)][\alpha(1) \beta(2)+\beta(1) \alpha(2)]\) e. \([1 s(1) 2 s(2)+2 s(1)] s(2)] \times\) \([\alpha(1) \beta(2)-\beta(1) \alpha(2)+\alpha(1) \alpha(2)]\)

Write the Slater determinant for the ground-state configuration of Be.

Refer to the first ionization energies and electron affinities of the first 11 elements (units of eV) shown in the following table. Why is the magnitude of the electron affinity for a given element smaller than the magnitude of the first ionization energy?