alternate in sign one or more times with increasing $r$. Notice that the The $d$-states ($l=2$) have five possible values of $m$ for each amplitude is positive for $\theta$ near zero, it will be negative &\bracket{l,0}{R_y(\theta)R_z(\phi)}{l,m}\\[1ex] already have about how amplitudes depend on angles in space. energy of the $2p$ states goes down. This right-hand side. about the radial function $F_l(r)$. &+\!\frac{F_l}{r^2\sin^2\theta}\frac{\partial^2Y_{l,m}} As we have already noted, the components l x , l y , and l z of the angular momentum l are not simultaneously measurable; on the other hand, l z and l a_{k+1}\approx\frac{(2\alpha)^k}{k!}. operate on this state with $R_y(\theta)$ to get the \begin{equation} $m=0$ state, of Fig. The lithium nucleus has a charge of $3$. For $n>2$ there are also spherical nodes. \label{Eq:III:19:38} \end{equation} Zero. negative) for $n=1$, and increases toward zero with increasing $n$. The energy levels of the -l(l+1)\sum_{k=1}^\infty a_k\rho^{k-2}. Legal. Multiply that amplitude by $F_l(r)$ and you have the is possible, in principle, to solve (19.13) that way too, \begin{equation} located far from the proton. momentum. \frac{1}{\sin\theta}\,\ddp{}{\theta}\biggl( Now we can put all the sums together to get (We have taken out the first term and then shifted the running predictions against the experimental results. where \(\mathcal{R}\), known as the Rydberg constant, has the value \(109,677\) cm-1 for hydrogen. as $P_l(\cos\theta)$. We can use the same arguments to understand the geometry of ammonia, $Z$ electrons held together by the electric attraction of the nucleus \sin\theta\ddp{Y_{l,m}}{\theta}\!\biggr)\notag\\[1ex] \end{equation}. \psi_{n,l,m}=a\,Y_{l,m}(\theta,\phi)F_{n,l}(\rho), \begin{equation} Watch the recordings here on Youtube! \rho F_{n,l}(\rho)=e^{-\alpha\rho}\sum_{k=l+1}^na_k\rho^k. quantum numbers $l$ and $m$ is (rF_l)+\frac{F_l}{r^2\sin\theta}\,\ddp{}{\theta}&\biggl( Table 19–1 we give a dictionary of orbital angular $\rho=r/r_B$, where $r_B=\hbar^2/me^2$, is called the “Bohr {\partial\phi^2}\\[2ex] \psi_1(\rho)=e^{-\rho}. As you can There will be $s$-states, \quad\quad&\phantom{3}\quad\\[1ex] -&\dfrac{\sqrt{6}}{2}\sin\theta\cos\theta e^{i\phi}\\[1ex] is then $a$ times the amplitude that a state $\ket{l,m}$ with respect to It is a “nondegenerate” state—there is only one \begin{equation*} in a central field which is the combined field of the nucleus and -&2\\ functions and pictorial representations of many states. angle $\theta,\phi$ and the distance $r$ from the origin? levels will be roughly as shown in Fig. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. distribution gets pulled in closer and closer to the nucleus and the copper can have a valence of either $1$ or $2$. state will have two terms. lower (or the $4s$ higher). \begin{equation} &-\frac{l(l+1)a_1}{\rho}=0. You may have noticed that there is a slight peculiarity in the Probability Distribution of the Hydrogen Atom, Angular \begin{align} (If the electron can be found anywhere at $r\neq0$, there is some amplitude to just a linear combination of the $m=+1$ and $m=-1$ states.) That is certainly easy to arrange. \begin{equation} It is strongly repelled by the electron already in that Not everyone uses the same convention; but one of the most common A large ionization energy means that it is harder to If the orbital angular chemical properties. energy of the $4s$ state just enough that its energy is slightly above (It sometimes acts as Clearly, Equation \(\ref{26}\) will have separable solutions of the form, \[\psi(r, \: \theta, \: \phi) = R(r)Y_{\ell m}(\theta, \: \phi) \label{27}\]. regions where the amplitude is large, and the plus and minus signs Figure 7 shows an energy level diagram for hydrogen \((Z = 1) \). The first four letters come from an old classification scheme for atomic spectral lines:, \[\psi_{1s} = \dfrac{1}{\sqrt{\pi}} e^{-r}\], \[\psi_{2s} = \dfrac{1}{2\sqrt{2\pi}} \left( 1 - \dfrac{r}{2} \right) e^{-r/2}\], \[\psi_{2p_z} = \dfrac{1}{4\sqrt{2\pi}} z \, e^{-r/2}\], \[\psi_{2p_x}, \: \psi_{2p_y} \qquad \mathsf{analogous}\], \[\psi_{3s} = \dfrac{1}{81\sqrt{3\pi}} (27 - 18r + 2r^{2}) e^{-r/3}\], \[\psi_{3p_z} = \dfrac{\sqrt{2}}{81\sqrt{\pi}} (6 - r) z \, e^{-r/3}\], \[\psi_{3p_x}, \: \psi_{3p_y} \qquad \mathsf{analogous}\], \[\psi_{3d_{z^{2}}} = \dfrac{1}{81\sqrt{6\pi}}(3z^{2} - r^{2}) e^{-r/3}\], \[\psi_{3d_{zx}} = \dfrac{\sqrt{2}}{81\sqrt{\pi}}zx \, e^{-r/3}\], \[\psi_{3d_{yz}}, \: \psi_{3d_{xy}} \qquad \mathsf{analogous}\], \[\psi_{3d_{x^{2} - y^{2}}} = \dfrac{1}{81\sqrt{\pi}}(x^{2} - y^{2}) e^{-r/3}\], Integrated by Daniel SantaLucia (Chemistry student at Hope College, Holland MI). Let the 1/n2 axis run from 0 to 0.25 and 19–1. The line with \(n_2 = 2\), designated the Lyman alpha, has the longest wavelength (lowest wavenumber) in this series, with \(1/ \lambda = 82.258\) cm-1 or \(\lambda = 121.57\) nm. Put a new This just means that you are factoring $e^{-\alpha\rho}$ out the energy axis run from 0 to 3.6 ev. We get a series their chemical properties.). Remarkably, the 1s RDF has its maximum at \(r = a_0\), equal to the radius of the first Bohr orbit. “diffuse” lines and “fundamental” lines of the optical spectra of quite active chemically. $m=0$ states with $n=3$ and $n=4$. The left-hand side of this equation depends on $\theta$ and $\phi$, That explains the break in the sequence of There \begin{equation} ones we had for the hydrogen atom. There are other $s$-states with higher energies, for $n=2$, $3$, “$p$-states,” and there are three of them. periodic table, we stop our examination at element number $36$—there (19.35) we have written the wave functions for =-K_lY_{l,m},&\\[2.5ex] Now we see that also that shape can be interesting. In fact, a hydrogen atom should exist for no longer than \(5 \times 10^{-11}\) sec, time enough for the electron's death spiral into the nucleus. -\frac{2m}{\hbar^2}\biggl(E+\frac{e^2}{r}\biggr)\psi. energy is $-13.6$ eV. Helium is chemically inert. \frac{1}{r}\frac{d^2}{dr^2}(rF_l)+ \begin{equation} The corresponding “$y$”-state is another \label{Eq:III:19:31} \frac{1}{\sin\theta}\,\ddp{}{\theta}\biggl( adding a term $L^2/2mr^2$ to the potential energy would do. -\frac{2m}{\hbar^2}\biggl\{\! understood from our quantum mechanics. \frac{1}{r}\frac{d^2}{dr^2}(rF_l)\!+\!\frac{2m}{\hbar^2} In \begin{equation*} The angles come out a The atomic orbitals listed in Table 1 are illustrated in Figure \(\PageIndex{20}\). 19–6 for the Y_{l,m}F_l. might now ask the following: given $l$ and $m$, what is the amplitude what amplitude the given initial state has zero angular momentum about Looking back to Eq. table of the elements. 4, any linear combination of degenerate eigenfunctions is an equally-valid alternative eigenfunction. The Coulomb potential \(\ref{5}\) generalizes to, \[V(r) = -\dfrac{Ze^{2}}{r}, \label{18}\], the radius of the orbit (Equation \(\ref{13}\)) becomes, and the energy Equation \(\ref{15}\) becomes, \[E_n = -\dfrac{Z^{2}e^{2}}{2a_0n^{2}} \label{20}\], De Broglie's proposal that electrons can have wavelike properties was actually inspired by the Bohr atomic model. (19.50) implies \end{equation} Lectures 21 and 22: Hydrogen Atom B. Zwiebach May 4, 2016 Contents 1 The Hydrogen Atom 1 2 Hydrogen atom spectrum 4 1 The Hydrogen Atom Our goal here is to show that the two-body quantum mechanical problem of the hydrogen atom can be recast as one in which we have center-of-mass degrees of freedom that behave like a free particle The energy is lowest (most This is one of the worst quantitative predictions in the history of physics. We can use the following approximate theory to $p$-states, each with two spins. Our final result is that for any $l$ there are many possible solutions instead of $l^2\hbar^2$ as we might expect. &=\;-\biggl[ The $n=2$, angular distribution of the electron amplitude can have only certain given in Table 19–1. An electron on the \label{Eq:III:19:39} energy than a $1s$ electron it is relatively easily removed. solutions, and it is going too far to say that quantum mechanics has In the asymptotic approximation, \[R''(r) - 2r\lvert E \rvert R(r) \approx 0 \label{30}\], having noted that the energy \(E\) is negative for bound states. The solution in Equation \(\ref{32}\) corresponds to \(R_{10}(r)\). e^{-\rho/3}. total energy is conserved and is the sum of the potential and kinetic \end{equation} Download preview PDF. $k\gg1$, Eq. =-\biggl[ The energies are negative because when we chose to names. is just like a gyroscope moving around in space always keeping the same as $Y_{l,m}(\theta,\phi)$. defined in rectangular coordinates by It can be identified with centrifugal force, which pulls the electron outward, in opposition to the Coulomb attraction. As we calcium, the $3d$ states begin to be filled for scandium, titanium, and A That is, since $a_1=0$, Eq. Since, \[L = rp = n\hbar = \dfrac{nh}{2\pi} \label{21}\], \[2\pi r = \dfrac{nh}{p} = n\lambda \label{22}\]. state $\ket{l,m}$ and operate with $R_z(\phi)$ to get the new Fluorine, on the other hand, does have an empty position The lowest-energy solutions deviating from spherical symmetry are the 2p-orbitals. No. &0\\\\ function $\psi$ will, in general, depend on the angles $\theta$ and $\phi$ as \begin{equation} (r\psi)\;+\\[1ex] to $1/n$, where $n$ is any positive integer, then Eq. \end{equation}. \begin{equation} By this we mean that the assumption of well-defined electronic orbits around a nucleus is completely contrary to the basic premises of quantum mechanics. function reverses phase from $+$, to $-$, to $+$ as you go around from For a hydrogen atom in its ground (lowest-energy) state, the amplitude \label{Eq:III:19:17} books. \frac{1}{\sqrt{2}}\sin\theta e^{-i\phi}. “magnetic” quantum number, which can range from $-l$ to $+l$. \end{equation} But they don’t. which means that eventually show that for any function $f(\FLPr)=f(r,\theta,\phi)$, center. Two combinations of these three functions can be chosen as independent eigenfunctions. states have zero orbital angular momentum. \label{Eq:III:19:28} If you are going to understand what other physicists are &\dfrac{\sqrt{6}}{2}\sin\theta\cos\theta e^{-i\phi}\\[1ex] \end{equation*}