The normalized wave function of the ground state of a hydrogen-like atom with nuclear charge Z e has the form ( 16.1 ) |Ψ 〉 = A e − β r , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/eqn16_1.tif"/>
In Section 1.3 , we developed the Bohr theory of the hydrogen-like atoms: H , H e + , L i 2 + https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C010_equ_0001.tif"/> , and so on. Here, we develop the modern quantum theory of these atoms. Again, we deal with a two-particle system comprised of an electron of mass 10.1 m e = 9.10938972 × 10 − 31 k g https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C010_equ_0002.tif"/>
Let  , A ^ , B ^ , C ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/ineqn10_1.tif"/> Ĉ be arbitrary linear operators. Prove that [ A ^ B ^ , C ^ ] = [ A ^ , C ^ ] B ^ + A ^ [ B ^ , C ^ ] , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/ineqn10_2.tif"/> [ A ^ , [ B ^ , C ^ ] ] + [ B ^ , [ C ^ , A ^ ] ] + [ C ^ , [ A ^ , B ^ ] ] = 0. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/ineqn10_3.tif"/>
Calculate the square A → ⋅ S → 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/ineqn13_1.tif"/> of the scalar product of an arbitrary vector A → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/ineqn13_2.tif"/> and of the spin vector S → = S x i → + S y j → + S Z k → . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/ineqn13_3.tif"/>
Abstract This chapter reiterates the quantum numbers for atomic orbitals, known from general chemistry, and places them into the context developed so far. It is sketched how the Schrodinger equation (SE) for the hydrogen atom hydrogen-like systems (one electron plus a nucleus of charge Z) is set up. When the nucleus is treated as a fixed point charge, the SE is only for the electron. The solutions of the SE can be obtained by switching to spherical polar coordinates, such that the variables are separable in terms of the electron distance from the nucleus, r, and two angles. The kinetic energy of the electron then has a radial component, and an angular component. The latter is associated with the angular momentum quantum number, which is codified by the letters s, p, d, f, and so forth. A step by step solution of the SE is provided later, in chapter 19.
Using the Planck’s formula for P n , show that The average number of photons is given by ( 4.1 ) 〈 n 〉 = 1 e x − 1 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/eqn4_1.tif"/> where x = ℏ ω k B T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/ineqn4_1.tif"/> , and k B is the Boltzmann constant. Show that for large temperatures ( T ≫ 1), the average energy is proportional to temperature, i.e., 〈 E 〉 = k B T . Calculate 〈 n 2 〉 and show that the ratio ( 4.2 ) α = 〈 n 2 〉 − 〈 n 〉 〈 n 〉 2 = 2 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/eqn4_2.tif"/>
The wave functions for the hydrogen atom are determined in the momentum representation, using momentum variables pr, ${p}_{\ensuremath{\theta}}$, ${p}_{\ensuremath{\varphi}}$ which are associated with the spherical polar variables r, $t(={e}^{i\ensuremath{\pi}cos\ensuremath{\theta}})$ and $v({e}^{i\ensuremath{\varphi}})$. The functions may be written as ${{\ensuremath{\varphi}}_{n,l,m}({p}_{r},{p}_{\ensuremath{\theta}},{p}_{\ensuremath{\varphi}})={\ensuremath{\alpha}}_{n,l}({p}_{r}){\ensuremath{\beta}}_{l}^{m}({p}_{\ensuremath{\theta}})\ensuremath{\delta}({p}_{\ensuremath{\varphi}}+m),}{{\ensuremath{\alpha}}_{n,l}({p}_{r})={\left(\frac{n}{i{p}_{0}}\right)}^{3l+z}\ensuremath{\Sigma}\stackrel{n\ensuremath{-}l\ensuremath{-}1}{k=0}\left({n\ensuremath{-}l\ensuremath{-}1}{k}\right)\frac{{2}^{k\ensuremath{-}l+1}}{(2l+k+1)(2l+k)\ensuremath{\cdots}(l+k+2)}{\left(\frac{\frac{i{p}_{0}}{n}}{{p}_{r}\ensuremath{-}\frac{i{p}_{0}}{n}}\right)}^{l+k+2},}{{\ensuremath{\beta}}_{l}^{m}({p}_{\ensuremath{\theta}})=\ensuremath{\Sigma}\stackrel{E(\frac{l}{2})}{j=E(\frac{m}{2})}{{a}_{j}{p}_{\ensuremath{\theta}}}^{\ensuremath{-}\frac{(m+1)}{2}}{J}_{\frac{j\ensuremath{-}(m\ensuremath{-}1)}{2}}({p}_{\ensuremath{\theta}}),}$ where $\ensuremath{\delta}$ is the Dirac delta function and ${p}_{0}=Z{a}_{0}^{\ensuremath{-}1}$ (a0 is the Bohr radius). ${J}_{\ensuremath{\nu}}({p}_{0})$ are Bessel functions of integer or half-integer order. The radial functions are shown to have poles up to order n+1 at pr=inp0.