XVII Hydrogen atom

We introduce the centre-of-mass coordinate $𝐑=\frac{m_e{𝐫}_e+m_p{𝐫}_p}{m_e+m_p}$ and the relative coordinate $𝐫={𝐫}_e-{𝐫}_p$. The Hamiltonian then takes the form

$$\widehat{H}=\frac{{\widehat{𝐏}}^2}{2M}+\frac{{\widehat{𝐩}}^2}{2\mu }-\frac{e^2}{4\pi {\epsilon }_0|𝐫|},$$

(333)

where $\widehat{𝐏}={\widehat{𝐩}}_p+{\widehat{𝐩}}_e$ is the total momentum, $M=m_e+m_p$ is the total mass, $\widehat{𝐩}=\frac{m_e{𝐩}_e-m_p{𝐩}_p}{m_e+m_p}$ is the momentum of the relative motion, and $\mu =\frac{m_e{m}_p}{m_e+m_p}$ is the reduced mass. Since $m_e\ll m_p$, $\mu \approx m_e$. We consider the atom at rest, $𝐏=0$. Then we have to solve the Schrödinger equation

$$E\psi (𝐫)=-\frac{{\mathrm{\hslash }}^2}{2\mu }\mathrm{\Delta }\psi (𝐫)-\frac{e^2}{4\pi {\epsilon }_{0}r}\psi (𝐫).$$

(334)

The Coulomb potential only depends on $r$ and hence is a central potential. Therefore, the solutions of the Schrödinger equation are of the form

$${\psi }_{nml}(𝐫)=R_n(r)Y_{lm}(\theta ,\varphi ),$$

(335)

where $Y_{lm}(\theta ,\varphi )$ are the spherical harmonics of Eq. (244).

The radial equation (243) now takes the form

$$-\frac{{\mathrm{\hslash }}^2}{2\mu r^2}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)+\left(\frac{{\mathrm{\hslash }}^{2}l(l+1)}{2\mu r^2}-\frac{e^2}{4\pi {\epsilon }_{0}r}\right)R=ER(r).$$

(336)

We introduce the rescaled radial coordinate

$$\rho =\sqrt{\frac{-8\mu E}{{\mathrm{\hslash }}^2}}r$$

(337)

and the new parameter

$$n=\sqrt{\frac{-\mu }{2E}}\frac{e^2}{4\pi {\epsilon }_0\mathrm{\hslash }}$$

(338)

and look for solutions $R(r)=F(\rho )e^{-\rho /2}$. Equation (336) then takes the form

$$\frac{d^{2}F}{d{\rho }^2}+\left(\frac{2}{\rho }-1\right)\frac{dF}{d\rho }+\left(\frac{n-1}{\rho }-\frac{l(l+1)}{{\rho }^2}\right)F=0.$$

(339)

This equation is known as Kummer’s differential equation. For integer values $n=l+1,l+2,l+3,\mathrm{\dots }$, the solutions are given by $F_{n,l}(\rho )={\rho }^l{L}_{n-l-1}^{2l+1}(\rho )$, where $L_s^k(\rho )$ are the Laguerre polynomials:

$$L_s^0(\rho )=e^{\rho }\frac{d^s}{d{\rho }^s}(e^{-\rho }{\rho }^s),L_s^k(\rho )={(-1)}^k\frac{d^k}{d{\rho }^k}{L}_{s+k}^0(\rho ).$$

(340)

Since $n=\sqrt{\frac{-\mu }{2E}}\frac{e^2}{4\pi {\epsilon }_0\mathrm{\hslash }}$ the energy levels of the Hydrogen atom only depend on the principal quantum number $n$ but not on $l$ or $m$:

$$E_n=-\frac{\mu }{2}{\left(\frac{e^2}{4\pi {\epsilon }_0\mathrm{\hslash }}\right)}^2\frac{1}{n^2}.$$

(341)

The energies can also be written as $E_n=-\mathrm{Ry}\frac{1}{n^2}$ where Ry$=\frac{e^2}{8\pi {ϵ}_0{a}_0}=13.6$ eV is the so-called Rydberg energy and $a_0=\frac{4\pi {ϵ}_0{\mathrm{\hslash }}^2}{\mu e^2}\approx 0.53\times {10}^{-10}$m is called the Bohr radius.

The lowest energy level has energy $E_1=-\mathrm{Ry}=-13.6$ eV. For energy $E>0$, the electrons are not bound to the nucleus. Hence the Rydberg energy is the amount of energy required to release the electron from the nucleus (this process is called ionisation).

The electron can change from one energy level $n$ to another level $n^{\prime }$ if it absorbs or emits a photon which carries the right amount of energy $\mathrm{\hslash }\omega =E_n-E_{n^{\prime }}$. This gives rise to discrete spectral lines. The most important lines are grouped into series, such as the Lyman series ($n^{\prime }=1$), the Balmer series ($n^{\prime }=2$), and the Paschen series ($n^{\prime }=3$).

For each $n$ the allowed values of $l$ are $0,1,2,\mathrm{\dots },n-1$, and for each $l$ there are $2l+1$ allowed values of $m$. Hence, the degeneracy of each energy level $E_n$ is ${\sum }_{l=1}^{n-1}(2l+1)=n^2$. Later we will learn that the electron has an intrinsic degree of freedom called spin so that the degeneracy of each energy level is really $2n^2$. On the other hand we neglected small perturbations which lift the degeneracy and result in a fine structure of the energy levels. They will be discussed in the third year module PHYS321, Atomic & Nuclear Physics.

The wavefunctions

$${\psi }_{nlm}(𝐫)=Y_{lm}(\theta ,\varphi )F_{n,l}(2r/(n{a}_0))$$

(342)

of the hydrogen atom are also called atomic orbitals. The azimuthal quantum number is denoted by a symbol s for $l=0$, p for $l=1$, d for $l=2$, and f for $l=3$. These symbols are then preceded by the principal wavenumber $n$, so that orbitals are denoted by 1s, 2s, 2p, 3s, 3p, 3d etc.

We consider the hydrogen atom in a constant magnetic field $𝐁=B_z𝐤$, which points into the $z$ direction. The total magnetic moment of the electron is [see Eqs. (274) and (280)]

$$\widehat{𝐦}=-\frac{e}{2m_e}(\widehat{𝐋}+2\widehat{𝐒}).$$

(348)

The interaction energy of the magnetic moment with the magnetic field is

$${\widehat{V}}_B=-𝐁\cdot \widehat{𝐦}=\frac{e{B}_z}{2m_e}({\widehat{L}}_z+2{\widehat{S}}_z).$$

(349)

The total Hamiltonian is given by

$$\widehat{H}=\frac{{\widehat{𝐩}}^2}{2\mu }-\frac{e^2}{4\pi {\epsilon }_0|𝐫|}+{\widehat{V}}_B.$$

(350)

The Schrödinger equation $E\psi =H\psi$ is solved by

	${\psi }_{nml\uparrow }$	$=$	${\psi }_{nml}(𝐫)\left({\displaystyle \genfrac{}{}{0pt}{}{1}{0}}\right),(m_s=1/2)$		(351)
	${\psi }_{nml\downarrow }$	$=$	${\psi }_{nml}(𝐫)\left({\displaystyle \genfrac{}{}{0pt}{}{0}{1}}\right),(m_s=-1/2).$		(352)

where ${\psi }_{nml}(𝐫)$ are the orbital wavefunctions of the hydrogen atom. These functions are eigenfunctions of ${\widehat{V}}_B$:

$${\widehat{V}}_B{\psi }_{nml,m_s}(𝐫)=(m+2m_s){\mu }_B{B}_z{\psi }_{nml,m_s}(𝐫),$$

(353)

where ${\mu }_B=e\mathrm{\hslash }/2m_e$ is the Bohr magneton. However, the degeneracy of the eigenvalues is now lifted,

$$E_{nml}=-\frac{e^2}{8\pi {ϵ}_0{a}_0}\frac{1}{n^2}+(m+2m_s){\mu }_B{B}_z.$$

(354)

The splitting of the levels depends linearly on the strength $B_z$ of the magnetic field.

The splitting of the orbital energies can be observed in the spectral lines of the hydrogen atom. This is called the Zeeman effect. The observable spectral lines is restricted by selection rules: $m$ can only change by $-1$, $0$, or $1$, because of properties of the emitted photons. Since $m_s$ cannot change, the spin splitting is not observed in the normal Zeeman effect.