Selection Rules

Let us now consider spontaneous transitions between the different energy levels of a hydrogen atom. Since the perturbing Hamiltonian (1093) does not contain any spin operators, we can neglect electron spin in our analysis. Thus, according to Sect. 9.4, the various energy eigenstates of the hydrogen atom are labeled by the familiar quantum numbers $n$, $l$, and $m$.

According to Eqs. (1122) and (1131), a hydrogen atom can only make a spontaneous transition from an energy state corresponding to the quantum numbers $n$, $l$, $m$ to one corresponding to the quantum numbers $n'$, $l'$, $m'$ if the modulus squared of the associated electric dipole moment

$$d^2 = \vert\langle n,l,m\vert e x\vert n',l',m'\rangle\ver... ...2 + \vert\langle n,l,m\vert e z\vert n',l',m'\rangle\vert^2$$ (1144)

is non-zero. Now, we have already seen, in Sect. 12.5, that the matrix element $\langle n,l,m\vert z\vert n',l',m'\rangle$ is only non-zero provided that $m'=m$ and $l'=l\pm 1$. It turns out that the proof that this matrix element is zero unless $l'=l\pm 1$ can, via a trivial modification, also be used to demonstrate that $\langle n,l,m\vert x\vert n',l',m'\rangle$ and $\langle n,l,m\vert y\vert n',l',m'\rangle$ are also zero unless $l'=l\pm 1$. Consider

$$x_\pm = x + {\rm i} y.$$ (1145)

It is easily demonstrated that

$$[L_z, x_\pm]= \pm \hbar x_\pm.$$ (1146)

Hence,

$$\langle n,l,m\vert[L_z,x_+]-\hbar x_+\vert n',l',m'\rangle... ...,(m-m'-1) \langle n,l,m\vert x_+\vert n',l',m'\rangle = 0,$$ (1147)

and

$$\langle n,l,m\vert[L_z,x_-]+\hbar x_-\vert n',l',m'\rangle... ...,(m-m'+1) \langle n,l,m\vert x_-\vert n',l',m'\rangle = 0.$$ (1148)

Clearly, $\langle n,l,m\vert x_+\vert n',l',m'\rangle$ is zero unless $m'=m-1$, and $\langle n,l,m\vert x_-\vert n',l',m'\rangle$ is zero unless $m'=m+1$. Now, $\langle n,l,m\vert x\vert n',l',m'\rangle$ and $\langle n,l,m\vert y\vert n',l',m'\rangle$ are obviously both zero if $\langle n,l,m\vert x_+\vert n',l',m'\rangle$ and $\langle n,l,m\vert x_-\vert n',l',m'\rangle$ are both zero. Hence, we conclude that $\langle n,l,m\vert x\vert n',l',m'\rangle$ and $\langle n,l,m\vert y\vert n',l',m'\rangle$ are only non-zero if $m'=m\pm 1$.

The above arguments demonstrate that spontaneous transitions between different energy levels of a hydrogen atom are only possible provided

$$l' = l\pm 1,$$ (1149)

$$m' = m, m\pm 1.$$ (1150)

These are termed the selection rules for electric dipole transitions (i.e., transitions calculated using the electric dipole approximation). Note, finally, that since the perturbing Hamiltonian does not contain any spin operators, the spin quantum number $m_s$ cannot change during a transition. Hence, we have the additional selection rule that $m_s'=m_s$.