Editing Rotational transition

{{Short description|Abrupt change in a quantum particle's angular momentum}}

In [[quantum mechanics]], a '''rotational transition''' is an abrupt change in [[angular momentum]].  Like all other properties of a quantum [[Elementary particle|particle]], [[angular momentum quantization|angular momentum is quantized]], meaning it can only equal certain discrete values, which correspond to different [[rotational energy]] states.  When a particle loses angular momentum, it is said to have transitioned to a lower rotational energy state.  Likewise, when a particle gains angular momentum, a positive rotational transition is said to have occurred.

Rotational transitions are important in physics due to the unique [[spectral lines]] that result. Because there is a net gain or loss of energy during a transition, [[electromagnetic radiation]] of a particular [[frequency]] must be absorbed or emitted. This forms [[spectral line]]s at that frequency which can be detected with a [[spectrometer]], as in [[rotational spectroscopy]] or [[Raman spectroscopy]].

== Diatomic molecules ==
Molecules have [[rotational kinetic energy|rotational energy]] owing to rotational motion of the nuclei about their [[center of mass]]. Due to [[Quantization (physics)|quantization]], these energies can take only certain discrete values. Rotational transition thus corresponds to transition of the molecule from one rotational energy level to the other through gain or loss of a [[photon]]. Analysis is simple in the case of [[diatomic molecules]].

=== Nuclear wave function ===
Quantum theoretical analysis of a molecule is simplified by use of [[Born–Oppenheimer approximation]]. Typically, rotational energies of molecules are smaller than [[molecular electronic transition|electronic transition]] energies by a factor of {{math|''m''/''M''}} ≈ 10<sup>−3</sup>–10<sup>−5</sup>, where {{math|''m''}} is electronic mass and {{math|''M''}} is typical nuclear mass.<ref>Chapter 10, ''Physics of Atoms and Molecules'', B.H. Bransden and C.J. Jochain, Pearson education, 2nd edition.</ref> From [[uncertainty principle]], period of motion is of the order of the [[Planck constant]] {{math|''h''}} divided by its energy. Hence nuclear rotational periods are much longer than the electronic periods. So electronic and nuclear motions can be treated separately. In the simple case of a diatomic molecule, the radial part of the [[Schrödinger Equation]] for a nuclear wave function {{math|''F''<sub>''s''</sub>('''R''')}}, in an electronic state {{math|''s''}}, is written as (neglecting spin interactions)
<math display="block">\left[- \frac{\hbar^2}{2\mu R^2} \frac{\partial}{\partial R} \left(R^2 \frac{\partial}{\partial R}\right)+ \frac{\langle \Phi_s|N^2|\Phi_s \rangle}{2\mu R^2}+ E_s(R)-E\right]F_s(\mathbf R) = 0 </math>
where {{math|''μ''}} is [[reduced mass]] of two nuclei, {{math|'''R'''}} is vector joining the two nuclei, {{math|''E''<sub>''s''</sub>(''R'')}} is energy [[eigenvalue]] of electronic wave function {{math|Φ<sub>''s''</sub>}} representing electronic state {{math|''s''}} and {{math|''N''}} is orbital [[momentum operator]] for the relative motion of the two nuclei given by
<math display="block"> N^2 = -\hbar^2 \left[ \frac{1}{\sin\Theta} \frac{\partial}{\partial \Theta}\left(\sin \Theta \frac{\partial}{\partial \Theta}\right)+ \frac{1}{\sin^2\Theta} \frac{\partial^2}{\partial \Phi^2} \right] </math>
The total [[wave function]] for the molecule is
<math display="block"> \Psi_s = F_s(\mathbf R)\Phi_s(\mathbf R,\mathbf r_1, \mathbf r_2, \dots, \mathbf r_N)</math>
where {{math|'''r'''<sub>''i''</sub>}} are position vectors from center of mass of molecule to {{math|''i''}}th electron.
As a consequence of the Born-Oppenheimer approximation, the electronic wave functions {{math|Φ<sub>s</sub>}} is considered to vary very slowly with {{math|'''R'''}}. Thus the Schrödinger equation for an electronic wave function is first solved to obtain {{math|''E''<sub>''s''</sub>(''R'')}} for different values of {{math|''R''}}. {{math|''E''<sub>''s''</sub>}} then plays role of a [[potential well]] in analysis of nuclear wave functions {{math|''F''<sub>''s''</sub>('''R''')}}.

[[File:Angular Momentum of a Diatomic Molecule.png|thumb|Vector addition triangle for orbital angular momentum of a diatomic molecule with components of orbital angular momentum of nuclei and orbital angular momentum of electrons, neglecting coupling between electron and nuclear orbital motion and spin-dependent coupling.Since angular momentum {{math|'''N'''}} of nuclei is perpendicular to internuclear vector {{math|'''R'''}}, components of electronic angular momentum {{math|'''L'''}} and total angular momentum {{math|'''J'''}} along {{math|'''R'''}} are equal.]]

=== Rotational energy levels ===
The first term in the above nuclear wave function equation corresponds to [[kinetic energy]] of nuclei due to their radial motion. Term {{math|{{sfrac|{{bra|Φ<sub>''s''</sub>}} ''N''<sup>2</sup> {{ket|Φ<sub>''s''</sub>}}|2''μR''<sup>2</sup>}}}} represents rotational kinetic energy of the two nuclei, about their center of mass, in a given electronic state {{math|Φ<sub>''s''</sub>}}. Possible values of the same are different rotational energy levels for the molecule.

[[Angular momentum|Orbital angular momentum]] for the rotational motion of nuclei can be written as
<math display="block"> \mathbf N = \mathbf J - \mathbf L </math>
where {{math|'''J'''}} is the total orbital angular momentum of the whole molecule and {{math|'''L'''}} is the orbital angular momentum of the electrons.
If internuclear vector {{math|'''R'''}} is taken along z-axis, component of {{math|'''N'''}} along z-axis – {{math|''N''<sub>''z''</sub>}} – becomes zero as
<math display="block"> \mathbf N = \mathbf R \times \mathbf P  </math>
Hence
<math display="block"> J_z = L_z </math>
Since molecular wave function Ψ<sub>s</sub> is a simultaneous [[eigenfunction]] of {{math|''J''<sup>2</sup>}} and {{math|''J''<sub>''z''</sub>}},
<math display="block"> J^2 \Psi_s = J(J+1) \hbar^2 \Psi_s  </math>
where J is called [[quantum number|rotational quantum number]] and {{math|''J''}} can be a positive integer or zero.
<math display="block"> J_z \Psi_s = M_j\hbar \Psi_s </math>
where {{math|−''J'' ≤ ''M''<sub>''j''</sub> ≤ ''J''}}.

Also since electronic wave function {{math|Φ<sub>''s''</sub>}} is an eigenfunction of {{math|''L''<sub>''z''</sub>}},
<math display="block"> L_z \Phi_s = \pm \Lambda\hbar \Phi_s </math>
Hence molecular wave function {{math|Ψ<sub>s</sub>}} is also an eigenfunction of {{math|''L''<sub>''z''</sub>}} with eigenvalue {{math|±Λ''ħ''}}.
Since {{math|''L''<sub>''z''</sub>}} and {{math|''J''<sub>''z''</sub>}} are equal, {{math|Ψ<sub>''s''</sub>}} is an eigenfunction of {{math|''J''<sub>''z''</sub>}} with same eigenvalue {{math|±Λ''ħ''}}. As {{math|{{abs|'''J'''}} ≥ ''J''<sub>''z''</sub>}}, we have {{math|''J'' ≥ Λ}}. So possible values of rotational quantum number are
<math display="block"> J = \Lambda, \Lambda +1, \Lambda+2, \dots </math>
Thus molecular wave function {{math|Ψ<sub>''s''</sub>}} is simultaneous eigenfunction of {{math|''J''<sup>2</sup>}}, {{math|''J''<sub>''z''</sub>}} and {{math|''L''<sub>''z''</sub>}}.
Since molecule is in eigenstate of {{ math|''L''<sub>''z''</sub>}}, expectation value of components perpendicular to the direction of z-axis (internuclear line) is zero. Hence
<math display="block"> \langle \Psi_s|L_x|\Psi_s\rangle = \langle L_x \rangle = 0 </math>
and
<math display="block"> \langle \Psi_s|L_y|\Psi_s\rangle = \langle L_y \rangle = 0 </math>
Thus
<math display="block"> \langle \mathbf J . \mathbf L \rangle = \langle J_z L_z \rangle = \langle {L_z}^2 \rangle </math>

Putting all these results together,
<math display="block"> \begin{align}
\langle \Phi_s |N^2|\Phi_s \rangle F_s(\mathbf R) &= \langle \Phi_s | \left(J^2 + L^2 - 2 \mathbf J \cdot \mathbf L\right) |\Phi_s \rangle F_s(\mathbf R) \\
&= \hbar^2 \left[J(J+1)-\Lambda^2\right] F_s(\mathbf R) + \langle \Phi_s | \left({L_x}^2 + {L_y}^2\right) |\Phi_s \rangle F_s(\mathbf R)
\end{align}</math>

The Schrödinger equation for the nuclear wave function can now be rewritten as
<math display="block">- \frac{\hbar^2}{2\mu R^2}\left[ \frac{\partial}{\partial R} \left(R^2 \frac{\partial}{\partial R}\right)- J(J+1)\right]F_s(\mathbf R)+[{E'}_s(R)-E]F_s(\mathbf R) = 0 </math>
where
<math display="block"> {E'}_s(R) = E_s(R) - \frac{\Lambda^2 \hbar^2}{2\mu R^2} + \frac{1}{2\mu R^2} \langle \Phi_s |\left({L_x}^2 + {L_y}^2\right)|\Phi_s \rangle </math>
E&prime;<sub>s</sub> now serves as effective potential in radial nuclear wave function equation.

==== Sigma states ====
Molecular states in which the total orbital momentum of electrons is zero are called [[molecular orbitals|sigma states]]. In sigma states {{math|1=Λ = 0}}. Thus {{math|1=''E''&prime;<sub>s</sub>(''R'') = ''E''<sub>s</sub>(''R'')}}. As nuclear motion for a stable molecule is generally confined to a small interval around {{math|''R''<sub>0</sub>}} where {{math|''R''<sub>0</sub>}} corresponds to internuclear distance for minimum value of potential {{math|''E''<sub>s</sub>(''R''<sub>0</sub>)}}, rotational energies are given by,
<math display="block"> E_r = \frac{\hbar^2}{2\mu {R_0}^2} J(J+1) = \frac{\hbar^2}{2I_0} J(J+1) = BJ(J+1) </math>
with
<math display="block"> J = \Lambda, \Lambda +1, \Lambda+2, \dots </math>
{{math|''I''<sub>0</sub>}} is [[moment of inertia]] of the molecule corresponding to [[mechanical equilibrium|equilibrium]] distance {{math|''R''<sub>0</sub>}} and {{math|''B''}} is called '''rotational constant''' for a given electronic state {{math|Φ<sub>''s''</sub>}}.
Since reduced mass {{math|''μ''}} is much greater than electronic mass, last two terms in the expression of {{math|''E''&prime;<sub>''s''</sub>(''R'')}} are small compared to {{math|''E''<sub>s</sub>}}. Hence even for states other than sigma states, rotational energy is approximately given by above expression.

=== Rotational spectrum ===
{{main|Rotational spectroscopy}}
When a rotational transition occurs, there is a change in the value of rotational quantum number {{math|''J''}}. Selection rules for rotational transition are,
when {{math|1=Λ = 0}}, {{math|1=Δ''J'' = ±1}} and
when {{math|Λ ≠ 0}}, {{math|1=Δ''J'' = 0, ±1}} as absorbed or emitted photon can make equal and opposite change in total nuclear angular momentum and total electronic angular momentum without changing value of {{math|''J''}}.

The pure rotational spectrum of a diatomic molecule consists of lines in the far [[infrared]] or [[microwave]] region. The frequency of these lines is given by
<math display="block"> \hbar \omega = E_r(J+1)-E_r(J) = 2B(J+1) </math>
Thus values of {{math|''B''}}, {{math|''I''<sub>0</sub>}} and {{math|''R''<sub>0</sub>}} of a substance can be determined from observed rotational spectrum.

== See also ==
* [[Vibrational transition]]

== Notes ==
{{reflist}}

== References ==
* {{cite book | author=B.H.Bransden C.J.Jochain | title=Physics of Atoms and Molecules | publisher=Pearson Education }}
* {{cite book | author=L.D.Landau E.M.Lifshitz | title=Quantum Mechanics (Non-relativistic Theory) | publisher=Reed Elsvier }}

{{DEFAULTSORT:Rotational Transition}}
[[Category:Chemical physics]]