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Quantum Mechanics of Rotational Motion

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Quantum Mechanics of Rotational Motion Chapter 17 Quantum Mechanics of Rotational Motion P. J. Grandinetti Chem. 4300 Would be useful to review Chapter 4 on Classical Rotational Motion. P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Rotational Angular Momentum Operators P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Rotational Angular Momentum of a System of Particles ⃗ System of particles: Ltotal divided into orbital and spin contributions ⃗ ⃗ ⃗ Ltotal = Lorbital + Lspin ⃗ ⃗ Imagine earth orbiting sun at origin with Lorbital and spinning about its center of mass with Lspin. For molecules, relabel contributions as translational and rotational ⃗ ⃗ ⃗ ; Ltotal = Ltrans + Jrot Í ⃗ Ltrans is associated with center of mass of rigid body translating relative to some fixed origin, Í ⃗ Jrot is associated with rigid body rotating about its center of mass. ⃗ ⃗ With no external torques, to good approximation, Ltrans and Jrot contributions are separately conserved. Approximate molecule as rigid body, and describe motion with Í 3 translational coordinates to follow center of mass Í 3 rotational coordinates to follow orientation of its moment of inertia tensor PAS, , i.e., Euler angles, 휙, 휃, and 휒 P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Rotational Angular Momentum Operators in the Body-Fixed Frame Angular momentum vector components for rigid body about center of mass, J⃗, are given in terms of a space-fixed frame and a body-fixed frame. ̂ ̂ ̂ ⃗ Ja, Jb, and Jc represent J components in body-fixed frame, i.e., PAS. Body-fixed frame components are given by 0 1 ) ) ) ̂ ` 휙 휙 휃 휙 휃 Ja = i * sin )휃 * cos cot )휙 + cos csc )휒 0 1 ) ) ) ̂ ` 휙 휙 휃 휙 휃 Jb = i cos )휃 * sin cot )휙 + sin csc )휒 ) ̂ ` Jc = i )휙 P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Rotational Angular Momentum Operators in the Space-Fixed Frame Angular momentum vector components for rigid body about center of mass, J⃗, are given in terms of a space-fixed frame and a body-fixed frame. ̂ ̂ ̂ ⃗ Jx, Jy, and Jz represent J components in space-fixed frame. Space-fixed frame components are given by 0 1 ) ) ) ̂ ` 휙 휙 휃 휙 휃 Jx = *i * sin )휃 * cos cot )휙 + cos csc )휒 0 1 ) ) ) ̂ ` 휙 휙 휃 휙 휃 Jy = *i cos )휃 * sin cot )휙 + sin csc )휒 ) ̂ ` Jz = *i )휙 P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Rotational Angular Momentum Operators Body-fixed frame are related to space-fixed frame components ̂2 ̂2 ̂2 ̂2 ̂2 ̂2 ̂2 J = Jx + Jy + Jz = Ja + Jb + Jc 4 5 )2 )2 )2 )2 ) Ĵ2 = *`2 csc2 휃 + csc2 휃 * 2 cot 휃 csc 휃 + + cot 휃 )휙2 )휒2 )휙휕휒 )휃2 )휃 ̂ ; ̂ `̂ ̂ ; ̂ `̂ ̂ ; ̂ `̂ Commutators in space-fixed frame: [Jx Jy] = i Jz [Jy Jz] = i Jx [Jz Jx] = i Jy ̂ ; ̂ `̂ ̂ ; ̂ `̂ ̂ ; ̂ `̂ Commutators in body-fixed frame: [Ja Jb] = *i Jc [Jb Jc] = *i Ja [Jc Ja] = *i Jb ̂ ; ̂ Space fixed components commutes with body-fixed components, e.g., [Jx Jb] = 0 < i = x; y; z [Ĵ2; Ĵ ] = 0 [Ĵ2; Ĵ ] = 0 and [Ĵ ; Ĵ ] = 0 where 훼 i i 훼 훼 = a; b; c Uncertainty principle says you can simultaneously know angular momentum vector length, 1 body-fixed frame component, and 1 space-fixed frame component. P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Rotational Angular Momentum Operators Eigenvalues of rotational angular momentum vector component operators: Í Space-fixed frame: ̂ ` ; ; ; ; ; ; ; Ji = MJ where MJ = *J *J + 1 § J * 1 J i = x y or z Í Body-fixed frame: ̂ ` ; ; ; ; ; 훼 ; ; J훼 = KJ where KJ = *J *J + 1 § J * 1 J = a b or c Eigenvalues of rotational angular momentum vector squared operator: Ĵ2 = J.J + 1/`2 : Rotational angular momentum quantum number, J can only take on integer values of J = 0; 1; 2; § : P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Rotational Kinetic Energy Operator Classical mechanics of rotational motion: kinetic energy in terms of principal components of moments of inertia tensor J2 J2 J2 K = a + b + c 2Ia 2Ib 2Ic Converting to QM, need solutions of Schrödinger equation L M Ĵ2 Ĵ2 Ĵ2 Ĥ .휃; 휙, 휒/ = a + b + c .휃; 휙, 휒/ = E .휃; 휙, 휒/ 2Ia 2Ib 2Ic .휃; 휙, 휒/, is function of molecule orientation. Euler angles, .휃; 휙, 휒/, define orientation of molecule’s moment of inertia tensor PAS relative to some inertial space-fixed frame. P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Rotational Kinetic Energy Operator Common to re-express Hamiltonian as ̂2 ̂2 ̂2 J J J 2휋 [ ] Ĥ = a + b + c = AĴ2 + BĴ2 + CĴ2 ` a b c 2Ia 2Ib 2Ic where ` ` ` ≡ ≡ ≡ A 휋 B 휋 C 휋 4 Ia 4 Ib 4 Ic are called rotational constants where A g B g C, and have dimensionality of frequency. Also common to re-express Hamiltonian as 2휋c [ ] Ĥ = 0 Ã Ĵ2 + B̃ Ĵ2 + C̃ Ĵ2 ` a b c where ` ` ` ̃ ≡ A ̃ ≡ B ̃ ≡ C A = 휋 B = 휋 C = 휋 c0 4 c0Ia c0 4 c0Ib c0 4 c0Ic are called wavenumber rotational constants, and have dimensionality of wave numbers. P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Classification of molecules Molecules grouped into 5 classes based on principal components. Name Diagonal values Examples Spherical Ia = Ib = Ic = I CH4 < Prolate Symmetric I|| = Ia Ib = Ic = I⟂ CH3F < Oblate Symmetric I⟂ = Ia = Ib Ic = I|| CHF3 < < Asymmetric Ia Ib Ic CH2Cl2, CH2CHCl Linear Ia = 0, Ib = Ic = I OCS, CO2 Molecules with two or more 3-fold or higher rotational symmetry axes are spherical tops. All molecules with one 3-fold or higher rotational symmetry axis are symmetric tops because principal moments about two axes normal to n-fold rotational symmetry axis (n g 3) are equal. P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Spherically Symmetric Molecules P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Spherically Symmetric Molecule Rotational Energy Eigenvalues For spherical symmetric molecule we have Ia = Ib = Ic = I ̂2 ̂2 ̂2 ̂2 ̂2 ̂2 ̂2 J J J Ja + J + Jc J Ĥ = a + b + c = b = 2Ia 2Ib 2Ic 2I 2I Schrödinger equation becomes Ĵ2 Ĥ .휃; 휙, 휒/ = .휃; 휙, 휒/ = E .휃; 휙, 휒/ 2I Eigenvalue of Ĵ2 is J.J + 1/`2 and J = 0; 1; 2; §. Rotational energy is J.J + 1/`2 E = = hc BJ̃ .J + 1/; J = 0; 1; 2; § J 2I 0 B̃ is wavenumber rotational constant ` ̃ ≡ B 휋 4 c0I No zero point energy. Why? Free rotation, like free translation has no zero point energy. P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Spherically Symmetric Molecule Rotational Energy Eigenstates Solutions to Schrödinger equation are 4 5 1_2 J 휙, 휃; 휒 2 + 1 .J/ 휙, 휃; 휒 J;M ;K . / = D ; . / J J 8휋2 *MJ *KJ .J/ D ; .휙, 휃; 휒/ are called Wigner D-functions, m1 m2 .J/ im 휙 .J/ im 휒 D ; .휙, 휃; 휒/ = e 1 d ; .휃/ e 2 m1 m2 m1 m2 .J/ where d ; .휃/ are called reduced Wigner-d functions and are tabulated in various texts. m1 m2 J = 0; 1; 2; § ; ; ; ; ; ; KJ = *J *J + 1 § 0 § J * 1 J body-fixed frame component ; ; ; ; ; ; MJ = *J *J + 1 § 0 § J * 1 J space-fixed frame component ` ⃗ KJ is quantum number for rotation about c in PAS. KJ is projection of J onto c of PAS frame. ` ⃗ MJ is quantum number for rotation about z in space-fixed frame. MJ is projection of J onto z of space-fixed frame. Energy is independent of KJ and MJ. P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Energy levels and degeneracies of spherical rotor molecule energy degeneracy ̃ Rotational state energy divided by hc0 gives quantity, F.J/, in wave numbers, E J = F̃ .J/ = BJ̃ .J + 1/ hc0 Each energy level, EJ, has gJ = .2J + 1/.2J + 1/ from range of both MJ and KJ values. Í ; ; ; ; KJ = *J *J + 1 § J * 1 J Í ; ; ; ; MJ = *J *J + 1 § J * 1 J ΔE in wave numbers between adjacent levels is EJ+1 * EJ ̃ ̃ = FJ+1 * FJ hc0 = B̃ .J + 1/.J + 2/* BJ̃ .J + 1/ = 2B̃ .J + 1/ P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Symmetric Molecules P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Prolate Symmetric Molecule think “cigar shaped” < In prolate symmetric molecule: Ia Ib = Ic. Define I|| = Ia & I⟂ = Ib = Ic Hamiltonian is ̂ ̂ ̂ 0 1 Ĵ2 J2 Ĵ2 Ĵ2 J2 + J2 ̂2 Ĥ a b c a b c J 1 1 ̂2 = + + = + = + * Ja 2Ia 2Ib 2Ic 2I|| 2I⟂ 2I⟂ 2I|| 2I⟂ ̂2 ̂ ̂2; ̂ Operators appearing in last Hamiltonian are J and Ja, which commute, [J Ja] = 0. ̂2 ̂ Wigner-D functions are also eigenfunctions of J and Ja: Prolate and spherically symmetric eigenstates have same form. KJ is quantum number for rotation about a axis in PAS frame. ` ⃗ KJ is projection of J onto a axis of PAS frame. | | If KJ = J prolate molecule is found rotating about axis nearly parallel to a axis. | | If KJ is small or zero prolate molecule is found rotating about b or c axes, primarily. P. J. Grandinetti Chapter 17: Quantum Mechanics of Rotational Motion Prolate Symmetric Molecules think “cigar shaped” Rotational energy of a prolate symmetric molecule is 0 1 `2 J.J + 1/ 1 1 2`2 EJ;K = + * KJ J 2I⟂ 2I|| 2I⟂ or with some rearranging becomes E ; J KJ ̃ ̃ ̃ ̃ 2 = F ; = BJ.J + 1/ + .A * B/K J KJ J hc0 Ã and B̃ are rotational constants defined as ` ` ̃ ≡ and ̃ ≡ B 휋 A 휋 4 c0I⟂ 4 c0I|| Since rotational energy is independent of orientation of J⃗ in isotropic space—space-fixed frame—rotational energy of prolate symmetric molecule is independent of MJ.
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