Eulerian Angles I. THE EULERIAN ANGLES. FIG. 1: First Rotation in the Euler Scheme. l2h2:euler2.tex. P(x,y,z)=P(x. N y. φ x x

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1 Eulerian Angles I. THE EULERIA AGLES There is no question that once we leave diatomic molecules, life gets difficult. Most chemistr concerns polatomics, so sticking with diatomics is possibl ecessivel pedantic. Be that as it ma, we need to discuss rotational aspects of polatomic species. To begin the stud of polatomic molecules we need to define rotations in three dimensions in a manner which is rigorous. We seek three independent angles, coordinates in a sense, which will suffice to allow us to write the Hamiltonian for the rotation in a clear, unambiguous, manner. Our Eulerian angles are not taken from Goldstein which used to be the gold, no pun intended, standard in this field) [1] Instead, we now shift to a more current standard, which is eemplified b Zare [2] We need three angles, three rotations, which will unambiguousl allow us to rotate a coordinate sstem from a starting to a final configuration. The order of operation will be prescribed, and remain inviolate. We start with the familiar polar angle from spherical polar coordinates, φ, which we take over completel as the first Euler angle. We rotate from the coordinate sstem space, space, z space to {,, z } where z is colinear with z space, since this first rotation is a rotation about the original z-ais. The new -ais is called the line of nodes, hence the subscript.) Again, from spherical polar coordinates, we take the positive direction as. z P,,z)=P,,z) φ FIG. 1: First Rotation in the Euler Scheme et, from the {,, z } sstem, we rotate about the -ais counterclockwise b an angle which we will call θ. This second rotation leads us from {,, z } to { r, r, z r }. Finall, we rotate about the new) z u ais, again counterclockwise, to go from { r, r, z r } to { f, f, z f } where the subscript f stands for final. This last angle is called χ. l2h2:euler2.te

2 2 z z u ϑ P,,z)=P,,z), u u FIG. 2: Second Rotation in the Euler Scheme showing the ais of rotation, ˆn θ z u P,,z)=P,,z) F, u χ F FIG. 3: Third Rotation in the Euler Scheme showing the ais of rotation, ˆn χ It is clear that we have for the first rotation cos φ sin φ 0 = sin φ cos φ 0 space space 1) z z space

3 and for the second rotation about the newl created ais, which is now called the line of nodes, u cos θ 0 sin θ u = ) z u sin θ 0 cos θ z and finall, once again about a z-ais, but this time the new one, f cos χ sin χ 0 f = sin χ cos χ 0 u u 3) z f z u The final coordinates are called the bod coordinates, compared to the original set, which is referred to as the space coordinates or lab coordinates). The product of these three rotations, in the proper invariable) order, will locate the bod coordinates. Thus, the bod coordinates are connected to the space coordinates through the transformation: cos χ sin χ 0 sin χ cos χ cos θ 0 sin θ sin θ 0 cos θ cos φ sin φ 0 sin φ cos φ ) We take these composite rotation matri multiplications in order, i.e., first the φ rotation, then the θ rotation ielding cos θ 0 sin θ cos φ sin φ sin φ cos φ 0 5) sin θ 0 cos θ which ields cos θ cos φ cos θ sin φ sin θ sin φ cos φ 0 sin θ cos φ sin θ sin φ cos θ and finall, including the χ rotation, we have cos χ sin χ 0 cos θ cos φ cos θ sin φ sin θ sin χ cos χ 0 sin φ cos φ sin θ cos φ sin θ sin φ cos θ We finall arrive at the overall transformation cos χ cos θ cos φ sin φ sin χ cos χ cos θ sin φ + cos φ sin χ cos χ sin θ sin χ cos θ cos φ sin φ cos χ sin χ cos θ sin φ + cos φ cos χ sin χ sin θ 6) sin θ cos φ sin θ sin φ cos θ The angle φ is the eact same angle as that used in spherical polar coördinates, i.e., is familiar. θ measures the tilt down from the original z-ais, and is also famliar from spherical polar coördinates. The new angle, χ, is uncommon. We have derived the transformation which takes the space coördinates into the bod coördinates, i.e., II. bod = Φ space bod space where Φ is a 33 matri above). Φφ, θ, χ) is a function of the three Euler angles. As notes above, this overall transformation is a compound of three different rotations, Rφ), Rθ), Rχ), in reverse order, i.e., Φ = Rχ) Rθ) Rφ)

4 As usual, these are written from right to left, i.e., we do Rφ) first, then Rθ), and finall Rχ). There must be an inverse transformation which takes the bod coördinates back into the space coördinates, i.e., Φ 1. This must have the form Φ 1 = R 1 φ) R 1 θ) R 1 χ) As with all rotations in three dimensions, the result of doing and then undoing must be the unit operation, i.e., no operation at all. Thus Φ 1 Φ = 1 4 III. IFITESIMAL ROTATIOS I QM Consider a sstem described b a set of eigenfunctions i >, such that where i > is a new set of basis functions. We then have i > = U op i > < i i > =< i U opu op i > where the superscript dagger implies transpose comple conjugate. This last equation is the same as the radius preserving rotation in space, the lengths and angles between the unprimed and the primed sstem are maintained during the linear transformation U op, if U op = Uop 1 = I. The matri representative of an operator will change upon this kind of transformation. Consider the operator Q op so that and This means that the operator has been transformed, i.e., < i Q op j > Q i,j < i U opu op Q op U opu op j >=< i U op Q op U op j > = Q i,j Q op = U op Q op U op = Q op = U op Q op U 1 op If the transformation U op a unitar transformation) can be written as U op = I op + ıɛr op where R op is going to be an infinitesimal rotation operator, then two such transformations, compounded, would have the form Uop 2 = I op + ı ɛ ) 2 2 R op i.e., if we did each of these for ɛ/2 the net would be as if we did the original for ɛ, i.e., U 2 op = I op + ı ɛ 2 R op) 2 = Uop If we did this over and over again, using smaller and smaller increments then we would eventuall have U op = lim I op + ı ɛ n op) n n R 7) This is a fanc wa of writing U op = e ıɛrop) 8) which can be verified b taking the Talor epansion of this eponential form Equation 8) and identifing each term as being present in the product s epansion Equation 7).

5 5 IV. FIITE VS. IFIITESIMAL ROTATIOS One can not associate a vector with a finite rotation represented b an orthogonal radius preserving) transformation. The direction would be the ais of rotation, and the magnitude would be the angle of rotation, or something related to it. But, since the rotations, in three dimensions, were not commutative, i.e., Rot 1 Rot 2 Rot 2 Rot 1, neither Rot 1 nor Rot 2 can be true vectors. On the other hand, for infinitesimal rotations, this association of a rotation with a vector is perfectl plausible. Consider the finite rotation about the z-ais cos φ sin φ 0 R z = sin φ cos φ V. when φ is infinitesimal, i.e., close to zero. Then the cosine is approimatel 1, and if the infinitesimal angle is δφ then one has R z = 1 δφ 0 δφ Consider R z φ) as rotating about the z-ais from φ 0 to φ 0 + φ i.e., which means that the effect of the rotation would be to substitute the value at φ 0 φ for the new value φ new, i.e., g φ) Rg φ ) Rg φ) = g φ δφ ) δφ φ FIG. 4: Rg)? Here, we have that Rgφ) picks up an earlier or older value of g R z φ) φ 0 >= φ new >= φ 0 φ >

6 6 in the space coördinate sstem. Then ) φ > φ new >= φ 0 > + φ + φ=φ 0 which could be written as so and we then write and remember that L z = ı h φ new >= 1 φ n φ 0 φ >= φ new >= R z φ) = e φ ) n ) e φ φ 0 > and we choose a reduced coördinate scheme where h = 1: R z φ) = e ıφlz VI. Let ψ = ψ,, z), and let R z δα) mean that ψ,, z) becomes ψ + δα, δα, z) which is the meaning of rotating about the z-ais be δα. Epanding ψ + δα, δα, z) in a Talor series, we have remember, ψ,, z) after rotation = ψ + δα, δα, z) = ψ,, z) + this alternate interpreation). Since δψ,, z) δα ψ + δα, δα, z) = ψ + δα, δα, z) = ψ,, z) ψ,, z) = + ψ,, z) ψ,, z) δα δα + [ 1 + δα )] ψ,, z) + [ 1 δα )] ψ,, z) L z = ı h ) we would have α = L ı For an infinitesimal rotation, we have again, having h = 1) and if α = n ) α n then and δα = α n which is R z = 1 + L z ı δα then in the limit, n goes to infinit we have R z = 1 + L ) n z α ı n e α Lz ı = e ıαlz If we use an ais other than the z-ais, then we need to define a vector whose direction is that ais. Call that vector ˆn, then for a general rotation R n α) = e ıα L ˆn

7 7 or VII. AOTHER DERIVATIO OF RESULT α = L ı From the polar coördinate definitions, we have ) 1 + sin2 φ cos 2 dφ = d φ 2 d ) 1 cos 2 dφ = d φ 2 d which leads to and so and which leads to ) = cos φ 9) ) = sin φ 10) = cos2 φ = sin2 φ L ˆk = ı in our case, and in general L ˆn = ı α where ˆn is the unit ais around which the rotation is taking place. If the sstem is invariant under this rotation about this ais, then the component of angular momentum L n is a constant of the motion. For our three dimensional problem, in Euler angles, we have L nˆ φ = ı L ˆn θ = ı θ L nˆ χ = ı χ VIII. EULER AGLE REPRESETATIO The Euler Angle rotations can be represented as Rφ, θ, χ) = e ıχ L ˆn χ e ıθ L ˆn θ e ıφ L ˆn φ

8 where this form has been chosen to represent the rotation of the phsical sstem, rather than the coördinate sstem. This rotation is equivalent to first a rotation of χ about the original z-ais, then a rotation of θ about the new Y-ais, and then finall a rotation of φ about the new z-ais. We are using the Euler angle representation which emplos three unit vectors ˆn φ, ˆn θ and ˆn χ associated with the three Euler angles. We need to epress these in terms of the Cartesian unit vectors in the space sstem, î, ĵ, and ˆk. We know that ˆn φ = ˆk since the first rotation is about the original space) z-ais. This means that ı = L z The new -ais after the first φ) rotation, is -sin φî + cos φĵ, as seen in Figure VIII. Thus 8 nϑ ^ sin ϕ cos ϕ cos ϕ sin ϕ ϕ mies with z ais under net rotation ϑ) FIG. 5: The φ rotation changes the ais of rotation for the second θ) rotation. ˆn θ = sin φî + cos φĵ After the first φ) rotation, the -ais ) is transformed into cos φî + sin φĵ which is now itself rotated in the second θ) rotation into sin θ cos φî + sin φĵ so ) ˆn χ = sin θ cos φî + sin φĵ + ˆk cos θ which means that ı = L z 11) ı θ = sin φl + cos φl 12) ı χ = sin θ cos φl + sin θ sin φl + cos θl z 13) From these equations, we can solve for the components of angular momentum directl, i.e., multipling Equation 12 through b sin φ we get L z = ı ı sin φ θ = sin2 φl + sin φ cos φl

9 9 n χ cos ϑ z ϑ sin ϑ cos ϑ sin ϑ ϑ FIG. 6: The θ rotation changes the ais of rotation for the third χ) rotation. multipling Equation 13 through b cos φ we obtain and adding, ields or or, in final form +ı cos φ sin θ χ = cos2 φl cos φ sin φl + cos φ cot θl z ı sin φ θ + ıcos φ sin θ χ = L + ı cos φ cot θ L = +ı sin φ 1 θ ı cos φ sin θ χ cot θ ) L = ı sin φ 1 θ cos φ sin θ χ cot θ )) or From Equation 12 we now obtain cos φl ı θ = sin φ +ı sin φ 1 θ ı cos φ sin θ χ cot θ )) 1 L = ı cos φ θ sin φ cos φ ı sin φ θ 1 + sin φı sin θ χ cot θ ) which is or, in final form L = ı cos φ 1 θ + sin φ sin θ χ cot θ )) L = ı cos φ 1 θ sin φı sin θ χ cot θ )

10 10 IX. THE HAMILTOIA ow we need to obtain L îl + ĵl + ˆkL z which will be used to form the Hamiltonian, presumabl b forming h2 2I L 2 h2 2I L 2 h2 2I z L 2 z but it is not clear what the relation is between the I, I, and I z on the one hand, and the I A, I B, and I C values from the principal ais transformation on the other hand. We quote the final Schrödinger Equation for smmetric tops as [3] 1 sin θ ) sin θ ψ θ θ ψ sin 2 θ 2 + cot 2 θ + C ) 2 ψ B χ 2 2 cotθ sin θ 2 ψ χ + W hb ψ = 0 Here, θ and φ are equivalent to the usual polar angles between an ais fied space and some ais fied in the molecule, and χ is the angle of rotation around the ais fied in the molecule. For a smmetric top, this chosen ais is natuall the molecular or smmetr ais. C is the totational constant for the smmetr ais and B is h2 2I B for the ais perpendicular to the smmetr ais. We note that this result is not derived in Townes and Schawlow, but referenced back to Kemble [4] where it is also not derived. [1] H. Goldstein, Classical Mechanics, Addison-Wesle Pub. Co., Inc. Reading, Massachusetts, 1959, page107 [2] R.. Zare, Angular Momentum, John Wile & Sons, ew York, [3] C. H., Townes and A. L. Schawlow, MicroWave Spectroscop, McGraw-Hill Book Co., ew York, 1955, page 61 [4] E. C. Kemble, The Fundamental Principles of Quantum Mechanics, McGraw-Hill Book Co., ew York, 1937