Cheistry 460 Fall 017 Dr. Jean M. Standard October 30, 017 Angular Moentu Properties Classical Definition of Angular Moentu In classical echanics, the angular oentu vector L is defined as L = r p, (1) where r corresponds to the position vector and p is the (linear) oentu vector. The cartesian coponents of the angular oentu vector, L x, L y, and L z, can be expressed in ters of the coponents of position and oentu using the definition of the cross product, L x = yp z zp y () L y = zp x xp z (3) L z = xp y yp x. (4) The square of the agnitude of the angular oent vector, L, is defined as for any vector as the su of the squares of the coponents, L = L x + L y + L z. (5) The agnitude L of the angular oentu vector is therefore the square root of the relation in Eq. (5), L = L x + L y + L # z 1/. (6) Angular Moentu Operators To construct quantu echanical operators for angular oentu, the basic rules for constructing operators are eployed: coordinates are transfored into ultiply by operators, and oenta are transfored into derivative operators using the relation ˆp k = i, k = x, y, or z. (7) k Using the rules for operator construction, the angular oentu coponent operators becoe ˆL x = ŷˆp z ẑˆp y = i y z z % ' (8) # y & ˆL y = ẑˆp x ˆxˆp z = i z x x % ' (9) # z & ˆL z = ˆxˆp y ŷˆp x = i x y y % '. (10) # x &
The operator for the square of the angular oentu also ay be constructed, ˆL = ˆLx + ˆL y + ˆL z. (11) Note that there is no ˆL operator in quantu echanics. Since quantu echanical operators ust be linear operators, the square root in the classical definition precludes the use of the agnitude L as an operator. Spherical Polar Coordinates The angular oentu is closely related to the angular variables θ and φ of the spherical polar coordinate syste. A point r, θ, φ ( ) in the cartesian axis syste is shown in Fig. 1. Figure 1. Diagra illustrating the spherical polar coordinates r, q, and h. The equations relating cartesian coordinates ( x, y, z) and spherical polar coordinates ( r, θ, φ) are x = r sinθ cosφ y = r sinθ sinφ z = r cosθ. (1) The ranges of the coordinates are 0 r, 0 θ π, and 0 < φ π. Solving for r, θ, φ yields ( ) in ters of ( x, y, z) r = ( x + y + z ) 1/ (13) θ = cos 1 z # x + y + z ( ) 1/ % ' ' &' (14) φ = tan 1 y % '. (15) # x &
The angular oentu coponent operators also ay be expressed in ters of spherical polar coordinates, 3 ˆL x = i sinφ % cotθ cosφ ' (16) # θ φ & ˆL y = i cosφ % cotθ sinφ ' (17) # θ φ & ˆL z = i φ. (18) The square of the angular oentu operator also ay be expressed in spherical polar coordinates, ˆ L = x + y + z = & ( ' θ + cotθ θ + 1 sin θ φ ) +. (19) * Coutators The coponents of the angular oentu operators do not coute, [ x, y ] = i z [ y, z ] = i x [ z, x ] = i y. (0) However, each of the coponents coute with the square of the angular oentu, [, x ] =, ˆ [ L y ] = ˆ [ L, z ] = 0. (1) Since and z coute, they can possess a set of siultaneous eigenfunctions. However, since x and y do not coute with z, they cannot possess the sae set of eigenfunctions as z and. Angular Moentu Eigenvalue Equations The angular oentu eigenvalue equations are ˆL Y l ( θ,φ) = l( l +1) Y l ( θ,φ) () ˆL z Y l ( θ,φ) = Y l ( θ,φ). (3) Here, l and are integers, with l = 0, 1,, and = 0, ±1, ±,, ± l. The integer l is known as the angular oentu quantu nuber and is known as the agnetic, or aziuthal, quantu nuber. The functions Y l θ,φ ( ) are known as spherical haronics, and they are discussed in ore detail below.
Eigenfunctions of Angular Moentu The eigenfunctions of the angular oentu operators are the spherical haronics, 4 Y l ( θ,φ) = ( 1) ( l +1) 4π ( l ) ( l + ) 1/ P l ( cosθ) e iφ. (4) In Eq. (4), the phase factor 1 functions, ( ) is oitted for < 0. The functions P l ( cosθ ) are associated Legendre P l 1 ( u) = l l ( 1 u) / d l+ du l+ The associated Legendre functions obey the recursion forula ( ) P l+1 ( l +1) u P l ( u) = l +1 ( u 1) l. (5) ( ) P l 1 ( u) + l + The first several associated Legendre polynoials are listed in the table below. ( u). (6) ( ) ( ) 1/ ( ) P 0 0 ( u) = 1 P 0 1 ( u) = u 0 1 P ( u) = 3u 1 1 P 1 ( u) = 1 u ( ) 1/ 1 P ( u) = 3u 1 u P u ( ) = 3 1 u As with any quantu echanical eigenfunctions, the spherical haronic functions are orthonoral, π ( θ,φ) Y l ( θ,φ) sinθ dθ dφ = δ l 'l δ '. (7) π 0 0 * Y l'' Raising and Lowering Operators The angular oentu raising and lowering operators ˆ L + and ˆ L are defined as ˆL + = ˆL x + i ˆL y (8) ˆL = ˆL x i ˆL y. (9) In spherical polar coordinates, the raising and lowering operators becoe ˆL + = e iφ θ icotθ % ' (30) # φ & ˆL = e iφ θ + icotθ % '. (31) # φ &
Soe useful coutators involving the raising and lowering operators are 5 ˆL, ˆL+ # = ˆL, ˆL # = 0 ˆL +, ˆL # = ˆL z ˆL z, ˆL+ # = ˆL + ˆL z, ˆL # = ˆL. (3) The raising and lowering operators + and have the following effects on the eigenfunctions Y l ( θ,φ), ˆL + Y l θ,φ ˆL Y l θ,φ ( ) = l( l +1) ( +1) # % 1/ Yl,+1 θ,φ ( ) = l( l +1) ( 1) ( ) (33) # 1/ % Yl, 1 ( θ,φ ). (34) Note that the raising operator raises the agnetic quantu nuber by one and the lowering operator lowers by one, but the operators have no effect on the angular oentu quantu nuber l.