1 Properties of Spherical Harmonics
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1 Proerties of Sherical Harmonics. Reetition In the lecture the sherical harmonics Y m were introduced as the eigenfunctions of angular momentum oerators lˆz and lˆ2 in sherical coordinates. We found that lˆzy m "my m [.] and lˆ2y m "( + )Y m. [.2] The sherical harmonics can be defined as N m Y m P m ( cosq) e imf [.] where is the quantum number of the orbital angular momentum and m the magnetic quantum number. There are analytical definitions for the normalization factor and the associated Legendre Polynomials ( cosq) that allow the m N m P calculations of the sherical harmonics. The sherical harmonics for and 2 are given by Y [.4] Y Y ± cosq sinqe ± if 2 [.5]
2 2 Chater Y ( cos 2 q ) Y 2 ± sinqcos 2 qe ± if. [.6] Y 2 ± sin 2 qe ± 2if 2 Note that the sign of the functions Y ± and Y 2 ± is defined differently than in the scrit of the lecture. The definition here is in agreement with most of the literature on sherical harmonics..2 Grahical Reresentation of Sherical Harmonics The sherical harmonics are often reresented grahically since their linear combinations corresond to the angular functions of orbitals. Figure.a shows a lot of the sherical harmonics where the hase is color coded. One can clearly see that Y m is symmetric for a rotation about the z axis. The linear combinations 2( Y m + ( ) m Y m ) 2 Y m cos( m Y and i 2( Y m ( ) m Y m ) 2 Y m sin( m are always real and have the form of tyical atomic orbitals that are often shown.. Proerties of Sherical Harmonics There are some imortant roerties of sherical harmonics that simlify working with them... Orthogonality and Normalization The sherical harmonics are normalized and orthogonal i.e. 2 * Y m Y 2 m 2 sin q ( d q ) df d m m 2 d 2 [.7] where the Kronecker delta is defined as
3 Proerties of Sherical Harmonics a) m 4 m m 2 m m m m 2 m m 4 -π -π/2 2 π/2 π 4 b) m 4 m m 2 m m m m 2 m m 4 -π -π/2 2 π/2 π 4 Figure.: Grahical Reresentation of the Sherical Harmonics a) Plot of the sherical harmonics Y m where the hase of the function is color coded. Note that Y m is always axially symmetric with resect to a rotation about the z axis since it deends only on the angle q. The hase of the function changes with a eriodicity of m. b) The linear combinations 2( Y + ( ) m Y m ) 2 Y m cos( m Y and i 2( Y m ( ) m Y m ) 2 Y m sin( m are all real and show only a hase of (ositive) and (negative) and corresond to the tyical orbital shaes.
4 4 Chater a b a b d ab. [.8] They form a comlete basis set of the Hilbert sace of square-integrable functions i.e. every such function can be exressed as a linear combination of sherical harmonics f f m m Y m. [.9] f m The coefficients can be calculated as f m 2 * f Y m sinq( dq) df. [.]..2 Product of Two Sherical Harmonics Since the sherical harmonics form a orthonormal basis set the roduct of two sherical harmonics can again be exressed in sherical harmonics. Let us first look at a simle examle Y Y cosq cosq 2 2 cos 2 q. [.] Comaring this to the sherical harmonics of Eqs. [.4]-[.6] it is immediately clear that we need the functions Y 2 and Y to exress the roduct. We can make an Ansatz Y Y c Y + c 2 Y 2 [.2] which leads to cos 2 q c + c 2 4 c From this it is immediately clear that 5 ( cos 2 q ) 5 c 2 + c cos 2 q. [.]
5 Proerties of Sherical Harmonics c 2 5 [.4] and 54 5 c. [.5] For a general roduct this is of course more comlicated but there are a few simle rules for the general roduct Y m Y 2 m 2. Since the deendence on f is always given by ex( im it is immediately clear that the roduct function has to have the magnetic quantum number m + m 2. Using similar arguments the orbital angular quantum number can be limited to the range 2 L + 2. In rincile it is not imortant to know these restrictions since the Clebsch-Gordan coefficients (or the Wigner j symbols) will do the selection automatically. The roduct can in general be written as the following linear combination Y m Y 2 m 2 L ( 2 + ) ( ) ( 2L + ) - 2 L m m 2 * YL 2 L [.6] where the Wigner j symbols are related to the Racah or Clebsch-Gordan coefficients by 2 L m m 2 ( ) 2 c ( m 2 m 2 L ) ( 2L + ). [.7] The Wigner j symbols or the Clebsch-Gordan coefficients can be found in tables in books about angular momentum or calculated using rograms like atlab acsyma or athematica. Written with Clebsch-Gordan coefficients we obtain for Eq. [.6]
6 6 Chater Y 2 m 2 Y m ( 2 + ) ( ) - ( 2L + ) Y L L c ( m 2 m 2 L )c ( 2 L ). [.8] symbols for To calculate the coefficients for the above examle we need the Wigner j [.9] and obtain Y 2 Y ( 2L + ) L Y L L Y + Y + Y 2 5. [.2] In the same way more comlex roducts can be calculated and decomosed in the sherical harmonics. This is an iterative way to calculate the functional form of higher-order sherical harmonics from the lower-order ones. We will discuss this in more detail in an exercise... Addition Theorem of Sherical Harmonics The sherical harmonics obey an addition theorem that can often be used to simlify exressions m Y m ( q f )Y m ( q 2 f 2 ) * 2 + P ( cosw) [.2] where w omega describes the angle between two unit vectors oriented at the olar coordinates ( q f ) and ( q 2 f 2 ) with cosw cosq cosq 2 + sinq sinq 2 cos( f f 2 ). [.22]
7 Proerties of Sherical Harmonics 7..4 Integrals Over Sherical Harmonics The integration over the roduct of three sherical harmonics can be simlified using the roduct rule of Eq. [.6] and the orthogonality of Eq. [.7]. This leads to 2 Y m Y 2 m 2 Y m sinq( dq) df L ( 2 + ) ( ) ( 2L + ) - 2 L 2 L m m 2 2 * Y L Y m sinq( dq) df ( 2 + ) ( ) ( 2 + ) m m 2 m [.2] a simle exression involving only a normalization constant and two Wigner j symbols..4 Literature () D.. Brink G. R. Satchler Angular omentum third edition Clarendon Press 99. (2) A. R. Edmonds Angular omentum in Quantum echanics Princeton University Press 96. (). E. Rose Elementary Theory of Angular omentum John Wiley & Sons Inc. New York 957.
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The oerators a and a obey the commutation relation Proof: [a a ] = (7) aa ; a a = ((q~ i~)(q~ ; i~) ; ( q~ ; i~)(q~ i~)) = i ( ~q~ ; q~~) = (8) As a s
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