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Hydrogen-like-wavefunctions

# Hydrogen-like-wavefunctions - TABLE 8.1 The Angular and...

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Unformatted text preview: TABLE 8.1 The Angular and Radial Wave Functions of a Hydrogen-like Atom Angular Part Y(0, (1)) Radial Part Rn, g(r) 1 1/2 Z 3/2 a- _ — /2 Y(s)= (4—?) R(15)— 2(a—0) 8 3/2 R(25) = ;(E) (2 — age-W2 2 2 0 Z 3/2 R33 =— — 6—6a+a2e“°/2 ( ) 9%(‘1 ) ( ) 3 1/2 1 Z 3/2 = — = — — —O'/2 Y(px) (411-) smﬂcosqb R(2p) 2 6(90) are 3 1/2 1 Z 3/2 a- Y(py) (497) smﬂsmda R(3p) 9 6(“0) (4 (7)012 3 1/2 Y(pz) — (E) cost? 5 1/2 1 (Z )3/2 Y dzz = —) 3c0529 — 1 R 351 = — — 0250/2 ( ) 16w ( ) < ) WE a0 15 1/2 - 2 Y(dx2_y2) E) 3m Bees 245 15 1/2 ' 2 ' Hairy) — (Kr—71') sm 631n2qb a = E d (113)1/2 . ““0 Y( H) — E smﬁcosBcosqﬁ 15 1/2 . . Y(dyz) = (E) smﬂcosﬂsmqﬁ I 2 — r/ao 1 e — r/cqJ was) = Rm >< Y(6,¢>) = 93/2 x T A B L E 1 0 - 3 The Hydrogen—like Wave Functions. Z is the nuclear Charge, (10 is the Bohr radius, and (r = Zr/ao. QUANTUM ELECTRON NUMBERS WAVE FUNCTION SHELL n I m ORBITAL 1b(n,l.m) I Z 312 K 1 0 0 1 — (— -0 S w an) e I Z ' 312 2 _ 2 _.., —(J'12 L 2 0 0 s 4 m (a0) ( a)e 1 Z. 312 —a-[2 2 l 0 2p: 4 V2? (00) (re cos 6 1 Z ' 312 _ 2 I -— ‘0” sm 3 cos 2 1 +1 *{rp 4V27T(ao) 0'6 (1) _ 1 z - 312 _ . . Zpy 4 m (:0) 0e "’2 5m 6 SH] (;b 1 Z 312 3 0 3 — 27—18 +2 2 “”3 M 0 s 81 V3; (00 ) ( 0' 0' )e V? g 312 _ —ma 3 1 0 3pz 81 V; (do) (6 0’)Ue cos 6 - 312 3P: ﬂ— (E) (6 — chew” sin 0 cos d) 3 1 +1 * [ 81 V77 00 _ V3 Z 3l2 —u _ _ 3py 81 V71- (:0) (6 — (ﬂare ’3 Sin 6 sm (1) l I Z I 3/2 - l3 2 2 _ ”- 9 — I 3 2 0 3d: 81 611' (do) 028 (3 cos ) ‘F , ar2 3d“ V2,. (Z ) aze’m sin 6 cos 6 cos d; 3 2 :1 * 1 8| Vrr an 1"" . 312 3d!” €3le; (:20) 02‘2"“ sin 6 cos 6 sin d) 312 3d12_y2 I,— (E) (ﬁe—”’3 sin2 0 C05 29') 3 2 +2 ,ﬁ Si VZ‘rr aol _ 1 Z 3’2 . A, . . . 3d“, 81 V5; (3;) Ole "3 5m2 6 sm 2d) 31'2 N 4 0 0 4s 1 _ (z) (24 — 180- + 30'2 — rT"/8)€“”“ an * For these terms linear combinations of the surface spherical harmonics are taken. rather than the harmonics themselves, in order to give Sines and cosines of <15. instead of exponentials containing 1'. [See Exercise 10-26.) ...
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Hydrogen-like-wavefunctions - TABLE 8.1 The Angular and...

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