Page 187 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

178 The hydrogen atom

Table 6.2. Real wave functions for the hydrogen-like atom. The parameter a ì
has been replaced by a 0

State Wave function

Spherical coordinates Cartesian coordinates
(Z=a 0 ) 3=2
1s p  e ÿ Zr=a 0
ð
(Z=a 0 ) 3=2 Zr
2s p  2 ÿ e ÿ Zr=2a 0
4 2ð a 0
(Z=a 0 ) 5=2 (Z=a 0 ) 5=2
2p z p  re ÿ Zr=2a 0 cos è p  ze ÿ Zr=2a 0
4 2ð 4 2ð
(Z=a 0 ) 5=2 (Z=a 0 ) 5=2
2p x p  re ÿ Zr=2a 0 sin è cos j p  xe ÿ Zr=2a 0
4 2ð 4 2ð
(Z=a 0 ) 5=2 (Z=a 0 ) 5=2
2p y p  re ÿ Zr=2a 0 sin è sin j p  ye ÿ Zr=2a 0
4 2ð 4 2ð
!
2 2
(Z=a 0 ) 3=2 Zr Z r
3s p  27 ÿ 18 ‡ 2 e ÿ Zr=3a 0
81 3ð a 0 a 2 0
2(Z=a 0 ) 5=2 Zr 2(Z=a 0 ) 5=2 Zr
3p z p  6 ÿ re ÿ Zr=3a 0 cos è p  6 ÿ ze ÿ Zr=3a 0
81 2ð a 0 81 2ð a 0
2(Z=a 0 ) 5=2 Zr 2(Z=a 0 ) 5=2 Zr
3p x p  6 ÿ re ÿ Zr=3a 0 sin è cos j p  6 ÿ xe ÿ Zr=3a 0
81 2ð a 0 81 2ð a 0
2(Z=a 0 ) 5=2 Zr 2(Z=a 0 ) 5=2 Zr
3p y p  6 ÿ re ÿ Zr=3a 0 sin è sin j p  6 ÿ ye ÿ Zr=3a 0
81 2ð a 0 81 2ð a 0
(Z=a 0 ) 7=2 2 (Z=a 0 ) 7=2 2 2
2 ÿ Zr=3a 0
3d z 2 p  r e (3 cos è ÿ 1) p  (3z ÿ r )e ÿ Zr=3a 0
81 6ð 81 6ð
2(Z=a 0 ) 7=2 2(Z=a 0 ) 7=2
2 ÿ Zr=3a 0
3d xz p  r e sin è cos è cos j p  xze ÿ Zr=3a 0
81 2ð 81 2ð
2(Z=a 0 ) 7=2 2(Z=a 0 ) 7=2
2 ÿ Zr=3a 0
3d yz p  r e sin è cos è sin j p  yze ÿ Zr=3a 0
81 2ð 81 2ð
(Z=a 0 ) 7=2 2 (Z=a 0 ) 7=2 2 2
2 ÿ Zr=3a 0
2
3d x ÿ y 2 p  r e sin è cos 2j p  (x ÿ y )e ÿ Zr=3a 0
81 2ð 81 2ð
(Z=a 0 ) 7=2 2 2(Z=a 0 ) 7=2
2 ÿ Zr=3a 0
3d xy p  r e sin è sin 2j p  xye ÿ Zr=3a 0
81 2ð 81 2ð