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Physics 21S - Elementary Modem Physics
Hydrogen- Like Wave Functions
Angular Momentum Quantum NumbersL2= n2(/2+/) (n,l, m,ms)L =nm n
= 1,2,3, ...zS2 = l.n2 1= 0,1,2,3, ... ,n-l
4 m =-1,-1 +1,... ,0, ... ,1S =nm m =+lz s s - 2
,
Wave function (ignoring spin):'!j;n,l,m (r, e,cj = 1\,,1 (r
)J;,m (e,cj
Spherical Harmonics
y =_1_0,0 .J4;
y. - [3 . 0 i~I,l =+\jS; sm e
1';0 = ~ 3 cosO, 41r
Y = ~ IS sin ' Oe2i~2,2 321r
Jgs +~~l ==t -sinOcosOe-', 81r~ 0 = ~ 5 (3cos' 0 -1), 161r
v --HfS . 30 3i133 -+ --sm e, 641r
v ~05. 20 0 2i132 = --sm cos e, 321r
1';,1 ==t~ 6~~ sinO(Scos2 0-I)ei
1';0 = ~ 7 (Scos30-3cosO), 161r
Energy
E=eZ2e4j.t (13.6 eV)Z2
n 2n2n2 n2
Characteristic radius:n,2 ao 0.OS29nma=--=- =V kZe2p, Z Z
v (r) rv an'
R = r e-r/2a2,1 r:-s2v6a-
R = 2 (1- 2r + 2r2 Je-r/3a3,0 3-J3a3 3a 27 a2
R3 I = 4.J2r (1-~)e-r/3a'27-J3a5 6a
n; = 2.J2r2 e-r/3a,2 81-JlSa7
R ...:._1_(I-~+~-~Je-r/4a4,0 - 4j;1 4a 8a2 192a3
R = .J5r (1-~+~Je-r/4a4,1 16-J3a5 4a 80a2
2 ( )R - r 1 r -r/4a4,2 - 320N -12a e
3
R - r -r/4a4 3 - r:::::-9 e, 768v3Sa9
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r-----------------------,~ a.}Q'-;J~',o ~\Llt-
....t~L ":J" ,1-;)s-: Radial Wave FunctionsR =_2_e-r/a
1,0 j;1
R =_I_(I_~)e-r/2a2,0 C3 2v2a- a
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Fundamental Equations
Schrodinger equation:. aqJlh- = HqJ
ar
Time-independent Schrodinger equation:
H1/! = E1/!.
Hamiltonian operator:
Momentum operator:p = -ihV
Time dependence of an expectation value:
d(Q} = !.- ([H, Q]) + (OQ)dt Ii at
Generalized uncertainty principle:
Heisenberg uncertainty principle:
Canonical commutator:
Pauli matrices:
(0 1)(Jx= 1 0 ' (0 -i)(Jy = i 0 ' (I 0 )(J:;= 0 -1
[x, p] = ili
Angular momentum:
