Chapter12_Hatom - Chapter 12 Hydrogen atom e2 V r)= 4#$ 0 r...

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9/21/13 1 V ( r ) = " e 2 4 #$ 0 r Chapter 12 Hydrogen atom x = r sin " cos # y = r sin " sin # z = r cos " Spherical Coordinates " 2 ( r , # , $ ) % ( r , # , $ ) = 1 r & 2 & r 2 ( r % ( r , # , $ )) + 1 r 2 ' 2 ( # , $ ) % ( r , # , $ ) " 2 ( # , $ ) = 1 sin 2 # % 2 %$ 2 + 1 sin # % %# sin # % %# & ' ( ) * +
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9/21/13 2 " ! 2 2 m 1 r # 2 ( r $ ( r , % , & )) # r 2 + 1 r 2 ' 2 ( % , ( ) $ ( r , % , & ) ) * + , - . + " e 2 4 /0 0 r $ ( r , % , & ) = E $ ( r , % , & ) We will use the separation of variables. First, the radial component will be separated from the angular part: " ( r , # , $ ) = R ( r ) Y ( # , $ ) E ! ( r , " , # ) = ! ! 2 2 m " 2 ( r , ! , " ) # ( r , ! , " ) + ! e 2 4 $% 0 r # ( r , ! , " ) Schrodinger’s equation for the hydrogen atom " ! 2 2 m Y ( # , $ ) 1 r % 2 ( rR ( r )) % r 2 + R ( r ) 1 r 2 & 2 ( # , $ ) Y ( # , $ ) ' ( ) * + , + " e 2 4 -. 0 r R ( r ) Y ( # , $ ) = E R ( r ) Y ( # , $ ) " ! 2 2 m Y ( # , $ ) 1 r % 2 ( rR ( r )) % r 2 + R ( r ) 1 r 2 & 2 ( # , $ ) Y ( # , $ ) ' ( ) * + , + " e 2 4 -. 0 r R ( r ) Y ( # , $ ) = E R ( r ) Y ( # , $ ) Multiply by r 2 / ( R ( r ) Y ( " , # )) " ! 2 2 m r R ( r ) # 2 ( rR ( r )) # r 2 + 1 Y ( $ , % ) & 2 ( $ , % ) Y ( $ , % ) ' ( ) * + , + " e 2 r 4 -. 0 = E r 2 Move the energy term to the left and regroup: " ! 2 2 m r R ( r ) # 2 ( rR ( r )) # r 2 + " e 2 r 4 $% 0 " E r 2 & ' ( ) * + " ! 2 2 m 1 Y ( , , - ) . 2 ( , , - ) Y ( , , - ) & ' ( ) * + = 0
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9/21/13 3 The terms must each be equal to a constant. Defined the constant as: " 2 ( # , $ ) Y ( # , $ ) Y ( # , $ ) = % l ( l + 1) This yields two equations: " ! 2 2 m r R ( r ) d 2 ( r R ( r )) d r 2 + " e 2 r 4 #$ 0 " r 2 E + ! 2 2 m l ( l + 1) = 0 " 2 ( # , $ ) Y ( # , $ ) = % l ( l + 1) Y ( # , $ ) " ! 2 2 m r R ( r ) # 2 ( rR ( r )) # r 2 + " e 2 r 4 $% 0 " E r 2 & ' ( ) * + " ! 2 2 m 1 Y ( , , - ) . 2 ( , , - ) Y ( , , - ) & ' ( ) * + = 0 Radial equation Angular equation " 2 ( # , $ ) Y ( # , $ ) = % l ( l + 1) Y ( # , $ ) " 2 ( # , $ ) = 1 sin 2 # % 2 %$ 2 + 1 sin # % %# sin # % %# & ' ( ) * + Remember 1 sin 2 " # 2 #$ 2 Y ( " , $ ) + 1
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