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Chapter 7 Lecture 3D

# Chapter 7 Lecture 3D - Chapter 7 3 Dimensional Quantum...

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Chapter 7- 3 Dimensional Quantum Mechanics 7-2 Fundamental Concepts ! ! 2 2 m " 2 # ( " r , t ) + V ( " r ) # ( " r , t ) = i ! \$ \$ t # ( " r , t ) Schrodinger Equation # ( " r , t ) 2 dV = 1 Normalization % \$ \$ t & ( " r , t ) + " " i J ( " r , t ) = 0 Continuity Equation Operator commutation relations [ x i , x i ] = 0 [ p i , p i ] = 0 [ x i , p i ] = i ! ! ij Cartesian Coordinates E = p x 2 2 m + p y 2 2 m + p z 2 2 m + V ( x , y , z ) ! ! 2 2 m ( " x 2 + " y 2 + " z 2 ) # ( r ) + V ( x , y , z ) # ( x , y , z ) = E # ( x , y , z ) " i = \$ \$ x i , # ( x , y , z ) = # ( x ) # ( y ) # ( z ) ! ! 2 2 m # '( y ) # ''( z ) " x 2 # ( x ) + # ( x ) # ''( z ) " y 2 # '( y ) + # '( y ) # ''( z ) " z 2 # ''( z ) ( ) + V ( x , y , z ) # ( x ) # '( y ) # ''( z ) = E # ( x ) # '( y ) # ''( z ) If the potential energy term can be factorized as V ( x , y , z ) = + V ( x ) + V ( y ) + V ( z ) then the 3 D equation can be solved by separation of var iables int o 3 1 D equations . ! ! 2 2 m " x 2 # ( x ) # ( x ) + " y 2 # ( y ) # ( y ) + " z 2 # ( z ) # ( z ) % & ( ) * + V ( x ) + V ( y ) + V ( z ) = E x + E y + E z Energy Partioned if V = V ( x ) + V ( y ) + V ( z ) ! ! 2 2 m " x 2 # ( x ) # ( x ) % & ( ) * + V ( x ) = E x , ! ! 2 2 m " y 2 # ( y ) # ( y ) % & ( ) * + V ( y ) = E y , ! ! 2 2 m " z 2 # ( z ) # ( z ) % & ( ) * + V ( z ) = E z ! ! 2 2 m # ( x )'' + V ( x ) # ( x ) = E x # ( x ) , ! ! 2 2 m # ( y )'' + V ( y ) # ( y ) = E y # ( y ), ! ! 2 2 m # ( z )'' + V ( z ) # ( z ) = E z # ( z ) # ( x , y , z ) = # l ( x ) # m ( y ) # n ( z ) E lmn = E l + E m + E n Energy Partitian

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7-3 Particle in a Symmetric Box ! ! 2 2 m " 2 " x 2 + " 2 " x 2 + " 2 " x 2 # \$ % & ( ) ( x , y , z , t ) + V ( x , y , z ) ) ( x , y , z , t ) = i !
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