Lecture 15A.pdf - Quantum Mechanics in 3-d Quantum Mechanics in 3-d Position operator r eˆx x eˆ y y eˆz z Momentum operator p i eˆx eˆ y eˆz i y

Lecture 15A.pdf - Quantum Mechanics in 3-d Quantum...

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Quantum Mechanics in 3-d
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Momentum operator: i z e y e x e i p z y x ˆ ˆ ˆ Position operator: z e y e x e r z y x ˆ ˆ ˆ Quantum Mechanics in 3-d Kinetic energy operator: 2 2 2 2 2 2 m m m p p T 2 2 2 2 2 2 2 Laplacian z y x ) , , ( 2 2 2 z y x V m H Hamiltonian: Wave function: ) , , ( ) ( z y x r
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Time dependent Schrodinger equation ) , ( ) ( ) , ( 2 ) , ( ) , ( 2 2 t r r V t r m t r H t t r i After separation of variables (r and t) ) ( ) ( ) , ( t r t r / ) ( iEt e t ) ( ) ( r E r H Time independent Schrodinger equation in 3-d ) ( ) ( ) ( 2 . . 2 2 r E r r V m e i ) ( ) ( 2 ) ( 2 2 2 2 2 2 2 r r V E m r z y x
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Factorization of Wave function ) ( ) ( ) ( 2 ) ( 2 2 r E r r V m r H Consider TISE Suppose ) , ( H H If ) ( ) ( ) , ( H H H then the wave function can be factorized as ) ( ) ( ) , ( so that ) ( ) ( ) ( E H ) ( ) ( ) ( E H and E E E
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Free Particle in 3-d ) ( ) ( ) ( 2 2 2 r E r r V m 0 ) ( r V ) ( ) ( ) ( ) , , ( ) ( z y
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