Chapter2_13

# Chapter2_13 - 2 1 − = = − − op op op op op a a a a H...

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PHY4604 R. D. Field Department of Physics Chapter2_13.doc University of Florida The Harmonic Oscillator (2) Hamiltonian Operator: The Hamiltonian operator for the simple harmonic oscillator is given by op op x op x m p m H ) ( 2 1 ) ( 2 1 2 2 2 ω + = since 2 2 2 1 2 2 1 ) ( x m Kx x V = = where K is the spring constant and m K / . The operators (p x ) op and x op obey the commutation relations h i x p op op x = ] , ) [( and h i p x op x op = ] ) ( , [ , where op op op op op op A B B A B A ] , [ . Raising and Lowering Operators: Define the two operators (a ± ) op as follows: ( ) op x op op p i x m m a ) ( 2 1 ) ( m h = ± , Note that op op a a ) ( ) ( + = and 1 ] ) ( , ) [( = + op op a a and 1 ] ) ( , ) [( = + op op a a . Also, 2 1 1 ) ) ( )( ) ( ( 2 1 ) ( ) ( + = + + = + op op op x op op x op op H x m p i x m p i m a a h h 2 1 1 ) ) ( )( ) ( ( 2 1 ) ( ) ( = + + = + op op op x op op x op op H x m p i x m p i m a a h h . Hence, h h h h 2 1
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Unformatted text preview: 2 1 ) ( ) ( ) ( ) ( − = + = + − − + op op op op op a a a a H and op op op a a H ) ( ] ) ( , [ + + = h and op op op a a H ) ( ] ) ( , [ − − − = h . Thus (a + ) op is a “raising operator” and shifts the energy by h + and (a-) op is a “lowering operator” and shifts the energy by h − ! Note that ( ) op op op a a m x ) ( ) ( 2 ) ( − + + = h and ( ) op op op x a a m i p ) ( ) ( 2 ) ( − + − = h ....
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## This note was uploaded on 05/29/2011 for the course PHY 4064 taught by Professor Fields during the Spring '07 term at University of Florida.

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