Chapter2_13 - 2 1 ) ( ) ( ) ( ) ( = + = + + op op op op op...

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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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