Chapter6_31 - The zero point energy h 2 1 = E is a...

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PHY3063 R. D. Field Department of Physics Chapter6_31.doc University of Florida The Harmonic Oscillator (4) Properties of the Ground State: We can use () op op op x a a m i p ) ( ) ( 2 ) ( + = ω h and () op op op a a m x ) ( ) ( 2 ) ( + + = h to calculate ( x)( p x ) for the ground state. We see that 0 | ) ) ( ) (( | 2 | | 0 0 0 0 >= + < >= >=< < + E a a E m E x E x op op op h since 0 | ) ( 0 >= E a op and 0 ) ( | 0 = < + op a E . Also, m E a a a a E m E x x E x op op op op op op 2 | ) ) ( ) )(( ) ( ) (( | 2 | | 0 0 0 0 2 h h >= + + < >= >=< < + + where I used 1 ] ) ( , ) [( = + op op a a which means 1 ) ( ) ( ) ( ) ( + = + + op op op op a a a a . Similarly, 0 | ) ) ( ) (( | 2 | ) ( | 0 0 0 0 >= < >= >=< < + E a a E m i E p E p op op op x x h and 2 | ) ) ( ) )(( ) ( ) (( | 2 | ) ( | 0 0 0 2 0 2 h h m E a a a a E m E p E p op op op op op x x >= < >= >=< < + + Thus, 2 2 2 2 h h h h = >= >< < m m p x x which satisfies the uncertainty principle ( saturates the lower bound ). Zero-Point Energy:
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Unformatted text preview: The zero point energy h 2 1 = E is a manifestation of the uncertainty principle as follows: 2 2 2 2 2 2 2 ) ( 2 1 ) ( 4 2 1 2 1 2 1 x m x m x m p m E x ∆ + ∆ = + = h , where I set x = ∆ x and ) 2 /( x p p x x ∆ = ∆ = h The minimum energy occurs when dE/d ∆ x = 0 which occurs at ) 2 /( m x h = ∆ and h 2 1 min = E ....
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