Chapter3_15 - | / | dt O d O t > < ....

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PHY4604 R. D. Field Department of Physics Chapter3_15.doc University of Florida Uncertainty Principle Examples Position and Momentum: For x op and (p x ) op we get 2 2 1 | ] ) ( , [ | ) )( ( h = > < op x op x p x i p x . There is a lower limit on how well one can simultaneously know the position x and momentum p x and the lower limit is 2 / h . Energy and Time: We know that for any observable O that dies not depend explicitly on time dt O d i H O op op > < >= < h ] , [ and we know that 2 4 1 2 2 ) ] , [ ( ) ( ) ( > < B A i B A Now suppose we choose A = O op and B = H op , where H op is the Hamitonian. We get dt O d H O i H O op op > < = > < 2 | ] , [ | ) )( ( 2 1 h Now we define E H and define
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Unformatted text preview: | / | dt O d O t &gt; &lt; . Then 2 ) )( ( h t E This is the energy-time uncertainty principle where t is the amount of time it takes for the expectation value of the observable O to change by one standard deviation as follows: t dt O d O &gt; &lt; . If E is small then the rate of change of all observables must be very gradual or, put the other way around, if any observable is changing rapidly then the uncertainty in the energy must be large....
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