Chapter3_15 - | | dt O d O t>< βˆ† β‰ βˆ† Then 2 h β‰₯...

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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 > < βˆ† ≑ βˆ† . Then 2 ) )( ( h β‰₯ βˆ† βˆ† t E This is the energy-time uncertainty principle where βˆ† 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 βˆ† > < ≑ βˆ† . If βˆ† 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 β€œuncertainty” in the energy must be large....
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