lecture8_scan-1 - What happens to the ket in the open...

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1 Chem 120A Lecture 6 1/29/2007 reading: R & S ch1 A & F ch1 Measurement Consider a 2 slit experiment. The ket which describes the state of the system after the two slits can be written as the following. ˆ | | | i i i i a s T s s s =< >< > If you measure at each slit, you remove interference. Measurement destroys the superposition. Measurement projects | Ψ > onto the 1 2 | ,| s s > > basis. 1 2 | s s > > are orthogonal, meaning 1 2 | 0 s s < >= . Thus when 1 2 | s s > > are normalized, Experiment implies that 1 | s > still exists, project and get with probability 2 1 | | a . Probabilities at detector is then, 2 2 1 1 1 2 2 2 2 2 ~| | | | | ~| | | | | P a y s P a y s < > < > In general where ˆ i M is a measurement operator, 1 1 1 2 2 2 1 1 1 1 1 2 2 ˆ | | ˆ | | ˆ | | | | | M s s M s s M a s a s a s = >< = >< Ψ >= > Ψ >= > + > Measurement changes quantum state!!! Operators are matrices! 1 1 1 1 2 2 2 2 ˆ ˆ | | | | | | | | | s s T s s s s s T s s s Ψ >= >< >< > + >< >< > 1 1 2 2 | | s a s a = > + > 1 1 1 1 1 2 2 1 1 2 2 | | | | | s s s a s s a s a s a < Ψ >=< > + < > < Ψ >= < Ψ >=
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2 E.G. measure position probability: 2 2 ˆ | | | ( ) | ( ) | ˆ | ( ) N N N r n P r r r n r < > = = Ψ < >= Ψ a a Last Double Slit Item
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Unformatted text preview: What happens to the ket in the open space? Does the wavefunction change as a function of position? More realistically we need an operator to account for electron motion through space. We will take this to be G . The evolution through the double slit apparatus can then be represented as below. 1,2 | | | | | | | i i i i i y s y G s s T s s G s = &lt; &gt;= &lt; &gt;&lt; &gt;&lt; &gt; T = transmission operator G = propagation operator 1,2 1,2 1,2 | | | | | | | | | | | | | | | | | ( ) | i i i i i i i i i i i y G s s T s s G s T y G s s G s T y G s s G s T y GG s y DS s = = = &lt; &gt;&lt; &gt;&lt; &gt;= &lt; &gt;&lt; &gt; &lt; &gt;&lt; &gt; &lt; &gt; &lt; &gt; where (DS) is the double slit operator. The last line is often referred to as a scattering amplitude....
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lecture8_scan-1 - What happens to the ket in the open...

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