Lecture10 - CHEM 356, Lecture 10, Fall 2009 1 Expectation...

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Unformatted text preview: CHEM 356, Lecture 10, Fall 2009 1 Expectation Values for the Particle–in–a–box. Position : The expectation value for ˆ ? in eigenstate of the translational energy operator can be calculated as ⟨ ? ⟩ = ∫ ∗ ( ? ) ˆ ? ( ? ) ?? = 2 ∫ ? sin 2 ( ? ) ?? = 1 ∫ ? [ 1 − cos 2 ? ] ?? = 1 { ? 2 2 − ∫ ? cos 2 ? ?? } or ⟨ ? ⟩ = 2 . Similarly, but with a few more mathematical steps, the expectation value of ˆ ? 2 can be obtained as ⟨ ? 2 ⟩ = 2 ∫ ? 2 sin 2 ? ?? = ( 2 ) 2 ( 4 2 2 3 − 2 ) As the variance equals the square of the uncertainty, let us use this relationship: (Δ ? ) 2 = ⟨ ? 2 ⟩ − ⟨ ? ⟩ 2 = ( 2 ) 2 [ 4 2 2 3 − 2 ] − 2 4 = ( 2 ) 2 [ 4 2 2 3 − 2 − 2 2 ] = ( 2 ) 2 [ 2 2 3 − 2 ] CHEM 356, Lecture 10, Fall 2009 2 or (Δ ? ) = 2 [ 2 2 3 − 2 ] 1 2 . ∙ Momentum : The expectation values for ˆ ? and ˆ 2 ? are calculated in exactly the same manner. For ˆ ? we have ⟨ ? ⟩ = 2 ∫ sin ? ( − ℏ ? ?? ) sin ? ?? = 2 ( − ℏ ) ∫ sin ? cos ? ?? , and hence ⟨ ? ⟩ = 0 , for every value of . Evaluation of ⟨ 2 ? ⟩ is carried out similarly; ⟨ 2 ? ⟩ = 2 ( − ℏ ) 2 ∫ sin ? ? 2 ?? 2 sin ? ?? = ( − ℏ ) 2 ( − 1) ( ) 2 2 ∫ sin 2 ? ?? , and so we see that ⟨ 2 ? ⟩ is given by ⟨ 2 ? ⟩ = ℏ 2 ( ) 2 . CHEM 356, Lecture 10, Fall 2009 3 Because ⟨ ?...
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This note was uploaded on 02/28/2011 for the course CHEM 356 taught by Professor Prof.iaskjd during the Fall '09 term at Waterloo.

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Lecture10 - CHEM 356, Lecture 10, Fall 2009 1 Expectation...

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