Fund Quantum Mechanics Lect & HW Solutions 87

Fund Quantum Mechanics Lect & HW Solutions 87 - 5.5...

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5.5. MULTIPLE-PARTICLE SYSTEMS INCLUDING SPIN 69 5.5.4 Wave function for multiple particles with spin 5.5.4.1 Solution complexsb-a Question: As an example of the orthonormality of the two-particle spin states, verify that (↑↑|↓↑) is zero, so that ↑↑ and ↓↑ are indeed orthogonal. Do so by explicitly writing out the sums over S z 1 and S z 2 . Answer: The inner product is by definition (↑↑|↓↑) = summationdisplay S z 1 = ± 1 2 ¯ h summationdisplay S z 2 = ± 1 2 ¯ h ( S z 1 ) ( S z 2 ) ( S z 1 ) ( S z 2 ) or writing out the second sum explicitly (↑↑|↓↑) = summationdisplay S z 1 = ± 1 2 ¯ h bracketleftBig ( S z 1 ) (+ 1 2 ¯ h ) ( S z 1 ) (+ 1 2 ¯ h ) + ( S z 1 ) ( 1 2 ¯ h ) ( S z 1 ) ( 1 2 ¯ h ) bracketrightBig or writing out the first sum also explicitly (↑↑|↓↑) = (+ 1 2 ¯ h ) (+ 1 2 ¯ h ) (+ 1 2 ¯ h ) (+ 1 2 ¯ h ) + (+ 1 2 ¯ h ) ( 1 2 ¯ h ) (+ 1 2 ¯ h ) ( 1 2 ¯ h ) + ( 1 2 ¯ h ) (+ 1 2 ¯ h ) ( 1 2 ¯ h ) (+ 1 2 ¯ h ) + ( 1 2 ¯ h ) ( 1 2 ¯ h ) (
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