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solmx1_04 - 1 Physics 325 Solution to First Midterm Exam...

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1 Physics 325 Solution to First Midterm Exam Winter 2004 PART I (10 pts) Consider a crude model for a helium atom ( Z = 2) in which the electrons do not interact with each other. Let ψ nlm ( r ) represent normalized single-electron energy eigenstates in the atom, and χ i ± represent the spin eigenstates of the ith electron. For each wave function listed below, determine whether or not that wave function is a valid wave function for the electrons in helium. Explain your reasoning in each case. 1.) (2 pts) ψ 100 ( r 1 ) ψ 200 ( r 2 ) χ 1+ χ 2+ Not valid, the total wave function must be anti-symmetric under exchange of the two particle labels: the spin wave function is symmetric, but the spatial wave function is neither symmetric nor anti-symmetric. Exchanging r 1 and r 2 gives a different state. 2.) (2 pts) ψ 100 ( r 1 ) ψ 100 ( r 2 ) χ 1+ χ 2 - Not valid, the spatial wave function is symmetric, but the spin wave function is neither symmetric nor anti-symmetric. 3.) (2 pts) ψ 100 ( r 1 ) ψ 100 ( r 2 ) χ 1+ χ 2 - - χ 1 - χ 2+ Valid, the spatial wavefunction is symmetric under the exchange of r 1 and r 2 while the spin wave function changes sign when the labels 1 and 2 are exchanged, making the total wave function anti-symmetric. 4.) (2 pts) ψ 100 ( r 1 ) ψ 210 ( r 2 ) - ψ 210 ( r 1 ) ψ 100 ( r 2 ) χ 1+ χ 2 - - χ 1 - χ 2+ Not valid, both the spatial and spin wave functions are anti-symmetric, making the total wave function symmetric. 5.) (2 pts) ψ 100 ( r 1 ) ψ 210 ( r 2 ) - ψ 210 ( r 1 ) ψ 100 ( r 2 ) χ 1+ χ 2 - + χ 1 - χ 2+ Valid, the spatial piece is anti-symmetric and the spin piece is symmetric, making the total wave function anti-symmetric.
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2 PART II (14 pts) A system contains two particles, one with spin 1 ( S 1 = 1), and the other with spin 2, ( S 2 = 2). The system is found to be in the state χ = S m s
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