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# ps3a - Quantum l/lechanics ll(34022 — Spring 2009...

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Unformatted text preview: Quantum l\/lechanics ll ((34022 — Spring 2009) Solutions: Problem Set 3 Prof. A4 Mueller TA: Fabio Dominguez 1. 13.7. (a) Since the two electrons are in the same spin state, their spatial wave function must be antisym— metric. this means they can’t be both in the one-particle ground state. The lowest energy is then the sum of the one—particle ground state energy and the one—particle first excited state energy. For the infinite square well potential this is E _ «2h? 27%? _ 5785? (1) _ 271m2 7an _ 2mm2 (b) The wave function must be antisymmetric The one—particle wave functions for the ground state and the ﬁrst excited state are 5 m: "J (“l f # sin—— Ql l/ 1 a} V/ a a i , 5 27m: 1‘ a: : — sin —' 3 1/2( l V a a ( > Then the normalized two—particle wave function is 2 , 7m: _ 27m . 27m: _ 7mm 117(3317332) — f (Sin l Sin 2 sm 1 sm 2) (4) a a a a. a (C) The ﬁrst excited state must have one particle in the ground state and the other in the second excited state. The energy is given by E «Eh? 9713712 579252 (f) : : l.) 2ma2 2771a2 me2 The two—particle wave function is , V2 _ 7mm , 37mg . 87ml . 7mg w(\$1,3:2) : sm sm 5111 \$111 . (0) a. a a a a 2‘. 13.12. For particles in a box, the requirement of the particles being conﬁned to a region of space imposes a quantization condition on the momentum of the particle (being momentum the variable conjugate ...
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