Fall 07 Phys 143l HW 11 Solutions

Fall 07 Phys 143l HW 11 Solutions - Physics 143 Fall Term...

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Unformatted text preview: Physics 143 Fall Term 2007 HOMEWORK 11 Due: November 28 Problem 1: (BFG Ch.10 Problem 1) Consider two particles in an infinite well. The quantum state of two particles in an infinite well is: Eq. 1022 or 1023 1 uS x , x u x u x u x u x NS 1 uA x , x u x u x u x u x NA Normalization: u u dx dx Therefore, 1 u x u x NS 1 NS |u | x |u | x 1 u x u x x u x u u x u x x u x u u x u x x u x u dx dx |u | x |u | x dx dx x u x u 2 |u | u u dx 1 NS The same for uA x , x , we obtain: 2 NA |u | u u dx 1 The wave function of single particle in an infinite well is: nx u sin L L Do the following integral: 2 mx nx sin u u dx sin dx L L L L L cos L cos L dx NS NA (If m n, plug in the above expressions, NS NA 2 2 |u | |u | , 1, m 0, m u u dx u u dx n n 21 21 0 0 2 2 21 21 1 1 4 0 P a g e | 13 Problem 2: (BFG Ch.10 Problem 5) Two identical free spinless bosons in a one-dimensional infinite well. For spinless boson, we shall use symmetric wave function: (from Problem 1) 1 u x u x u x u x uS x , x NS The wave function of single particle in this infinite well is: nx u sin 2a For the ground state, both bosons occupy the ground state of single particle: (m 1 u x u x u x u x uS x , x 4 Notice that m n here, so NS 4 but not 2 uS x , x n case) 1 x x sin sin 2a 2a a For the first excited state, one of boson occupies the first excited state of single particle: (m 1 u x u x u x u x uS x , x 2 Columbia University (NY) 4.3 University of TexasAustin n case uS x , x 1 2a sin x x sin 2a a sin x x sin 2a a Problem 3: (BFG Ch.10 Problem 7) Two identical free spin-up fermions in a one-dimensional infinite well. Two fermions with same spin, due to Pauli Principle, should occupy different energy state and have antisymmetric wave function: 1 uA x , x u x u x u x u x 2 For the ground state, one occupies the ground state and the other occupies the first excited state of single particle: 1 uA x , x u x u x u x u x 2 2a The wave function vanishes at the following point x , x (0, 0), (a, a), (a, 2a), (2a, a), (2a, 2a) uA x , x 1 sin x x sin 2a a sin x x sin 2a a (Actually, under the following condition, the wave function will also vanishes: x either x or x is equal to 0 or 2a, or, x P a g e | 23 Problem 4: (BFG Ch.10 Problem 8) Three identical free spinless bosons in a one-dimensional infinite well. For spinless boson, we shall use symmetric wave function and for the ground state, all the bosons occupy the ground state of single particle: uS x , x Nu x u x u x x x x N sin sin sin 2a 2a 2a For the first excited state, one of the bosons occupies the first excited state of single particle: uS x , x u x u x u x u x u x u x Nu x u x u x uA x , x N sin x x x sin sin a 2a 2a x x x sin sin sin 2a 2a a sin x x sin 2a a sin x 2a November 30, 2007 P a g e | 33 ...
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