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Problem 5.8 Suppose you had three particles, one in state √(x), one in state ½¿(x), and one in state √(x). Assuming &a, b, and ½c are orthonormal, construct the three-particle states (analogous to Equations 5.19, 5.20, and 5.21) representing (a) distinguishable par- ticles, (b) identical bosons, and (c) identical fermions. Keep in mind that (b) must be completely symmetric, under interchange of any pair of particles, and (c) must be com- pletely anti-symmetric, in the same sense. Comment: There's a cute trick for constructing completely antisymmetric wave functions: Form the Slater determinant, whose first row is ¥a(x₁), ¥b(x₁), ¥c(x₁), etc., whose second row is a (x2), ¥b(x2), ½c(x2), etc., and so on (this device works for any number of particles).

Problem 5.8 Suppose you had three particles, one in state √(x), one in state ½¿(x), and one in state √(x). Assuming &a, b, and ½c are orthonormal, construct the three-particle states (analogous to Equations 5.19, 5.20, and 5.21) representing (a) distinguishable par- ticles, (b) identical bosons, and (c) identical fermions. Keep in mind that (b) must be completely symmetric, under interchange of any pair of particles, and (c) must be com- pletely anti-symmetric, in the same sense. Comment: There's a cute trick for constructing completely antisymmetric wave functions: Form the Slater determinant, whose first row is ¥a(x₁), ¥b(x₁), ¥c(x₁), etc., whose second row is a (x2), ¥b(x2), ½c(x2), etc., and so on (this device works for any number of particles).

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Problem 5.8 Suppose you had three particles, one in state √(x), one in state √(x), and one in state √(x). Assuming √a, b, and c are orthonormal, construct the three-particle states (analogous to Equations 5.19, 5.20, and 5.21) representing (a) distinguishable par- ticles, (b) identical bosons, and (c) identical fermions. Keep in mind that (b) must be completely symmetric, under interchange of any pair of particles, and (c) must be com- pletely anti-symmetric, in the same sense. Comment: There's a cute trick for constructing completely antisymmetric wave functions: Form the Slater determinant, whose first row is a (x₁), ¥b(x₁), &c(x₁), etc., whose second row is a (x2), Yb(x2), ½c(x2), etc., and so on (this device works for any number of particles).6
Transcribed Image Text:Problem 5.8 Suppose you had three particles, one in state √(x), one in state √(x), and one in state √(x). Assuming √a, b, and c are orthonormal, construct the three-particle states (analogous to Equations 5.19, 5.20, and 5.21) representing (a) distinguishable par- ticles, (b) identical bosons, and (c) identical fermions. Keep in mind that (b) must be completely symmetric, under interchange of any pair of particles, and (c) must be com- pletely anti-symmetric, in the same sense. Comment: There's a cute trick for constructing completely antisymmetric wave functions: Form the Slater determinant, whose first row is a (x₁), ¥b(x₁), &c(x₁), etc., whose second row is a (x2), Yb(x2), ½c(x2), etc., and so on (this device works for any number of particles).6
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