Is there a linear combination that preserves p j

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Is there a linear combination that preserves indistinguishability? P. J. Grandinetti (Chem. 4300) Identical Particles in Quantum Mechanics Nov 17, 2017 6 / 20
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Particle Exchange Operator Introduce new operator to carry out particle exchange. ̂ Ψ( x 1 , ⃗ x 2 ) = Ψ( x 2 , ⃗ x 1 ) and ̂ Ψ( x 2 , ⃗ x 1 ) = Ψ( x 1 , ⃗ x 2 ) Obviously, Ψ( x 1 , ⃗ x 2 ) and Ψ( x 2 , ⃗ x 1 ) are not eigenstates of ̂ . Calculating [ ̂ , ̂ ( 1 , 2 )] : [ ̂ , ̂ ( 1 , 2 )]Ψ( x 1 , ⃗ x 2 ) = ̂ ̂ ( 1 , 2 )Ψ( x 1 , ⃗ x 2 ) - ̂ ( 1 , 2 ) ̂ Ψ( x 1 , ⃗ x 2 ) = ̂ E Ψ( x 1 , ⃗ x 2 ) - ̂ ( 1 , 2 )Ψ( x 2 , ⃗ x 1 ) = E Ψ( x 2 , ⃗ x 1 ) - E Ψ( x 2 , ⃗ x 1 ) = 0 , Since [ ̂ , ̂ ( 1 , 2 )] = [ ̂ , ̂ ( 2 , 1 )] = 0 , eigenstates of ̂ and ̂ are the same. But Ψ( x 1 , ⃗ x 2 ) and Ψ( x 2 , ⃗ x 1 ) are not eigenstates of ̂ . Eigenstates of ̂ are some linear combinations of Ψ( x 1 , ⃗ x 2 ) and Ψ( x 2 , ⃗ x 1 ) that preserve indistinguishability. P. J. Grandinetti (Chem. 4300) Identical Particles in Quantum Mechanics Nov 17, 2017 7 / 20
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Symmetric and Anti-Symmetric Wave functions We can examine the eigenvalues of ̂ ̂ Φ = 𝜆 Φ Since ̂ 2 = 1 then ̂ 2 Φ = Φ but also ̂ 2 Φ = ̂ 𝜆 Φ = 𝜆 2 Φ = Φ Thus 𝜆 2 = 1 and find 2 eigenvalues of ̂ to be 𝜆 = ± 1 . We can obtain these 2 eigenvalues with 2 possible linear combinations Φ S = 1 2 [ Ψ( x 1 , ⃗ x 2 ) + Ψ( x 2 , ⃗ x 1 ) ] , 𝜆 = + 1 , symmetric combination and Φ A = 1 2 [ Ψ( x 1 , ⃗ x 2 ) - Ψ( x 2 , ⃗ x 1 ) ] . 𝜆 = - 1 , anti-symmetric combination Easy to check that ̂ Φ S = Φ S and ̂ Φ A = -Φ A . Anti-symmetric wave function changes sign when particles are exchanged. Wouldn’t that make particles distinguishable? No, because sign change cancels when probability or any observable is calculated | ̂ Φ A | 2 = | - Φ A | 2 = | Φ A | 2 P. J. Grandinetti (Chem. 4300) Identical Particles in Quantum Mechanics Nov 17, 2017 8 / 20
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Symmetric and Anti-Symmetric Wave functions Using similar approach we find that wave function for multiple indistinguishable particles must also be either symmetric or anti-symmetric with respect to exchange of particles. If you follow similar procedure for 3 identical particles you find Φ S = 1 6 [Ψ( 1 , 2 , 3 ) + Ψ( 1 , 3 , 2 ) + Ψ( 2 , 3 , 1 ) + Ψ( 2 , 1 , 3 ) + Ψ( 3 , 1 , 2 ) + Ψ( 3 , 2 , 1 )] and Φ A = 1 6 [Ψ( 1 , 2 , 3 ) - Ψ( 1 , 3 , 2 ) + Ψ( 2 , 3 , 1 ) - Ψ( 2 , 1 , 3 ) + Ψ( 3 , 1 , 2 ) - Ψ( 3 , 2 , 1 )] P. J. Grandinetti (Chem. 4300) Identical Particles in Quantum Mechanics Nov 17, 2017 9 / 20
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Fermions or Bosons Mathematically, Schrödinger equation will not allow symmetric wave function to evolve into anti-symmetric wave function and vice versa. Therefore, particles can never change their symmetric or anti-symmetric behavior under particle exchange. Furthermore, particles with half-integer spins s = 1 2 , 3 2 , 5 2 , are always found to have anti-symmetric wave functions with respect to particle exchange. These particles are classified as fermions .
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