X 2 x 1 x 2 single particle hamiltonians must have

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x 2 ) = ̂ ( x 1 ) + ̂ ( x 2 ) Single particle Hamiltonians must have same form for particles to be identical. Schrödinger Eq. solutions for non-interacting particles can be written Ψ total = Ψ( x 1 )Ψ( x 2 ) Ψ( x 1 ) and Ψ( x 2 ) are individual particle wave functions. Simple proof: ̂ ( x 1 , ⃗ x 2 total = [ ̂ ( x 1 ) + ̂ ( x 2 ) ] Ψ( x 1 )Ψ( x 2 ) = E Ψ( x 1 )Ψ( x 2 ) = ̂ ( x 1 )Ψ( x 1 )Ψ( x 2 ) + Ψ( 1 ) ̂ ( x 2 )Ψ( x 2 ) = E Ψ( x 1 )Ψ( x 2 ) = E ( 1 )Ψ( x 1 )Ψ( x 2 ) + Ψ( x 1 ) E ( 2 )Ψ( x 2 ) = E Ψ( x 1 )Ψ( x 2 ) = [ E ( 1 ) + E ( 2 )] Ψ( x 1 )Ψ( x 2 ) = E Ψ( x 1 )Ψ( x 2 ) with total energy of E = E ( 1 ) + E ( 2 ) . P. J. Grandinetti (Chem. 4300) Identical Particles in Quantum Mechanics Nov 17, 2017 16 / 20
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Pauli exclusion principle If 2 non-interacting particles are fermions, e.g., e - , then we need to construct antisymmetric wave function, Φ A ( x 1 , ⃗ x 2 ) = 1 2 [ Ψ( x 1 )Ψ( x 2 ) - Ψ( x 2 )Ψ( x 1 ) ] to preserve indistinguishability of 2 electrons. Remember! There is zero probability of 2 fermions having same coordinates, x 1 = x 2 . In case of non-interacting fermions we find stronger constraint that two fermions cannot occupy identical wave functions, that is, same quantum states, also known as the Pauli exclusion principle . P. J. Grandinetti (Chem. 4300) Identical Particles in Quantum Mechanics Nov 17, 2017 17 / 20
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Slater determinant For N identical and non-interacting fermions occupying N different quantum states, which we labeled as Ψ a , Ψ b , , Ψ n , the anti-symmetric wave function can be expressed as a determinant Φ A ( x 1 , ⃗ x 2 , , ⃗ x N ) = 1 N ! | | | | | | | | | Ψ a ( x 1 ) Ψ b ( x 1 ) Ψ n ( x 1 ) Ψ a ( x 2 ) Ψ b ( x 2 ) Ψ n ( x 2 ) Ψ a ( x N ) Ψ b ( x N ) Ψ n ( x N ) | | | | | | | | | also known as a Slater determinant . Slater determinants enforce anti-symmetric wave functions. Any time 2 columns or rows are identical the determinant is zero. For example, if we try to place 3 identical electrons into only 2 different states, Ψ a and Ψ b , we find Φ A ( x 1 , ⃗ x 2 , ⃗ x 3 ) = 1 N ! | | | | | | | Ψ a ( x 1 ) Ψ b ( x 1 ) Ψ b ( x 1 ) Ψ a ( x 2 ) Ψ b ( x 2 ) Ψ b ( x 2 ) Ψ a ( x 3 ) Ψ b ( x 3 ) Ψ b ( x 3 ) | | | | | | | = 0 P. J. Grandinetti (Chem. 4300) Identical Particles in Quantum Mechanics Nov 17, 2017 18 / 20
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Example Approximate ground state wave function of 3 electrons in lithium atom as non-interacting fermions using H-like spin orbitals. Solution: First let’s consider the hard way. Start with earlier expression for 3 identical fermions 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 )] Next, define wave functions, e.g., 𝜓 ( 1 , 2 , 3 ) = 1s 𝛼 ( 1 ) 1s 𝛽 ( 2 ) 2s 𝛼 ( 3 ) , and 𝜓 ( 1 , 3 , 2 ) = 1s 𝛼 ( 1 ) 1s 𝛽 ( 3 ) 2s 𝛼 ( 2 ) Leave it as an exercise to find the other 4. Finally, plug all 6 wave functions into expression for Φ A above.
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