The Hamiltonian for a system of n particles that are identical, that is, particles that can be substituted for each other with no physical change, must be completely symmetric under any interchange of its arguments. Let Ψ(1, 2, …, n) be any solution of the Schrödinger equation, depending on the coordinates (spatial and spin) of the n identical particles 1 to n, and let P be any permutation of the n numbers 1 to n. Then PΨ(1, 2, …, n) ≡ Ψ(P1, P2, …, Pn) is a function which depends on the coordinates of particle Ψ in the same manner as the original function Ψ depended on the coordinates of particle i. Then using the symmetry of H, we can show that the operator P commutes with the Hamiltonian ( 2-1 ) H ( P Ψ ) = P ( H Ψ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493645/749bab20-4a79-4f5a-8c66-d9e9a36a87b9/content/eq89.tif"/>
The Hamiltonian for a system of n particles that are identical, that is, particles that can be substituted for each other with no physical change, must be completely symmetric under any interchange of its arguments. Let Ψ(1, 2, …, n) be any solution of the Schrödinger equation, depending on the coordinates (spatial and spin) of the n identical particles 1 to n, and let P be any permutation of the n numbers 1 to n. Then PΨ(1, 2, …, n) ≡ Ψ(P1, P2, …, Pn) is a function which depends on the coordinates of particle Ψ in the same manner as the original function Ψ depended on the coordinates of particle i. Then using the symmetry of H, we can show that the operator P commutes with the Hamiltonian ( 2-1 ) H ( P Ψ ) = P ( H Ψ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493645/749bab20-4a79-4f5a-8c66-d9e9a36a87b9/content/eq89.tif"/>
H a , a =1,2,..., N are identical copies of a one particle Hamiltonian acting in different Hilbert spaces and V ab , a < b are identical copies of the interaction Hamiltonian between two particles We try a wave function | ψ t > = ⊗ k - 1 N | ψ k t > https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1428.tif"/> Substituting this into the Schrodinger equation gives Σ k ψ 1 t ⊗ … ⊗ ( i d ψ k t / d t ) ⊗ … ⊗ ψ n t = = Σ k ψ 1 t ⊗ ... ⊗ H k ψ k t ⊗ ... ⊗ ψ n t + + Σ a < b ψ 1 t ⊗ ... ⊗ V a b ψ a t ... ⊗ ψ b t ⊗ ... ⊗ ψ n t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1429.tif"/> Taking the inner product on both sides with ψ 1 t ⨶.. ψ kt ⨶...⨶ ψ nt where a hat above a symbol means omission of that symbol gives the approximate equations i d | ψ k t > / d t = H k | ψ k t > + Σ m ≠ k < I ⊗ ψ m t | V k m | ψ k t ⊗ ψ m t > https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1430.tif"/> 256 In terms of kernels, we can express this as i d ψ k t ( r ) / d t = H k ψ k t ( r ) + ( Σ m ≠ k ∫ V ( r , r ′ ) | ψ m t ( r ′ ) | 2 d 3 r ′ ) ψ k t ( r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1431.tif"/> This equation however does not take into account the Pauli exclusion principle. For taking that into account, we must try an antisymmetrized wave function | ψ t > = Σ σ ∈ S n | ψ σ 1 , t > ⊗ ... ⊗ | ψ σ n , t > https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1432.tif"/> Substituting and taking the inner products after assuming orthonormality of the component wave functions gives us i d | ψ k t > / d t = H k | ψ k t > + Σ m ≠ k < I ⊗ ψ m t | V | ψ k t ⊗ ψ m t > − Σ m ≠ k < ψ m t ⊗ I | V | ψ k t ⊗ ψ m t > https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1433.tif"/> which in coordinate form can be expressed as i d ψ k t ( r ) / d t = H k ψ k t ( r ) + Σ m ≠ k ( ∫ V ( r , r ′ ) | ψ m t ( r ′ ) | 2 d 3 r ′ ) ψ k t ( r ) − Σ m ≠ k ( ∫ V ( r , r ′ ) ψ ¯ m t ( r ′ ) ψ k t ( r ′ ) d 3 r ′ ) ψ m t ( r ) = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1434.tif"/> These equations are special cases of nonlinear Schrodinger equations of the form i d ψ k t ( r ) / d t = ∇ 2 ψ k t ( r ) / 2 m + V 0 ( r ) ψ k t ( r ) + Σ m V k m ( r , ψ 1 t , ... , ψ N t ) ψ m t ( r ) , k = 1 , 2 , ... , N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1435.tif"/> The charge density is ρ ( t , r ) = − e Σ k | ψ k t ( r ) | 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1436.tif"/> and its rate of increase is given by ρ t ( r ) = - e Σ k ( ψ ¯ k t , t ( r ) . ψ k t ( r ) + ψ ¯ k t ( r ) . ψ k t , t ( r ) = ( i e / 2 m ) Σ k [ ψ k t ( r ) ∇ 2 ψ ¯ k t ( r ) - ψ ¯ k t ( r ) ∇ 2 ψ k t ( r ) ] = d i v ( ( i e / 2 m ) Σ k [ ψ k t ( r ) ∇ ψ ¯ k t ( r ) - ψ ¯ k t ( r ) ∇ ψ k t ( r ) ] ) = - d i v J ( t , r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1437.tif"/> 257 where the current density J ( t,r ) is given by J ( t , r ) = ( − i e / 2 m ) Σ k [ ψ k t ( r ) ∇ ψ ¯ k t ( r ) − ψ ¯ k t ( r ) ∇ ψ k t ( r ) ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1438.tif"/> It should be noted that this expression for the current density is the same as that derived using the linear Schrodinger equations but the wave functions ψ kt are calculated by solving the nonlinear Schrodinger equation. It should be noted that in deriving the above formula for the current density, we have made used of the relation V ¯ k m ( r , ψ 1 , … , ψ N ) = V m k ( r , ψ 1 , … , ψ N ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1439.tif"/> which is readily verified.
The postulates of quantum mechanics and their consequences, which have been developed to describe a single particle, are still valid when applied to a system of N particles. The system can be described by a wave function Ψ which depends on the time and on the 3 N coordinates of all the particles: 10.1 Ψ = Ψ ( r → 1 , … r → i , … r → N , t ) = Ψ ( x 1 , y 1 , z 1 , … , x i , y i , z i , … x N , y N , z N , t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn10_1_B.tif"/>
Spherically symmetric systems, where the potential energy U(r) is independent of the direction of r → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/inline-eqn7_1_B.tif"/> , are usually treated using spherical polar coordinates (r,θ,φ), related to the Cartesian coordinates as shown by Eqs.(2.20) and ( 2.21) . Therefore, in view of the appropriate form ( 2.36) of the Hamiltonian, the Schrodinger wave equation for Ψ (r.θ,φ) reads: 7.1 H ^ Ψ = − h 2 m [ 1 r 2 ∂ ∂ r ( r 2 ∂ Ψ ∂ r ) + 1 r 2 sin θ ∂ ∂ θ ( sin θ ∂ Ψ ∂ θ ) + 1 r 2 sin 2 θ ∂ 2 Ψ ∂ φ 2 ] + U ( r ) Ψ = E Ψ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn7_1_B.tif"/> In contrast to the motion of a particle whose potential energy depends on Cartesian coordinates, where the Schrödinger equation can be separated into three equations pertaining to motions along the axes x, y, z, which are independent of one another (see Problem 5.4.2 ), Eq.(7.1) also separates into equations pertaining to motions along the coordinates r, θ and φ, but the three types of motion are no longer independent. The separation of variables is made in two stages, first substituting into the energy eigenvalue equation the trial solution: 7.2 Ψ ( r , θ , φ ) = R ( r ) Y ( θ , φ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn7_2_B.tif"/> This transforms Eq.(7.1) in the form: 7.3 h 2 2 m [ 1 r ∂ ∂ r ( r 2 ∂ R ∂ r ) ] + r 2 [ E − U ( r ) ] = − h 2 2 m 1 Y [ 1 sin θ ∂ ∂ θ ( sin θ ∂ Y ∂ θ ) + 1 sin 2 θ ∂ 2 Y ∂ φ 2 ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn7_3_B.tif"/> where one side is independent of θ and φ , and the other side is independent of r , so that they must be separately equal to some constant, which is conveniently chosen as L 2 / 2 m. In this way we separate the radial motion, described by an ordinary differential equation where ∂ /∂r is replaced by d/dr: 7.4 [ − h 2 2 m 1 r 2 d d r ( r 2 d d r ) + U ( r ) + L 2 2 m r 2 ] R ( r ) = E R ( r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn7_4_B.tif"/> 243 from the rotational motion, described by the angular equation: 7.5 − h 2 [ 1 sin θ ∂ ∂ θ ( sin θ ∂ ∂ θ ) + 1 sin 2 θ ∂ 2 ∂ φ 2 ] Y= L 2 Y or L ^ 2 Y= L 2 Y https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn7_5_B.tif"/> which is the eigenvalue problem for orbital angular momentum, represented by the operator ( 2.33) . This separation corresponds to the classical splitting of kinetic energy among the two types of motion. The second stage of the separation process assumes that: 7.6 Y( θ , φ )= P ( θ ) Φ ( φ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn7_6_B.tif"/> such that Eq.(7.5) becomes: 7.7 h 2 [ sin θ P ( θ ) ∂ ∂ θ ( sin θ ∂ P ( θ ) ∂ θ ) + L 2 h 2 sin 2 θ ] = − h 2 1 Φ ( φ ) ∂ 2 Φ ( φ ) ∂ φ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn7_7_B.tif"/> Since one side is independent of φ , while the other is independent of θ , they must each be equal to a constant, which we call L z 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/inline-eqn7_2_B.tif"/> , and this yields two other ordinary differential equations: 7.8 − h 2 d 2 d φ 2 Φ ( φ ) = L z 2 Φ ( φ ) or L ^ z 2 Φ ( φ ) = L z 2 Φ ( φ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn7_8_B.tif"/> where L ^ z https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/inline-eqn7_3_B.tif"/> is defined by Eq.(2.25) , and: 7.9 1 sin θ d d θ ( sin θ d P ( θ ) d θ ) ( L 2 h 2 − L z 2 h 2 sin 2 θ ) P ( θ ) = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn7_9_B.tif"/>
We look for a Hamiltonian to use in the time-dependent Schrödinger equation, and because this is first order in time we stay first order in space trying to treat all coordinates on a similar basis. Write 4.1 https://www.w3.org/1998/Math/MathML"> i ∂ ψ ∂ t = H ψ = − i α i ∂ ψ ∂ x i + m β ψ i = 1 , 2 , 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429184550/8ac61231-bcf3-4e9e-a16f-9d6f78005bbb/content/math4_1_B.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> where ψ is now a column and the α i and β are square matrices.
In this Chapter we discuss the nonrelativistic theory of the elastic scattering of spin 1 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493645/749bab20-4a79-4f5a-8c66-d9e9a36a87b9/content/eq1266.tif"/> particles of mass μ in a spin-dependent potential which contains an ℓ · s term. The Schrödinger equation for the particle is ( 16-1 ) ( − ħ 2 2 μ ∇ 2 + V(r) + 2 ℓ ⋅ s U ( r ) − E ) Ψ = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493645/749bab20-4a79-4f5a-8c66-d9e9a36a87b9/content/eq1267.tif"/>
It is demonstrated that in a system of N similar particles, the permissible wave functions are either symmetrical or antisymmetrical with respect to interchanges of the particles, if the following principles are assumed to be valid: (1) The interchange of two like particles produces no change in any measurable property of the system. In particular, if $\ensuremath{\psi}$ is to be a permissible wave function, $\ensuremath{\psi}\overline{\ensuremath{\psi}}$ must be unaltered by such an interchange. (2) All the wave functions obtained from a given permissible wave function by permuting the similar particles are also permissible wave functions for the same eigenvalue.