Exchange interaction

In physics, the exchange interaction (with an exchange energy and exchange term) is a quantum mechanical effect that only occurs between identical particles. Despite sometimes being called an exchange force in an analogy to classical force, it is not a true force as it lacks a force carrier. Ψ A ( r → 1 , r → 2 ) = 1 2 [ Φ a ( r → 1 ) Φ b ( r → 2 ) − Φ b ( r → 1 ) Φ a ( r → 2 ) ] {displaystyle Psi _{A}({vec {r}}_{1},{vec {r}}_{2})={frac {1}{sqrt {2}}}}     (1) Ψ S ( r → 1 , r → 2 ) = 1 2 [ Φ a ( r → 1 ) Φ b ( r → 2 ) + Φ b ( r → 1 ) Φ a ( r → 2 ) ] {displaystyle Psi _{S}({vec {r}}_{1},{vec {r}}_{2})={frac {1}{sqrt {2}}}}     (2)   E ± = E ( 0 ) + C ± J e x 1 ± B {displaystyle E_{pm }=E_{(0)}+{frac {Cpm J_{ex}}{1pm B}}}     (3) C = ∫ Φ a ( r → 1 ) 2 ( 1 R a b + 1 r 12 − 1 r a 1 − 1 r b 2 ) Φ b ( r → 2 ) 2 d 3 r 1 d 3 r 2 {displaystyle C=int Phi _{a}({vec {r}}_{1})^{2}left({frac {1}{R_{ab}}}+{frac {1}{r_{12}}}-{frac {1}{r_{a1}}}-{frac {1}{r_{b2}}} ight)Phi _{b}({vec {r}}_{2})^{2},d^{3}r_{1},d^{3}r_{2}}     (4) B = ∫ Φ b ( r → 2 ) Φ a ( r → 2 ) d 3 r 2 {displaystyle B=int Phi _{b}({vec {r}}_{2})Phi _{a}({vec {r}}_{2}),d^{3}r_{2}}     (5) J e x = ∫ Φ a ∗ ( r → 1 ) Φ b ∗ ( r → 2 ) ( 1 R a b + 1 r 12 − 1 r a 1 − 1 r b 2 ) Φ b ( r → 1 ) Φ a ( r → 2 ) d 3 r 1 d 3 r 2 {displaystyle J_{ex}=int Phi _{a}^{*}({vec {r}}_{1})Phi _{b}^{*}({vec {r}}_{2})left({frac {1}{R_{ab}}}+{frac {1}{r_{12}}}-{frac {1}{r_{a1}}}-{frac {1}{r_{b2}}} ight)Phi _{b}({vec {r}}_{1})Phi _{a}({vec {r}}_{2}),d^{3}r_{1},d^{3}r_{2}}     (6) α ( 1 ) {displaystyle alpha (1)} β ( 2 ) {displaystyle eta (2)} ± α ( 2 ) {displaystyle alpha (2)} β ( 1 ) {displaystyle eta (1)}     (7) E e x − C + 1 2 J e x + 2 J e x ⟨ s → a ⋅ s → b ⟩ = 0 {displaystyle E_{ex}-C+{frac {1}{2}}J_{ex}+2J_{ex}langle {vec {s}}_{a}cdot {vec {s}}_{b} angle =0}     (8) E e x = C − 1 2 J e x − 2 J e x ⟨ s → a ⋅ s → b ⟩ {displaystyle E_{ex}=C-{frac {1}{2}}J_{ex}-2J_{ex}langle {vec {s}}_{a}cdot {vec {s}}_{b} angle }     (9)   − 2 J a b ⟨ s → a ⋅ s → b ⟩ {displaystyle -2J_{ab}langle {vec {s}}_{a}cdot {vec {s}}_{b} angle }     (10) H H e i s = − 2 J a b ⟨ s → a ⋅ s → b ⟩ {displaystyle {mathcal {H}}_{Heis}=-2J_{ab}langle {vec {s}}_{a}cdot {vec {s}}_{b} angle }     (11)   J a b = 1 2 ( E + − E − ) = J e x − C B 2 1 − B 4 {displaystyle J_{ab}={frac {1}{2}}(E_{+}-E_{-})={frac {J_{ex}-CB^{2}}{1-B^{4}}}}     (12)   E − − E + = 2 ( C B 2 − J e x ) 1 − B 4 {displaystyle E_{-}-E_{+}={frac {2(CB^{2}-J_{ex})}{1-B^{4}}}}     (13) H H e i s = 1 2 ( − 2 J ∑ i , j ⟨ S → i ⋅ S → j ⟩ ) = − ∑ i , j J ⟨ S → i ⋅ S → j ⟩ {displaystyle {mathcal {H}}_{Heis}={frac {1}{2}}left(-2Jsum _{i,j}langle {vec {S}}_{i}cdot {vec {S}}_{j} angle ight)=-sum _{i,j}Jlangle {vec {S}}_{i}cdot {vec {S}}_{j} angle }     (14) A s c = J e x ⟨ S 2 ⟩ a {displaystyle A_{sc}={frac {J_{ex}langle S^{2} angle }{a}}}     (15) A b c c = 2 J e x ⟨ S 2 ⟩ a {displaystyle A_{bcc}={frac {2J_{ex}langle S^{2} angle }{a}}}     (16) A f c c = 4 J e x ⟨ S 2 ⟩ a {displaystyle A_{fcc}={frac {4J_{ex}langle S^{2} angle }{a}}}     (17) E = − ∑ i ≠ j J i j ⟨ S i z S j z ⟩ {displaystyle E=-sum _{i eq j}J_{ij}langle S_{i}^{z}S_{j}^{z} angle ,}     (18) In physics, the exchange interaction (with an exchange energy and exchange term) is a quantum mechanical effect that only occurs between identical particles. Despite sometimes being called an exchange force in an analogy to classical force, it is not a true force as it lacks a force carrier. The effect is due to the wave function of indistinguishable particles being subject to exchange symmetry, that is, either remaining unchanged (symmetric) or changing sign (antisymmetric) when two particles are exchanged. Both bosons and fermions can experience the exchange interaction. For fermions, this interaction is sometimes called Pauli repulsion and is related to the Pauli exclusion principle. For bosons, the exchange interaction takes the form of an effective attraction that causes identical particles to be found closer together, as in Bose–Einstein condensation. The exchange interaction alters the expectation value of the distance when the wave functions of two or more indistinguishable particles overlap. This interaction increases (for fermions) or decreases (for bosons) the expectation value of the distance between identical particles (compared to distinguishable particles). Among other consequences, the exchange interaction is responsible for ferromagnetism and the volume of matter. It has no classical analogue. Exchange interaction effects were discovered independently by physicists Werner Heisenberg and Paul Dirac in 1926. The exchange interaction is sometimes called the exchange force. However, it is not a true force and should not be confused with the exchange forces produced by the exchange of force carriers, such as the electromagnetic force produced between two electrons by the exchange of a photon, or the strong force between two quarks produced by the exchange of a gluon. Although sometimes erroneously described as a force, the exchange interaction is a purely quantum mechanical effect unlike other forces. Quantum mechanical particles are classified as bosons or fermions. The spin–statistics theorem of quantum field theory demands that all particles with half-integer spin behave as fermions and all particles with integer spin behave as bosons. Multiple bosons may occupy the same quantum state; however, by the Pauli exclusion principle, no two fermions can occupy the same state. Since electrons have spin 1/2, they are fermions. This means that the overall wave function of a system must be antisymmetric when two electrons are exchanged, i.e. interchanged with respect to both spatial and spin coordinates. First, however, exchange will be explained with the neglect of spin. Taking a hydrogen molecule-like system (i.e. one with two electrons), we may attempt to model the state of each electron by first assuming the electrons behave independently, and taking wave functions in position space of Φ a ( r 1 ) {displaystyle Phi _{a}(r_{1})} for the first electron and Φ b ( r 2 ) {displaystyle Phi _{b}(r_{2})} for the second electron. We assume that Φ a {displaystyle Phi _{a}} and Φ b {displaystyle Phi _{b}} are orthogonal, and that each corresponds to an energy eigenstate of its electron. Now, we may construct a wave function for the overall system in position space by using an antisymmetric combination of the product wave functions in position space: