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Introduction

Identical particles cannot be distinguished by means of any intrinsic properties. This can lead to effects that have no classical analog. Two particles are identical if there are no interactions that can distinguish them. Therefore a physical observable must be symmetrical with respect to the interchange of any pair of two particles. The time-dependent Schrödinger equation for two identical particles is 

As H ( r₁ , r₂ ) = H ( r₂ , r₁ ), there are two fundamentally different kinds of solutions, namely Y ( r₁ , r₂ ) = Y ( r₂ , r₁ ) and Y ( r₁ , r₂ ) = -Y ( r₂ , r₁ ). The symmetric solution Y ( r₁ , r₂ ) = Y ( r₂ , r₁ ) describes particles that are called bosons. Particles that are described by the antisymmetric solution Y ( r₁ , r₂ ) = -Y ( r₂ , r₁ ) are called fermions. Electrons, protons and neutrons are fermions, and photons and pions are bosons. Atoms, being aggregates of tightly bound particles, are either fermions or bosons. Particles with integer spin are always bosons while particles with half-integer spin are always fermions. 

Fermions

Two identical fermions, e.g. two electrons with the same spin orientation, cannot occupy the same point in space, nor can they have the same value of momentum. Identical fermions must obey this exclusion principle. The picture on the right shows a probability distribution of two identical fermions. The dip between the two maxima reflects the fact that the two fermions cannot occupy the same point in space. The animations show the time development of the probability distribution of the two fermions, for the case that there is no interaction potential between the fermions. Upper part: Double-slit (in white), no magnetic field. Lower part: Double-slit (in white), confined magnetic field (in red). The magnetic flux through the red area is chosen such that for a single particle the shift in the diffraction pattern due to the Aharonov-Bohm effect is 180°. 

The animation shows the interchange of the maxima and minima of the probability distribution, due to the Aharonov-Bohm effect. 

The salient features of the intensity measured by detector D are the same as for the case where the source S emits a single particle. 

Angular distribution for fermions for B = 0 and B = B'. 

The fermionic nature of the wave reveals itself in the cross-correlated signal C of the detectors D and D'. Just as for a detector measuring the angular distribution, the positions of the detectors will be specified by the angles q₁ and q₂. 

Signals generated by the correlator C as a function of the detector positions. 

The correlated two-particle signal also shows the Aharonov-Bohm effect, as can be clearly seen by substracting the B = 0 and B = B' signals. 

Absolute value of the difference of the B = 0 and B = B' signals.

Bosons

Two non-interacting identical bosons can occupy the same point in space, or they can have the same value of momentum. The picture on the right shows a probability distribution of two identical bosons. 

The animations show the time development of the probability distribution of the two non-interacting bosons. To illustrate the differences between the fermions and the bosons, the latter are also assumed to carry electrical charge. Upper part: Double-slit (in white), no magnetic field. Lower part: Double-slit (in white), confined magnetic field (in red). The magnetic flux through the red area is chosen such that for a single particle the shift in the diffraction pattern due to the Aharonov-Bohm effect is 180°. 

The animation shows the interchange of the maxima and minima of the probability distribution, due to the Aharonov-Bohm effect. 

The salient features of the intensity measured by detector D are the same as for the case where the source S emits a single particle. 

Angular distribution for bosons for B = 0 and B = B'. 

Comparing the angular distributions for fermions and bosons shows that, due to the exclusion principle, the interference fringes of the former are less pronounced, as illustrated in the next figure. 

Angular distribution for fermions and bosons for B = B'. 

The bosonic nature of the wave reveals itself in the cross-correlated signal C of the detectors D and D'. Just as for a detector measuring the angular distribution, the positions of the detectors are specified by the angles q₁ and q₁. 

Signals generated by the correlator C as a function of the detector positions. 

The symmetry of the wave function leads to an increase of the probability for finding two bosons close to each other. Indeed, the left picture shows that the probability is the largest for q₁ = q₂ = 0, i.e. if the bosons are on top of each other. The correlated two-particle signal also shows the Aharonov-Bohm effect, as can be clearly seen by subtracting the B = 0 and B = B' signals. 

Absolute value of the difference of the B = 0 and B = B' signals.

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