Question

Aharonov-Bohm using density matrix?

Answer 1

pure state |Φi. The density matrix can then enter into the theory in one of two ways:
1. The system is composed of entangled subsystems. Tracing over the Hilbert space of a subsystem gives a density matrix which contains the statistics of measurements performed only on the remainder of the system:
ρ(Y ) = TrX [|Φ(X, Y )i hΦ(X, Y )|] . (1)
2. The system is not prepared as an ensemble of identical states, but occurs in an ensemble for which the state |Φii appears randomly with probability pi:
ρ = X pi |Φii hΦi| . (2)
i
In both cases, the properties of the density matrix have significant differences from an equivalent classical probability distribution. For entangled states, the en- tropy of a subsystem can exceed the entropy of the overall state, which cannot pure state |Φi. The density matrix can then enter into the theory in one of two ways:
1. The system is composed of entangled subsystems. Tracing over the Hilbert space of a subsystem gives a density matrix which contains the statistics of measurements performed only on the remainder of the system:
ρ(Y ) = TrX [|Φ(X, Y )i hΦ(X, Y )|] . (1)
2. The system is not prepared as an ensemble of identical states, but occurs in an ensemble for which the state |Φii appears randomly with probability pi:
ρ = X pi |Φii hΦi| . (2)
i
In both cases, the properties of the density matrix have significant differences from an equivalent classical probability distribution. For entangled states, the en- tropy of a subsystem can exceed the entropy of the overall state, which cannot

Answer 2

we discussed the Aharonov–Bohm and Aharonov–Casher effects, and showed that persistent currents are a consequence of the Aharonov–Bohm effect. In a conducting (metal or semiconductor) ring of size , the ground state of the itinerant electron system does not carry current in the absence of magnetic field. But when the ring is threaded by a magnetic flux, Φ, time-reversal invariance is broken. The wave function of an electron gains an Aharonov–Bohm (or Berry) phase ϕ when moving along the ring in one direction and −ϕ when moving in the opposite direction. Here, , where  (the unit of flux quantum). Due to this chirality, the ground state carries a current . As a result of gauge invariance, the current is a periodic function of ϕ with period 1. In addition, the current is an antisymmetric function of ϕ. Persistent current is an equilibrium property: It is nondissipative and it is a property of all the states below the Fermi energy.