QuantumPhysics.dvi

(Wang) #1

Landau level satisfy the differential equation


(


∂z ̄


+


mωB


̄h


z


)

ψ(0)(z, ̄z) = 0 (7.33)


Its general solution is straightforward (since as a differential equation in ̄z, you may think


ofzas a constant coefficient), and we get


ψ(z, ̄z) =φ(z)ψ(0)(z,z ̄) ψ(0)(z,z ̄) = exp


{


mωB


̄h


|z|^2


}

(7.34)


whereφ(z) is an arbitrary complex analytic function ofz. Actually,φ(z) cannot has poles


since they would lead to non-square integrable wave functions. Also, assuming thatψ(z,z ̄)


is single-valued, it follows thatφ(z) must be single-valued, and thus cannot have branch


cuts. Therefore,φ(z) must be single-valued and holomorphic throughout the complex plane.


A basis for such functions is obtained by polynomials (and square integrable Taylor expand-


able functions by completeness). Notice that all such polynomial functions are obtained by


applyinga†−repeatedly to the ground stateψ(0)of botha+anda−, since we have


ψ(n−)∼(a†−)n−ψ(0)(z, ̄z)∼zn−ψ(0)(z,z ̄) (7.35)


This agrees with general principles that the group theory ofa−,a†−must precisely reproduce


the normalizable spectrum.


7.4 The Aharonov-Bohm Effect


The classical equations of motion governing a charged particle in an electro-magnetic field


involve the fieldsBandE, even though both the Lagrangian and the Hamiltonian must be


formulated not in terms ofBandE, but rather in terms of the gauge potentials Φ andA.


The quantum equations are based on the Hamiltonian (or on the Lagrangian if one uses path


integral methods), and necessarily involves the gauge potentials ΦandA. This difference


leads to measurable effects, of which the most striking are the Aharonov-Bohm effects.


7.4.1 The scattering Aharonov-Bohm effect


Consider a beam of particles of chargee, moving in a plane perpendicular to an infinite


impenetrable straight solenoid with a non-zero magnetic flux ΦB. One may realize impene-


trability by requiring the presence of a potentialV(r) which is infinite inside the solenoid,


and vanishes outside. Although the particles can propagate onlyoutsidethe solenoid, where


B= 0, the presence of the solenoid produces measurable interference effects. Outside the

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