Advanced Solid State Physics

The conjugate variable is

∂L

∂Φ ̇

=CΦ = ̇ Q.

Then the Hamiltonian via a Legendre transformation:

H=QΦ ̇−L=

Q^2

2 C

+

(Φ−Φe)^2

2 L 1

−Ejcos

(

2 πΦ

Φ 0

)

Now the replacement of the conjugate variable, i.e. the quantization:

Q→−i~

∂

∂Φ

The result is the wanted Schrödinger equation for the given system:

−

~^2

2 C

∂^2 Ψ(Φ)

∂Φ^2

+

(Φ−Φe)^2

2 L 1

Ψ(Φ)−Ejcos

(

2 πΦ

Φ 0

)

Ψ(Φ) =EΨ(Φ)

3.1.3 Quantisation of a charged particle in a magnetic field (with spin)

For now we will neglect that the particle has spin, because in the quantisation we will treat the case
of a constant magnetic field pointing in the z-direction. In this simplified case there is no contribution
to the force from the spin interaction with the magnetic field (because it only adds a constant to the
Hamiltonian which does not affect the equation of motion).
We start with the equation of motion of a charged particle (without spin) in an EM-field, which is
given by the Lorentz force law.

F=m

∂^2 r(t)

∂t^2

=q(E+v×B) (7)

One can check that a suitable Lagrangian is

L(r,t) =

mv^2

2

−qφ+qv·A (8)

whereφandAare the scalar and vector Potential, respectively. From the Lagrangian we get the
generalized momentum (canonical conjugated variable)

pi=

∂L

∂r ̇x

=mvx+qAx

p=mv+qA (9)

v(p) =

1

m

(p−qA)