Solutions of the Landau Problem;  vortices 
(negative circulation is classical)

r = Sqrt[x^2 + y^2]

Axial probability current  (for  | 
                m0 >  states):

jϕ[ρ_, 
        m_] := (m - ρ^2)*ρ^(2*m)/(π*m!)*
        Exp[-ρ^2]

Check normalization:

Assuming[{m ∈ Integers, 
        m ≥ 0}, 
      Integrate[
        2*π*x*x^(2*m)/(π*m!)*Exp[-x^2], {x, 
          0, ∞}]]

Check total angular momentum:

Assuming[{m ∈ Integers, 
        m ≥ 0}, 
      Integrate[2*π*jϕ[x, m]*x, {x, 0, ∞}]]

<< Graphics`PlotField`

m = 1
PlotVectorField[{-y/r^2*jϕ[r, m], 
        x/r^2*jϕ[r, m]}, {x, -2.5, 2.5}, {y, -2.5, 2.5}, 
      PlotPoints -> 25, 
      ColorFunction -> (RGBColor[#, 0, 
              0.7 - 0.3*#] &)]
Plot[jϕ[Abs[x]*x^2, m], {x, -5, 5}, PlotPoints -> 30, 
      PlotRange -> {-0.1, 0.1}]

m = 2
PlotVectorField[{-y/r^2*jϕ[r, m], 
        x/r^2*jϕ[r, m]}, {x, -2.5, 2.5}, {y, -2.5, 2.5}, 
      PlotPoints -> 25, 
      ColorFunction -> (RGBColor[#, 0, 
              0.7 - 0.3*#] &)]
Plot[jϕ[Abs[x]*x^2, m], {x, -5, 5}, PlotPoints -> 30, 
      PlotRange -> {-0.1, 0.1}]

Probability current - 
          densities as functions of distance from the origin (negative is classical)

Plot[{jϕ[Abs[r], 1]*r, 
        jϕ[Abs[r], 4]*r, jϕ[Abs[r], 10]*r}, {r, 0, 5}, 
      PlotPoints -> 30, 
      PlotRange -> {-0.4, 0.2}]

Gauge A = (0, B x, 0)

n - th Harmonic oscillator wavefunction:

ψ[n_, 
        x_] := π^{-1/4}*(1/(Sqrt[n!]))*
        Nest[(1/(Sqrt[2])) (x*# - (∂_x) #) &, Exp[-x^2/2], 
          n]

Normalization check:

Integrate[
      Abs[ψ[5, x]^2], {x, -∞, ∞}]

\(\(Plot\ of\ the\ currents\ \((j = \(\(-x\)\(*\)\) | ψ\( | \^2\))\)\)\(:\)\)

f = Simplify[ψ[2, x]]
G1 = 
      PlotVectorField[{0, -x*Abs[f[[1]]]^2}, {x, -3, 3}, {y, 0, 
          4}, Frame -> True, PlotPoints -> 15, 
        HeadWidth -> 1.5, HeadLength -> 0.01, 
        ColorFunction -> (RGBColor[#, 0, 
                0.7 - 0.3*#] &)]
G2 = 
      Plot[-x*Abs[f]^2, {x, -3, 3}, 
        PlotRange -> {-1, 1}]
Show[{G1, G2}]