Department of physics, NTNU TFY4340 Mesoscopic Physics Spring 2010

Exercise 7

Assistance: Tuesday March 16.

a) Show that the Schrödinger equation (SE), Hϕ = Eϕ, for a 1D harmonic oscillator,

H = −\frac{h̄^{2}}{2m} \frac{d^{2}}{dy^{2}} + \frac{1}{2}mω_{0}^{2}y^{2},

can be written in the form

[−\frac{d^{2}}{dξ^{2}} + ξ^{2}] ϕ(ξ) = εϕ(ξ),

by introduction of dimensionless variables ξ = y\sqrt{mω_{0}/h̄} and ε = 2E/h̄ω_{0}.

b) Consider a 2DEG subject to a uniform perpendicular magnetic field B = Bẑ and a harmonic confining potential V (y) = mω_{0}^{2}y^{2}/2. Use the Landau gauge A = −yBx̂, introduce the ansatz

ψ_{n,k}(x, y) = e^{ikx} ϕ_{n}(y),

and show that the resulting equation for ϕ_{n} becomes

[−\frac{d^{2}}{dη^{2}} + (η − κ)^{2} + α^{2}η^{2}] ϕ_{n}(η) = εϕ_{n}(η)

in the dimensionless variables

η = y/l_{B} ; κ = kl_{B} ; α = ω_{0}/ω_{c} ; ε = 2E/h̄ω_{c}.

Here, l_{B} = \sqrt{h̄/eB} is the magnetic length, and ω_{c} = eB/m is the cyclotron frequency.

c) Show that the equation for ϕ_{n}(η) can be written as a 1D harmonic oscillator equation, that the solutions in original variables are

ψ_{n,k}(x, y) = e^{ikx} ϕ_{n}(y − kL_{B}^{2}),

with

L_{B}^{2} = \frac{ω_{c}^{2}}{ω_{c}^{2} + ω_{0}^{2}} l_{B}^{2},

and that the energy spectrum is

E_{n,k} = h̄Ω (n + \frac{1}{2}) + \frac{h̄^{2}k^{2}}{2M_{B}} ,

with Ω^{2} = ω_{c}^{2} + ω_{0}^{2} and M_{B} = m Ω^{2}/ω_{0}^{2}.

d) Find the (probability) current density

j = \frac{h̄}{m}Im (ψ^{∗}∇ψ) + \frac{e}{m}A|ψ|^{2}

for the states ψ_{n,k} found above. (Hint: One of the components of j is zero, the other component will be an expression that contains ϕ_{n}^{2}.)

e) Find the total (probability) current along the channel, I_{x} = ∫ dyj_{x}(y).

(Hint: Take advantage of the symmetry properties of ϕ_{n}, and assume that the functions ϕ_{n} are normalized.) Check the two limiting cases

(1) ω_{0} → 0 , ω_{c} ≠ 0
(2) ω_{c} → 0 , ω_{0} ≠ 0