{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

sol09 - Solid State Theory Solution Sheet 9 FS 11 Prof M...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Solid State Theory Solution Sheet 9 FS 11 Prof. M. Sigrist Exercise 9.1 Lowest Landau level in the Corbino geometry a) Because A z = 0 and z A = 0 it follows that ( ∇× A ) x = ( ∇× A ) y = 0. Furthermore, for r 6 = 0 we have ( ∇ × A ) z = 1 2 ∂x B + ν Φ 0 πr 2 x + 1 2 ∂y B + ν Φ 0 πr 2 y = B. (1) On the other hand, the magnetic flux through the origin is Φ = lim 0 Z r< B · d σ = lim 0 Z r< ∇ × A · d σ = lim 0 Z r = A · d γ = lim 0 2 π 1 2 B + ν Φ 0 π = ν Φ 0 (2) where we used Stokes’ theorem in the second line. b) H = 1 2 m * ( p x + eB 2 c y ) 2 + ( p y - eB 2 c x ) 2 + U ( r ) = 1 2 m * p 2 x + p 2 y - eB c ( xp y - yp x ) + eB 2 c r 2 + U ( r ) = ~ 2 2 m * - 1 r r r∂ r - 2 φ r 2 - i∂ φ l 2 + r 2 l 2 2 + U ( r ) = ~ 2 2 m * - 1 r r r∂ r + i∂ φ r - r 2 l 2 2 + U ( r ) (3) Using the ansatz for the wave function of the lowest Landau level we obtain the radial Schr¨ odinger equation given on the exercise sheet: ~ 2 2 m * - 1 r ∂r r ∂r + m r - r 2 l 2 2 + U ( r ) - E m r α e - r 2 4 l * 2 = 0 . (4) Furthermore, we obtain r∂ r ψ m = α - r 2 2 l * 2 ψ m (5) from which it follows that 1 r r r∂ r ψ m = 1 r 2 α - r 2 2 l * 2 2 ψ m - 1 l * 2 ψ m . (6) Defining E * m = 2 m * E m / ~ 2 we can write the radial Schr¨ odinger equation as - α r - r 2 l * 2 2 + m r - r 2 l 2 2 + 1 l * 2 + C * 1 r 2 + C * 2 r 2 4 l 4 - E * m ψ m = 0 (7) 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
0 2 4 6 8 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 r/l ψ m ( r ) m = 0 m = 2 m = 5 m = 9 Figure 1: Radial part of the wave function for several m ’s. or as 1 r 2 m 2 + C * 1 - α 2 + r 2 4 1 l 4 (1 + C * 2 ) - 1 l * 4 + α + 1 l * 2 - m l 2 - E * m ψ m = 0 . (8) In order to satisfy this equation for all r > 0 the expressions in the first two brackets have to vanish. This gives the relation between m and α and between l
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}