Supersymmetric quantum mechanics of the 2D Kepler problem

Transcription

1 quantum spectrum of the 2D Supersymmetric of the 2D Juan Mateos Guilarte 1,2 1 Departamento de Física Fundamental Universidad de Salamanca 2 IUFFyM Universidad de Salamanca Summer Lecture Notes, Spain, Colombia, 2010

2 Outline quantum spectrum of the 2D 1 quantum 2 spectrum of the 3 2D 4 5

3 quantum spectrum of the 2D classical action S = = d t classical { m d x k d xk + α r } 2 d t d t d t L d xk d t, xk ; Change of variables, r = + x k x k, k = 1, 2 α > 0, [α] = ML3 T 2 = energy length x k = 1 1 xk, t = mα mα t ; [xk] = 2 M2 L 4 T 2, [t] = M 3 L 6 T 3 { 1 dx k S = dt dxk + 1 } = dt L dxk 2 dt dt r dt, xk Hamiltonian formalism: p k = mαp k, H = mα 2 H p k = dxk dx k, H = p k L = 1 dt dt 2 pkpk 1 r [p k] = M 1 L 2 T, [H] = M 2 L 4 T 2

4 quantum spectrum of the 2D angular momentum classical invariants L = x 1p 2 x 2p 1, [L] = ML 2 T 1, {H, L} = 0 vector of Runge-Lenz A 1 = p 2L x1 r, {H, A k} = 0, {A 1, A 2} = 2 x2 A2 = p1l r, [Ak] = pnpn 1 L = 2E L r SO3 symmetry {L, A 1} = A 2, {L, A 2} = A 1, {M k, M j} = L 1 M k = 2E Ak, [Mk] = ML2 T 1 M 3 = L 3, {M a, M b} = ε abcm c, a, b, c = 1, 2, 3 C 2 = M1 2 + M2 2 + M3 2 = 1 A A L 2 2E

5 quantum spectrum of the 2D quantum Hamiltonian quantum invariants p k ˆp k = i x k, x k ˆx k = x k, [ˆx k, ˆp j] = iδ kj Ĥ = r, = x k x k quantum angular momentum ˆL = i x 1 x 2 x 2 x 1, [Ĥ, ˆL] = 0 quantum Runge-Lenz vector  1 = 1 2 ˆp2ˆL + ˆLˆp 2 ˆx 1 r,  2 = 1 2 ˆp1ˆL + ˆLˆp 1 ˆx 2 r [ˆL,  1] = i  2, [ˆL,  2] = i  1, [Ĥ,  1] = [Ĥ,  2] = 0

6 quantum spectrum of the 2D SO3 dynamical symmetry SO3 quantum symmetry ˆp 2 [Â 1, Â 2] = 2i 1 + ˆp ˆL, 2 r [ ˆM a, ˆM b] = i ε abc ˆM c ˆM 1 = 1 Â 1, ˆM 2 = 1 Â 2, 2E 2E ˆM 3 = ˆL Casimir operator and the Hamiltonian Ĉ 2 = ˆM ˆM ˆM 3 2 = 1 2Ĥ Â Â 2 2 = 2Ĥ ˆL D Kepler spectrum Ĥψ j = 1 2 Â2 1 + Â ˆL Ĥ = Ĉ jj ψj = j + 1 ψj, j N j = n 2, ˆ Hψ n = mα2 2 2 n ψn

7 quantum spectrum of the 2D Spectral quantum Irreducible SO3 representations Ĉ 2 j; m = 2 jj + 1 j; m ˆM 3 j; m = m j; m, m : j, j + 1,, j 1, j highest weight eigen-wave function  + = 2 e [i iϕ 2 r ϕ 1 2 r ϕ r i ] e iϕ 2r ϕ  +ψ jjr, ϕ = 0, ψ jjr, ϕ = f jre ijϕ f j j r = r 2r f jr f jr = r j 2r exp{ 2 2j j + 1 } Ladder operator  = 2 e [ i iϕ 2 r ϕ 1 2 r ϕ r + i ] e iϕ 2r ϕ  ψ jjr, ϕ = 2 j2j 1 4j 2j + 1 r r j 1 2r exp{ 2 2j + 1 }eij 1ϕ

8 quantum spectrum of the 2D Supercharges ˆQ = i ˆψ k + W x k x k 2D N = 2 SUSY QM = e W ˆQ 0e W, [W] = ML 2 T 1 ˆQ 0 = i ˆψ k x k, ˆQ 2 o = 0 = ˆQ 2, [ˆQ] = M 1 L 2 T Fermionic operators { ˆψ k, ˆψ l} = 0 = { ˆψ k, ˆψ l }, { ˆψ k, ˆψ l } = δkl, k, l = 1, 2, [ ˆψ k] = 1 Hilbert space: the fermionic Fock space H = L 2 R 2 C 4 = H 0 H1 H2, ˆN = ˆψ k ˆψ k ˆψ k 0 = 0, k = 1, 2, ˆψ k 0 = 1 k, ˆN 0 = 0, ˆN 1 k = 1 k ˆψ = = ˆψ 1 1 2, ˆN = Ψ x = f 0 x f k x 1 k + f 12 x k=1

9 quantum spectrum of the 2D quantum super-hamiltonian Super-Hamiltonian and supersymmetry Ĥ = 1 2 {ˆQ, ˆQ } = Ĥ 0 I4 2 W ˆψ x ˆψ k l = Ĥ 2 I4 + 2 W ˆψk ˆψ l k x l x k x l [Ĥ, ˆQ ] = [Ĥ, ˆQ] = 0 = [Ĥ, ˆN] Ordinary Schrödinger operators Ĥ 0 Ĥ H0 = 1 2 Ĥ 2 Ĥ H2 = W W + W 2 + W W W Matrix Schrödinger operator Ĥ 1 Ĥ Ĥ0 H1 = 1 2W 1 2 W 2 1 W Ĥ 0 2 2W SUSY-Hodge decomposition and ˆQ-complex H = ˆQH ˆQ H KerĤ,

10 quantum spectrum of the 2D Wipf et al superpotential W x = 2 r = 2 x, SUSY Kepler Hamiltonian W x k = 2 x k r, SUSY Kepler supercharge: ˆQ = i ˆψ k x k i 2 ĥ ĥ = W 2 x ˆψ k = xk k r ˆψ k, ĥ 2 {ĥ = 0,, ĥ} 2 W = 2 δ kl xkxl x k x l r r 2 = 1, ĥ 2 = 0 SUSY Hamiltonian Ĥ = I r ˆX, ˆX = [ĥ, ĥ ] = I 4 2ˆN + 2ĥ ĥ Ordinary Schrödinger operators Ĥ 0 = r Ĥ 2 = r

11 quantum spectrum of the 2D SUSY Kepler Hamiltonian Matrix Schrödinger operator: ˆN = 1 2 Ĥ 1 = + 2 x x2 2 2x 1x 2 r 3 r 3 2x 1x 2 r x2 1 x2 2 r 3

12 quantum spectrum of the 2D SUSY quantum invariants Spin operator and total angular momentum Ŝ = i ˆψ ˆψ 1 2 ˆψ ˆψ 2 1 = i Ĵ = ˆL + Ŝ = i x 1 x 2 x 2 x i ˆψ 1 ˆψ 2 ˆψ 2 ˆψ 1 [Ĵ, ˆN] = [Ĵ, ˆQ] = [Ĵ, ˆQ ] = [Ĵ, Ĥ] = 0, [Ĵ, ĥ] = [Ĵ, ˆX] = 0 SUSY Runge-Lenz-Wipf vector x 1 ˆp2 Ĵ + Ĵˆp 2 r ˆX, Â 2 = 1 x 2 ˆp1 Ĵ + Ĵˆp 1 2 r ˆX Â 1 = 1 2 [Â k, ˆQ] = [Â k, Ĥ] = [Â k, ˆN] = 0 [Â 1, Â 2 ] = 2i SO3 Lie algebra: ˆM 1, ˆM 2, ˆM 3 = Ĵ Ĥ 2 2 Ĵ, ˆM k = Â k 22 2 E [ ˆM a, ˆM b ] = i ε abc ˆM c, a, b, c = 1, 2, 3 Ĉ = 1 [ ] ˆM ˆM ˆM 3 2

13 quantum spectrum of the 2D Spectrum of the SUSY master equation Ĉ = ˆM ˆM 2 2 Ĵ2 Ĥ ˆN ˆQˆQ Ĵ ˆN ˆQ ˆQ bosonic zero mode: KerĤ ˆQ f 0 x 0 = 0 f 0 x = e 2 2 r SUSY paired bound states Ĥ ˆQH = 1 2 ˆQˆQ = 2 1 2ˆN ˆN 2 + Ĉ 2 Ĥ ˆQ H = 1 2 ˆQ ˆQ = 2 3 2ˆN ˆN 2 + Ĉ 2 E 0 E 1 E 2 ˆQ ˆQ ˆQ ˆQ ˆQ ˆQ E + 1 E + 2 E + 3, Ĉ 2 = jj + 1, E 0j = j + 1 2

14 quantum spectrum of the 2D ˆψ 1 = ˆψ 2 = , ˆψ1 =, ˆψ2 = Fermi operators ĥ = 1 r 0 x 1 x x x 1 ĥ ĥ = 1 r 2, ĥ = 1 r 0 x1 2 x 1x x 1x 2 x x1 2 + x2 2 x x x 2 x 1 0

15 Supercharges quantum spectrum of the 2D ˆQ = i ˆQ = i 0 D 1 D D D 1 D D D 2 D 1 0,, Dk = k 2 x k r D k = k + 2 x k r

16 quantum spectrum of the 2D ˆN down/up Ĥ-eigenstates ˆQ ˆQ = Splitting the Hamiltonian 0 D 1D 1 D 1D D 2D 1 D 2D D 1D 1 + D 2D 2 ˆN up/down Ĥ-eigenstates D 1 D 1 + D 2 D ˆQˆQ = 0 D 2 D 2 D 2 D D 1 D 2 D 1 D 1 0