Double Well Potential: Perturbation Theory, Tunneling, WKB (beyond instantons)

Transcription

1 Potential: Perturbation Theory, Tunneling, WKB (beyond instantons) May 17, 2009

2 Outline One-dimensional Anharmonic Oscillator Perturbation Theory of Non-linealization Method Hydrogen in Magnetic Field

3 H = d2 dx 2 + m2 x 2 + gx 4, x R m 2 0 is anharmonic oscillator m 2 < 0 is double-well potential

4 2 4 V(x)= 5x + x x The double-well potential

5 Idea is to combine in a single approach: Perturbation Theory near the minimum of the potential Ψ(x) = e αx2 (1+β 1 x 2 +β 2 x 3...), α > 0 (ground state) correct WKB behavior at large distances (inside of the domain of applicability) Tunneling between classical minima The art of interpolation... it leads to a solution! Solution: for any real x an eigenfunction is known with a certain relative accuracy δ Ψapprox Ψexact Ψ approx

6 What is known about eigenfunctions: For real m 2, g 0 any eigenfunction Ψ(x; m 2, g) is entire function in x Any eigenfunction has finitely many simple real zeros (the oscillation theorem) and infinitely many complex zeros situated on the imaginary axis A Eremenko, A Gabrielov (Purdue), B Shapiro (Stockholm), 2008

7 Main object to study is the logarithmic derivative y = Ψ (x) Ψ(x) here ϕ(x) is the phase. = ϕ (x), Ψ(x) = e ϕ(x)

8 Riccati equation In general, y is odd and y y 2 = E m 2 x 2 gx 4, n 1 y = x x i i=1 + y reg (x) here x i are nodes and y reg (0) = 0. Ground state: n = 0 (no nodes), y = y reg y has no singularities at real x and y(0) = 0. y(x) = 0 > extremes of Ψ(x) If m 2 (m 2 ) crit, single maximum at x = 0 If m 2 < (m 2 ) crit, two maxima and one minimum at x = 0

9 Asymptotics Asymptotics: y reg = g 1/2 x x + m2 2g 1/2 x y reg = Ex + E2 m 2 3 x + n + 1 x x 4gE + m4 1 8g 3/2 x x m2 2g x 3 + 2E(E2 m 2 ) 3g x x 0 1 x

10 Asymptotics or, for phase ϕ reg = g1/2 x 2 x 3 + m2 + m4 1 x +(n+1)log x 4gE 2g1/2 8g 3/2 x +m2 g x first two terms are H-J asymptotics (classical action), the third term also (but not its coeff) is defined by quadratic fluctuations 1 x ϕ reg = E 2 x2 + E2 m 2 12 x 4 + 2E(E2 m 2 ) 3g x x 0 It makes (physics) sense of pert theory at m 2 0 (around a minimum of potential)

11 Interpolation (ground state) Let us interpolate perturbation theory at small distances and WKB asymptotics at large distances ψ 0 = { c 2 gx exp A + ax2 /2 + bgx 4 } 2 (D 2 + gx 2 ) 1/2 where A, a, b, c, D are free (variational) parameters ( hard to modify) Very Rigid expression!

12 If we fix then b = 1 3, a = D2 3 + m2, c = 1 D ψ 0 = { 1 D 2 + gx exp A + (D2 + 3m 2 )x 2 /6 + gx 4 } /3 2 (D 2 + gx 2 ) 1/2 the dominant and the first two subdominant terms in the expansion of y at x are reproduced exactly A,D are still two free parameters which we can vary. Our approximation has no complex zeroes on imaginary x axis but symmetrical branch cuts going along imaginary axis to ±i (a meaning of square root)

13 If ψ 0 is taken as variational function then for all studied m 2 from -20 to +20 and g = 2 the variational energy reproduces 7-10 significant digits correctly!! but the accuracy drops down at m 2 < 0 (from 10 at m 2 = 0 to 7 s.d. at m 2 = 20) Can we fix it?

14 Perturbation Theory and Variational Method Take a trial function ψ 0 (x) normalized to 1, then restore the potential V 0, energy E 0 ψ 0 (x) ψ 0 (x) = V 0 E 0 and construct the Hamiltonian H 0 = p 2 + V 0. Variational energy E var = ψ 0 Hψ 0 = ψ 0 H 0 ψ 0 } {{ } =E 0 = E 0 + E 1 (V 1 = V V 0 ) + ψ 0 (H H 0 ) }{{} ψ 0 } V V {{ 0 } =E 1

15 Variational energy can be considered as the first two terms in a perturbation theory, it seems natural to require a convergence of this PT series By calculation of next terms E 2,E 3,... one can evaluate an accuracy of variational calculation (i) and improve it iteratively (ii) (if the series is convergent, of course) Our Perturbation Theory from Ψ 0 is definitely convergent (perturbation is subordinate) How to estimate the radius of convergency? (end of remark)

16 One more, physics property must be introduced into the approximation: at m 2 the barrier grows, tunneling between wells decreases, the wavefunction has two maxima (corresponding to two minima of the potential) and one minimum at origin which value tends to zero ψ 0 = { 1 (D 2 + gx 2 exp A + (D2 + 3m 2 )x 2 /6 + gx 4 } /3 ) 1/2 (D 2 + gx 2 ) 1/2 cosh αx (D 2 + gx 2 ) 1/2 (following the E.M. Lifschitz prescription, Ψ ± = Ψ(x + α) ± Ψ(x α)) in total, we have now three free parameters, A, D, α.

17 More accurate: for m 2 < 0 the expansion of y at x = 0 does not make sense of Perturbation Theory expansion, it is expansion near false minimum (near maximum) potential. The expansion near potential minimum at x min = ± well defined and it should be taken It is that behind the Lifschitz prescription m 2 2g is

18 With this modification for all studied m 2 from -20 to +20 and at g = 2 the variational energy reproduces 9-11 significant digits correctly!! 1-2 orders of magnitude improvement

19 Perturbation Theory of Non-linealization Method (Logarithmic perturbation theory) Take Riccati equation instead of Schroedinger equation y y 2 = E V, y = (log Ψ) and develop PT there. If Ψ 0 is given, let V = V 0 + λv 1 where V 0 = Ψ 0 /Ψ 0, then perturbation theory y = λ n y n, E = λ n E n

20 For nth correction λ n y n 2y 0 y n = E n Q n ; Q 1 = V 1 Q n n 1 = y i y n i, n = 2,3,... i=1 Multiply both sides by Ψ 2 0, (Ψ 2 0 y n ) = (E n Q n )Ψ 2 0 Boundary condition: Ψ 2 0 y n 0 at x (no particle current)

21 E n = Q nψ 2 0 dx Ψ2 0 dx y n x = Ψ 2 0 (E n Q n )Ψ 2 0 dx M. Price (1955), Ya.B. Zel dovich (1956)...Y.Aharonov-C.K.Au (1979), A.T. (1979)... d = 1 ground-state

22 g = 2, m 2 = 1 D = A = α = * * * E var = E var ( E 2 ) = Ẽ var = E var + E var = all digits are correct the next correction E 3 is 10 14

23 g = 2, m 2 = 1 D = A = α = * * * E var = E var = Ẽ var = E var + E var = all digits are correct the next correction E 3 is 10 13

24 40 30 y x 4 5 Logarithmic derivative y 0 as function of x for double-well potential with m 2 = 1,g = x y The first correction y 1 for m 2 = 1,g = 2

25 d Where 2 Ψ dx 2 x=0 = 0? = When E = 0 (classical motion stops to feel the presence of two minima) E(m 2 = (m 2 ) crit = ,g = 2) = 0 for m 2 > (m 2 d ) crit, 2 Ψ dx 2 x=0 < 0 (single-peak distribution) For 0 > m 2 > (m 2 ) crit the potential is double well one, but wavefunction is single peaked, no memory about two minima, particle prefers to stay near unstable equilibrium point! for m 2 < (m 2 d ) crit, 2 Ψ dx 2 x=0 < 0 (double-peak distribution) as it should be in WKB domain

26 g = 2, m 2 = 20 D = A = α = * * * E var = E var ( E 2 ) = Ẽ var = E var + E var = all digits are correct the next correction E 3 is 10 8

27 15 Yo X 5 Logarithmic derivative y 0 as function of x for double-well potential m 2 = 20,g = 2

28 First Excited State Similar expansions for x and x 0 (with addition log x ). ψ 1 = { 1 (D 2 + gx 2 ) exp A + (D2 + 3m 2 )x 2 /6 + gx 4 } /3 (D 2 + gx 2 ) 1/2 sinh αx (D 2 + gx 2 ) 1/2 (following the E.M.Lifschitz presciption) in total, we have three free parameters, A,D,α. For all studied m 2 from -20 to +20 and g = 2 the variational energy reproduces 9-11 significant digits correctly!! (similar to the ground state)

29 g = 2, m 2 = 20 D = A = α = * * * E var = E var ( E 2 ) = Ẽ var = E var + E var = all digits are correct the next correction E 3 is 10 10

30 Energy Gap E = E first excited state E ground state E = 211/4 ( m 2 5/4 e 2 m 2 3/ ) π 12 2 m 2 3/ m at g = 2 it is an asymptotic expansion... J Zinn-Justin et al, E Shuryak et al, 1994 : 12 - two loop contribution is it three loop one? what about a next term, four-loop contribution?

31 g = 2, m 2 = 20 E var = E (1) var E (2) var = = one instanton = (5.3% deviation) one instanton + correction = (0.36% deviation) one instanton+twocorrections = (0.22% deviation)

32 (i) What about excited states? (ii) How to modify the function ψ 0,1? ψ (n) 0 = P n (x 2 { ) (D 2 + gx 2 exp A + (D2 + 3m 2 )x 2 /6 + gx 4 } /3 ) n+1/2 (D 2 + gx 2 ) 1/2 cosh αx (D 2 + gx 2 ) 1/2 P n (x 2 (n) ) ψ 0 where P n is a polynomial of nth degree with positive roots found through conditional minimization (ψ (n) 0,ψ(l) 0 ) = 0, l = 0,1,2,...(n 1)

33 and for negative parity states ψ (n) 1 = P n (x 2 { ) (D 2 + gx 2 exp A + (D2 + 3m 2 )x 2 /6 + gx 4 } /3 ) n+1 (D 2 + gx 2 ) 1/2 sinh αx (D 2 + gx 2 ) 1/2 P n (x 2 (n) ) ψ 0 where P n is a polynomial of nth degree with positive roots found through conditional minimization (ψ (n) 1,ψ(l) 1 ) = 0, l = 0,1,2,...(n 1)

34 (iii) How to find corrections to the excited states, to the functions ψ (n) 0,1? Perturbation Theory of Non-linealization Method, A.T. 79 If V = V 0 + λv 1 where V 0 = Ψ 0 /Ψ 0, then perturbation theory Ψ = (P n (x 2 ) + λp (1) (n) n ) ψ 0 exp( λϕ 1 λ 2 ϕ ), E = λ n E n with constraint p (1,2,...) n 1 are polynomials of (n 1)th degree

35 Zeeman Effect on Hydrogen H = 2 r + γ2 ρ 2, x R 3 where r = x 2 + y 2 + z 2, ρ = x 2 + y 2 and γ magnetic field. For Ground State: ( y) y 2 = E V, y = log Ψ

36 For phase ϕ = γρ x (no more terms are known so far!) and ϕ = r + a 2,0 r 2 + a 0,1 ρ 2 + a 3,0 r 3 + a 1,1 rρ a n,k r n (ρ 2 ) k +... x 0

37 Interpolation (ground state): 1 ψ 0 = (D 2 + αz 2 + 4γ 2 ρ 2 ) 1/2 exp { A + ar + bz2 + cρ 2 + γ 2 rρ 2 } (D 2 + αz 2 + 4γ 2 ρ 2 ) 1/2 where A,a,b,c,D 2,α are variational parameters. Relative accuracy in total energy for γ = is not less than 10 6!

38 HAPPY BIRTHDAY MISHA!