Chebyshev Schrödinger Heisenberg H = 1 2m p2 + V (x), m = 1, h = 1 1/36

1 V (x) = { 0 (0 < x < L) (otherwise) ψ n (x) = 2 L sin (n + 1)π x L, n = 0, 1, 2,... Feynman K (a, b; T ) = e i EnT/ h ψ n (a)ψ n (b), E n = 1 2 { (n + 1)π L } 2 2/36

3/36 K (a, b; T ) = + = 1 2πiT L n= {e i 2T (a b+2nl)2 e i 2T (a+b 2nL)2} x x dx = 1, 0 = exp i N (x j x j 1 ) 2 2 t j=1

4/36 1 2πi t n= {e i 2 t (x j x j 1 +2nL) 2 e i 2 t (x j +x j 1 2nL) 2} Feynman Schrödinger Feynman Schrödinger

2 Chebyshev 2 ψ n (x) = L Chebyshev 1 1 sin (n + 1)π x L =ψ 0 (x)u n (η(x)), η(x) = cos π x L U n (η) = sin(n + 1)θ sinθ, θ = π x L U m (η)u n (η) 1 η 2 dη = π 2 δ m n 5/36

6/36 ηu n (η) = 1 2 U n+1(η) + 1 2 U n 1(η) Hamiltonian H = 1 2m 2 x = RD2 η, R = 1 ( π ) 2, Dη = 1 η 2 L 2 η Hamiltonian E 0 = R E 0 = 0, H H = H R H = R ( D η η 1 η 2 )( D η η 1 η 2 )

7/36 (Odake-Sasaki) [H, [H, η]] = ηr 0 (H) + [H, η]r 1 (H), R 0 (H) = R(4H + 3R), R 1 (H) = 2R η (sinusoidal) η e iht ηe iht = a (+) e iα +(H)t + a ( ) e iα (H)t α ± (H) = 1 2 { } R 1 (H) ± R1 2(H) + 4R 0(H)

8/36 a (±) = {±[H, η] ηα (H)} 1 α + (H) α (H) a (+) ψ n = A n ψ n+1, a ( ) ψ n = C n ψ n 1 ( B n = 0) η ψ n = A n ψ n+1 + B n ψ n + C n ψ n 1, C n = A n 1 B n

Hamiltonian SUSYQM Schrödinger Odake-Sasaki Heisenberg 9/36

10/36 3 Schrödinger SUSYQM ψ 0,+ (x;a) = e W (x;a) A(a) = 1 2 ( x + W (x;a) ), A(a)ψ 0,+ (x;a) = 0 Hamiltonian(H + (a)ψ 0,+ (x;a) = 0 : E 0 (a) = 0) H + (a) = A (a)a(a), H (a) = A(a)A (a) H + ψ E,+ (x) = Eψ E,+ (x), H ψ E, (x) = Eψ E, (x) Aψ E,+ (x) = E ψ E, (x), A ψ E, (x) = E ψ E,+ (x)

A(a)ψ 0,+ (x;a) = 0 11/36 A(a)A (a) = A (a 1 )A(a 1 ) + R(a 1 ), a 1 = f (a) H + (a) ψ n,+ (x;a) = A (a)a (a 1 )... A (a n 1 ) En (a)e n 1 (a 1 ) E 1 (a n 1 ) ψ 0,+(x;a n ), E n (a) = n k=1 R(a k ), a n = f (a n 1 ), a 0 = a, n = 1, 2, 3,... A(a)

12/36 A(a)ψ λ (x;a) = λψ λ (x;a) = ψ λ (x;a) ψ 0,+ (x;a)e 2λx ψ ik/ 2 (x;a)ψ ik/ 2 (x ;a)dk δ(x x ) ψλ (x;a)ψ λ (x;a)dx =?

13/36 4 Path-integral solutions for shape-invariant potentials using point canonical transformations R. De, R. Dutt and U.Sukhatme, Phys. Rev, A 46, 6869 (1992) V (r,θ) = c2 1/4 2r 2 sin 2 θ + d2 1/4 2r 2 cos 2 θ Energy-dependent Green s function Duru-Kleinert formalism

Path integration of one-dimensional three-body problem with three-body forces D. C. Khandekar and S. V. Lawande, J. Math. Phys. 18, 712 (1977) Duru-Kleinert point canonical transformations(pct) Supersymmetry and Quantum Mechanics F. Cooper, A. Khare and U. Sukhatme, Phys. Rep. 251, 267 (1995) 14/36

15/36 5 Heisenberg Hψ n (x) = E n ψ n (x), E 0 (= 0) < E 1 < E 2 < ψ n (x) = ψ 0 (x)p n (η(x)) [ ψ n (x)ψ n (x)dx = δ n,n ] P n (η) η n {P n (η) n = 0, 1, 2,...} P 1 (η) = 0 ηψ n = A n ψ n+1 + B n ψ n + A n 1 ψ n 1

(sinusoidal coordinate) η(x) [H, [H, η]] = ηr 0 (H) + [H, η]r 1 (H) + R 1 (H) Heisenberg operator e iht ηe iht = a (+) e iα +(H)t + a ( ) e iα (H)t R 1(H) R 0 (H), α ± (H) = 1 2 { } R 1 (H) ± R1 2(H) + 4R 0(H), a (±) = { ( ±[H, η] η + R ) } 1(H) α (H) R 0 (H) 16/36 1 α + (H) α (H).

17/36 e iht η ψ n e ie nt =A n ψ n+1 e i(e n+1 E n )t + B n ψ n + A n 1 ψ n 1 e i(e n 1 E n )t e iht η ψ n e ie nt =a (+) ψ n e iα +(E n )t + a ( ) ψ n e iα (E n )t ψ n R 1(E n ) R 0 (E n ) a (+) ψ n = A n ψ n+1, a ( ) ψ n = A n 1 ψ n 1, B n = R 1(E n ) R 0 (E n )

18/36 E n+1 E n = α + (E n ), E n 1 E n = α (E n ), α ± (E n ) = 1 2 { } R 1 (E n ) ± R1 2(E n) + 4R 0 (E n ), ψ n = a(+) A n 1 ψ n 1 = 1 {a (+) } n ψ 0. n 1 A k k=0

19/36 a ( ) a ( ) ψ n = A n 1 ψ n 1 a ( ) α = α α = α = α α = (α α ) n n 1 k=0 A 2 k α n n 1 k=0 A k ψ n,

20/36 w(α α) 0 w(u)u n du = n 1 k=0 A 2 k dα dα π w(α α) α α = 1 but... α H α =?

21/36 6 1. 2. ( ) = ( ) 3. H ( )( ) η a (±) 1 2 3 a = a ( ) f (H), a = f (H)a (+)

22/36 a a ψ n = { f (E n )} 2 A 2 n 1 ψ n { f (E n )} 2 A 2 n 1 = E n a a ψ n = E n ψ n = H = a a a α = α α = α H α = α α α α

23/36 α = c n ψ n = a α = c n A n 1 f (E n ) ψ n 1 n=1 α α = α c n ψ n c n = γ n = α c n 1 = f (E n )A n 1 n { f (E k )A k 1 } 2 = k=1 αn γn, n k=1 E k

α = α n γn ψ n, γ n = n k=1 E k 1. E k = k = γ n = n! α = α n n! ψ n α α = e α α dα dα π w(α α) α α = 1, 24/36 w(α α) = e α α

2. E k = k 2 + νk = γ n = n!γ (n + ν + 1)/Γ (ν + 1) α = α n n!γ (n + ν + 1)/Γ (ν + 1) ψ n α α = Γ (ν + 1)(α α ) n n!γ (n + ν + 1) = 0 F 1 (ν + 1, α α ) = Γ (ν + 1) (α α ) ν/2 I ν(2 α α ) w(α α)? = (α α) ν/2 Γ (ν + 1)I ν (2 α α ) 25/36

w(α α) dα dα π w(α α) α α = 1 α = u e iθ dα dα π = du dθ 2π dα dα π w(α α) α α = 0 0 du w(u) du w(u)u n = γ n 26/36 u n γ n ψ n ψ n

27/36 γ n = n!γ (n + ν + 1) Γ (ν + 1) n!γ (n + ν + 1) = =Γ (ν + 1) w(u) = 0 0 du w(u)u n 1 Γ (ν + 1) 0 ds dt e s t s n t n+ν [st = u ] e (t+t/u) t ν 1 = 2uν/2 Γ (ν + 1) K ν(2 u )

28/36 E n = n(n + ν) w(u) = 2uν/2 Γ (ν + 1) K ν(2 u ) dα dα π w(α α) α α = 1

29/36 dα dα w(α α) α α α α π dα dα 2(α α) ν/2 = π Γ (ν + 1) K ν(2 α α ) Γ (ν + 1) (α α) ν/2 I ν(2 α Γ (ν + 1) α ) (α α ) ν/2 I ν(2 α α ) Γ (ν + 1) = (α α ) ν/2 I ν(2 α α ) = α α

7 a ( ) a = a ( ) f (H) α α = α n γn ψ n, γ n = n k=1 E k ψ n = {a(+) } n ψ 0 = n A k 1 k=1 α = (a ) n γn ψ 0, a = f (H)a (+) (a α) n 30/36 γ n ψ 0

31/36 Schrödinger 1. E n = n α = e a α ψ 0 2. E n = n(n + ν) α = 0 F 1 (ν + 1, a α) ψ 0 dα dα π w(α α) α α = 1

32/36 α F e βh α I = N 1 i=1 dα i dα i π w(α i α i) N α j e ϵh α j 1, j=1 ϵ = β N, α N = α F, α 0 = α I α j e ϵh α j 1 = α j α j 1 (1 ϵα j α j 1 + O(ϵ 2 )) α j α j 1 (1 ϵα j α j 1) = e (1 ϵ)α j α j 1 = e e ϵ α j α j 1

33/36 α j α j 1 (1 ϵα j α j 1) (α j = α j 1) n (1 ϵα j α j 1) = [ = γ n (α j α j 1) n γ n ( 1 γ ) n ϵ γ n 1 (e ϵ α j α j 1) n n! ( ) ]

34/36 = = dαj dα j 0 π du w(u) w(α j α j) α j+1 (1 ϵh) α j α j (1 ϵh) α j 1 (α j+1 α j 1) n γ n (α j+1 α j 1) n u n γ 2 n ( 1 γ n γ n 1 ϵ α F (1 ϵh) N α I = ) 2 ( 1 γ ) 2 n ϵ γ n 1 (α N α 0) n γ n ( 1 γ ) N n ϵ γ n 1

35/36 ( lim 1 γ ) N n ϵ = e β γ n γ n 1 N γ n 1 lim α F (1 ϵh) N α I = N (α F α I) n γ n e βe n γ n = n k=1 E k = γ n γ n 1 = E n

36/36 8 Schrödinger Heisenberg Duru-Kleinert