(define i 0+1i)
(define pi (* 4 (atan 1)))
(define (sq x) (* x x))
(define (pw x n)(if (= n 0) 1 (* x (pw x (- n 1)))))
(define (ph x t p) (* (p t) (exp (- (sq (* (p t) x))))))
(define (int t p) (let ((dx 0.05))
  (let inj ((s 0)(x -10)) (if (> x 10) (/ (* s dx) (sqrt pi))
     (inj (+ s (ph x t p)) (+ x dx))))))

; (define (p t) (+ 1 (* t i)))
; not convergent unless |t| < 1

(define (p t) (/ (* 2 (sqrt t)))) ; heat solution
(define (p t) (/ (sqrt (+ 1 (* 4 i t))))) ; Schrödinger solution
(define (ex x t) (exp (- (sq (* (p t) x)))))
(define (dpdt x t) (let ((dt 0.000001)) (/ (- (ph x (+ t dt) p)(ph x t p)) dt)))
(define (d2pdx2 x t) (let ((dx 0.00001)) (/
   (- (+ (ph (+ x dx) t p) (ph (- x dx) t p)) (* 2 (ph x t p))) (sq dx))))

(define (zz r) (list (/ (dpdt r 1) (d2pdx2 r 1)) (/ (dpdt r 2) (d2pdx2 r 2))))

(zz 1) ; => (1.0000018469521585 0.999976322231711)
(zz 1.3) ; => (1.000027412021858 0.9999822685508571)
(zz 2.1) ; => (1.0000026843589063 0.9999395481127061)


(define (pp t) (* -1/4 (/ (sqrt t) (sq t))))
(define (dpdt2 x t) (- (* (pp t)(ex x t)) (* 2 (sq (p t)) (pp t) (sq x) (ex x t))))
(define (dpdt3 x t) (* (pp t) (- 1 (* 2 (sq (* (p t) x))))(ex x t)))
(define (d2pdx23 x t) (let ((sw (lambda (x t) (- (* (pw (p t) 3) 2 x (ex x t)))))
    (dx 0.000001)) (/ (- (sw (+ x dx) t) (sw x t)) dx))) 
(define (d2pdx27 x t) (- (* 2 (pw (p t) 3) (- 1 (* 2 (sq (* x (p t)))))(ex x t))))

(d2pdx27 2/3 3/4) ; => -0.23355976366975803
(dpdt3 2/3 3/4) ;   => -0.2335597636697579