Question

Preview File Edit View Go Tools window Help 크 Scanned from a husky station.pdf v Q Search For any eigenfunction ψ, of the infinite square well, show that (x)- L/2 and that 2(nT)의 where L is the well dimension 18

Please include every step in detail so I can understand. Thank you

Answer 1

Since, Psi_n = squareroot 2/L sin (n pi x/L) < x > = < Psi_n Vertical-bar x Vertical-bar Psi_n > = integral^L_0 2/l sin^2 (n pi x/L) middot x dx < x > = 2/L integral/L0 (1 - cos (2 n pi x/L)/2) middot x dx < x > = 2/L middot 1/2 integral^L_0 (1 - cos (2 n pi x/L)) middot x dx < x > = 1/L [[x^2/2]^L_0 - (x integral cos (2 n pi x/L) middot dx - integral 1 middot integral cos (2 n pi x/L) middot dx middot dx)] < x > = 1/L [L^2/2 - (x sin (2 n pi x/L)/(2 n pi/L) Vertical-bar ^L_0 - integral sin (2 n pi x/L)/(2 n pi/l) middot dx)] < x > = 1/L [L^2/2 - (0 + cos (2 n pi x/L)/(2 n pi/l)^2 Vertical-bar ^L_0)] < x > = 1/L [L^2/2 - (cos (2 n pi L/L) - cos 0/(2 pi n/L)^2)] < x > = L/2 Now < x^2 > = < Psi_n Vertical-bar x^2 Vertical-bar Psi_n > = integralL0 2/l sin^2 (n pi x/L) x^2 dx