Question

Provide a complete derivation of separation of variables for three identical non-interacting particles 1, 2, and 3 of mass m with coordinates X1, X2, and x3 respectively in a box of length L. You may use the results from one- dimensional particle in a box as presented in class and your text. a. What is T(x1,x2,X3) in terms of T(X1), T(X2), and T(x3)? b. What is V(X1,82,83) in terms of V(xi), V(x2), and V(x3)? c. What is H(X),X2,X3) in terms of H(X1), H(X2), and H(x3)? d. What is P(x1,x2,83) in terms of '(x1), (x2), and P(x3)? e. What is E in terms of E1, E2, and E3? f. What is the normalization constant N in terms of Ni, N2, and N3?

Answer 1

The multiparticle system Hamiltonian is given by H H (R1, M2, Mg..., Knot): 3 b + V (1,972, Mg.m., t) 2 ,2,3,..., innt) + V (2,2, 3, ไป 2m As there are 3 particle H(219 22, 23,1); } Þi + V (4) Me, Ag, t ) - 2m But as they are not interacting hence vero, 22, 23, t? : { v(xist) HCN1142, , t): Š Hi (21,t) (a) The Kinectic terms will be 2m T = T2 -t V2 T3= ат 2m

(6) Vi Cai,t).) 0,oxicl 1o, elsewhere NZL PARTICLE IN ID (c) H(2., 12, 13, t) - Hi («,,t + H2692,t) + Hz (az,t) (d) 4,2,(209.22,78) = (2)*sim ( mi24 ) sin ( hits) sin ( 91123) 41,2,3 (21) 22 323 )= 4,170) 42 () 43 () - [ € ) sin (ra)]sim (Tue)] [ sin ( 11 13 ) 2 sin Intl 172

(e) E = { Ei E;= n2t2 ,1-1,2,3 am L2 (f) NPN, 2 + N2 - 1 3.N2 - 1 N-N2: N3 3) N = N2 = Nz -

We know schrödinger ean is given by 2m H4 = -52 024 + v4 Total Hamiltonian 4 is the total wave function 4 CM,1, 22, 23) – 40207.4(02).4(23) [H +(91)-46422.4(237] conta com tom) + (x,x2,13) + V(21,0 22,13) 4(2,52773) + [v (m1).4,60)]4.43 + (U(na).4874143 + [V(23) 4314, 42

H(21, 22, 23) 41 42 43 = CH. 41) 4243 + (H₂4₂) 434 + (1343) 4,42 Comparing LHS & RMS -KZ 224 am From ID Box we know that wave function is given by 4- ve Sin (972). 46. 47. 43 = )* sin ( ni T L ) Sin C marme) sin (not "3

And energy foom 10 box we know is ginen E = n²a2h2 am12 E (Total emergeng) - T24? ( m2 +n? +13) 2mL2 Et Ezt Ez And the normalisation constant comes from ( 14 12 dx = 1 N2 + N2 + N = 1 As in here N = N₂ N3 N, 2 N₂ = N3 : 1