Question

Problem 5.7 Two noninteracting particles (of equal mass) share the same harmonic oscillator potential, one in the ground state and one in the first excited state. (a) Construct the wave function, y (x1, x2), assuming (1) they are distinguishable, (ii) they are identical bosons, (iii) they are identical fermions. Plot IV (x1,x2)|in each case (use, for instance, Mathematica's Plot3D). (6) Use Equations 5.23 and 5.25 to determine ((x1 - x2) for each case. (C) Express each y (x1, x2) in terms of the relative and center-of-mass coordinates r = x1 - x2 and R = (x + x2)/2, and integrate over R to get the probability of finding the particles a distance Ir apart: P(Ir)) = 2 / ly(R,r)? DR (the 2 accounts for the fact that r could be positive or negative). Graph P (r) for the three cases. (d) Define the density operator by n(x) = 8(x - x;); i= (n(x)) dx is the expected number of particles in the interval dx. Compute (n(x)) for each of the three cases and plot your results. (The result may surprise you.)

Answer 1

(a) It is given that the two non-interacting particles, one in state goound and anothee in first excited (a) Particles are distingushable The ground state to (x) = (mus, & huerta The first excitin sale, 4,co) - Content that the main e t ng YCH, uz) - %(4) 4, (x). (6) particles are identical bosous 4G, me) = ¢ (4:02) 4 () + 4, (2) 40-)) ce packeles me identical farmious Cokey falis ebony 40.62, 0) t (tocm) ti cms) – 4. c) 4,64) Probability density 141 = 14(2) 4 (2) I principle 7 = mos e vecant* *t) (Campus

141(.6) 4 (1) + $.42) 4cm). (434) fcm) + 4ic4) *cp) +24 cm34.com) 4, (,) & (32) = + (med et utro) almace niso tax dique mins] 'tong para) (****#*,ne) - + mange er et CRT+23) (Mitra 7,72 22 14*1 + (tocm) 4, v.) – 4 Coa) 41cm)| -{(4%C4746) + 4*cm) yon) – a 4.ca] tcas) 4,20 4,602) = + magne tico tris) ( 24, 28 plot is attached chose mus = 1 ( present plot).

()= Plot3D [Exp[- (x12 + x22)] x2?, {x1, -2, 2}, {X2, -2, 2}, AxesLabel * {x1, x2, "|\(x1,x2)|2"}, PlotLabel + "Distinguishable Particles" Distinguishable Particles Ourl= 0.20 (x1.x2) 20.15 0.101 0.05 0.00

(x12 + x2?)] (x1 + x2), {x1, -2, 2}, {X2, -2, 2), AxesLabel - {x1, x2, "|(x1,x2)/2"}, PlotLabel + "Identical Bosons"] Identical Bosons Outle= 0.20 (x1,x2) 20.15 0.104 0.051 0.004

plot3D Exp- In{}= Plot3D (x12 + x2 )] (x1 - x2), {x1, -2, 2), (x2, -2, 2), AxesLabel + {x1, x2, "|(x1,x2)2"}, PlotLabel + "Identical Fermions" Identical Fermions Outle= 0.20 (x1,x2) 20.15 0.104 0.05 0.001

(b) for distingustable particles, <cm-42") = S leal? (x,-au da, dan storey (*) at curat te Do ilgamation in mathematica, (Code aksley) 36 - no$) for identical bosous, <64-) ) = 5 1 461 (41-as) de, dez - 4 months SS e o cant**8) -x3(unta) diete foo identical femious <(% -9,²)=(14tl (n_n)? da, da = conges Services cantos) (x-108 din ding - 1

buvo a cono) Integrate[Integrate [Exp[- () (+ x2?)] (x1 - x2)? x2°, (x1, - Infinity, Infinity)], {x2, - Infinity, Infinity}] Ouvi:ConditionalExpression ( 26, Re ( 1 ) 2 0] Integrate[Integrate[Exp[-(mm) (x12 + x2?)] (x1 - x2)? (x1 + x2)?, {x1, - Infinity, Infinity)], {x2- Infinity, Infinity)] Oui - ConditionalExpression[m, Re 9) 20] Intesa como ) Integrate[Integrate[Exp[- ((x12 + x2?)] (x1 - x2)*, (x1, - Infinity, Infinity)], {x2, - Infinity, Infinity}] Outf-)= ConditionalExpress

19 r = ,-xz R= Cu tax) for destngushable particles, 10 mue cama e le Cai+23) Th R+%= " R-/="2. x + 5 = (R+8) + (R-7)e with=28 14 tn, 4th = 22 2-32 = =ac&?+ %) = axt+Ž -12 28. [CH])= a S 140,5) de - 2 fingeren (14:30) (-25 de -499 Se* (*) (73 de at - integration done plot is attached (choose me on)

WiMAGEN B.(lm) - 2 14,6)* die * +40) 7 a Site zeep (2247) et de - Jeg - plot attached #ClW) = a S 1. 6, ele de * te sale + ) + de - Jy mo weten te , plot allached (1) nCu), 8Cu - vu) + 8Cx-x) <nce) - S[Ercan) +86an)] pien, 13 de for distengushable, <news) - Campus est et (scara) + sca-no) e in con* 5) to Ми0 с. We

* * (-3)* Integrate[Exp[-(**) (22.1)) (R-1), «R, - Infinity, Infinity] m2 m2 (n + mr2w), Re M. ) 20) Outla)= ConditionalExpressio h3/27 (13/2 - lai Integrate[Exp[-(-i) (22 ]] ()?, {R, - Infinity, Infinitys] Out - ConditionalExpression (162/2013 , Re( . )) > ) neste cana) Integrate[Exp[-(-a) (22 ]] «r»?, R, - Infinity, Infinity] Outl= ConditionalExpression

mul ཁ(ཀད་ད༦ ༌ས ༼ ༧ ཊྛ༌བྷཱུ༌ en mW 211 - – ཡུ* – ཡ Fe: རྨ ཀྵ - (ཀ* + ཡ༤) for identical bosons, （ཀ) པཙ%a-༣) f6 སྣ་ - ཅན་ = ༈ (ཀའ་, 6, e བན (༤༠༣) ཨུས - ༈ (ཀt ༧ ( ) – - ༈ (གས+ {r 《 (1+ བཝ ༤༡. རས ༈ ཀ – C4 བཀས )

LIMINATION HITRATTATI for identical feonions, <)， ༼ཀg ༢ * ༼ཀ༽ ས ༣ ༣) ཀར - ༈ བཙཀ་ ཡཔIF * (1+ བྷཝ ) – (ད་དུ* *# )

1.- Plot (1+r2) Exp[***, {', -2, 2), AxesLabel -&r, "P(1/1)"), Plotlabel "Distinguishable Particles" Distinguishable Particles PII) 0.45 OL Oull - 0.35 0.30

3- Plot le expli*), 45-2, 29, are -2, 2), AxesLabel + {r, "P(Irl)"), PlotLabel + "Identical Bosons"] Identical Bosons P(II) Outl=

s- Plotller? Expl}, {r, -2, 2), -2, 2), AxesLabel + {r, "P(r)"}, Plotlabel - "Identical Fermions" Identical Fermions P(II) Our-

In- + 2 x?) Exp(-x?], {%, -2, 2), AxesLabel + {x, "<n(x) >"}, PlotLabel "Distinguishable Particles" Distinguishable Particles <n(x)> Our

aos polo 1.0 2 7) evl<*), **-2, 93,4 In + 2x2) Exp(-x?], {X, -2, 2), AxesLabel + {x, "<n (x) >"}, PlotLabel + "Identical Bosons" Identical Bosons <n(x)> Our-

Infos + 2x2) Exp(-x?], {X, -2, 2), AxesLabel + {x, "<n (x) >"}, PlotLabel + "Identical Fermions" Identical Fermions <n(x)> Our-