karlov-kirichenko-kvantovaya-mekhanika-2016 (810755), страница 29
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D # 5 "#'" ! A " '-( F-!" -! f (ξ) 0 "( * ξ =! '-! -! ! !!*-" ! 05!# "!"#5M' (' (Δ ≡ C2 − 4 A2 B 2 0./! 4 5!! !; " !&M14C2 .AIdx−∞AA−∞"( ' 2(B O?-!( "4 264 &M/- -! f (ξ) 0 "#'( " ! ( ! ; ! K f (ξ) = Ψ∗ Q̂+ Q̂ΨdV, A2 < "#.Ψ(−∞) = Ψ(+∞) = 0.f (ξ) = (Q̂Ψ)∗ (Q̂Ψ)dV.'ddx [x, p̂x ] = i ! Ĉ = % - ! " !& !5 "# * !-5 !"( ! * 5*"4MΔx = Δx2 = x2 , Δp = Δp2 = p2 .NQ̂+ = ξ Â+ − iB̂ + = ξ Â − iB̂.% 264 ! ξKK ξ 3 & !!"#5& ! <; !! !" 5! "(M∞22x |Ψ(x)| d x + ξf (ξ) =−∞2∞−∞∞ 2dΨdΨ∗ dΨ +ΨΨ∗x d x. dx + ξdx−∞dxdxAK<5& !" & - ! -! "( AA∞' (x2 |Ψ(x)|2 d x = x2 .A −∞%!& !" & - ! 0'! ( ! !(Q(KKNQ( - !(Mξ2∞−∞ ! II 05-5& M∞∞dΨ ∞d2 Ψd2 Ψd x = ξ 2 Ψ∗−ξ 2Ψ∗ 2 d x = −ξ 2Ψ∗ 2 d x.dxdx −∞dxdx−∞−∞dΨ∗ dΨdxKKO=0A% ( 5 "( ! "# "-d2 Ψ1d 2p̂2− 2 = 2 −iΨ = x2 Ψ,dxAdx! -! !" A ! ∞−∞Ψ∗p̂2x12Ψ dx =2' 2(px .,"( ! -! -! ! ! 1 !"(! ( " -! IA ! ( fmin = 0 D- f = fmin !! ( ξ = ξm % "!6!"# ! 5!"# 5( 26 f (ξ) ( -! " fmin = 0 ! 05!# 5" !"# " ! (xΨ(x) + ξANAOdξ !"(!'= 2ξp2x2(− 1 = 0,"' (fmin = x2 −ξ = ξm =22p2x .22p2x ,ξm =IBIAx2I.2ξm2p2x ; I ' (= 2 x2 ,−x2IN4x2 IO.D! -! ; IO ! "4 264 - 6""(! !( ! "#' .!;( " !& 2 %&"( ! "- 6""(!"#5 "!# - "5* 226! 6 '-& .!- * &D - AK "!# "464 ""' "!G ; 90! ( ;!# cf (ξ) f (ξm ) 0.< "# -!! ( -! x = 0 px = 0 ! "(Δx = x2 , Δpx = p2x ,2Ψ = Ψ0 exp' "( 26 f (ξ) "! -!<! ' IA "! !;' 2( ' 2 (xpx 2 4.−76 -!5( -! " I ! ()"# '- !& 26 !! ( "df (ξ)I= 0,dxΨ = Ψ0 exp9 0' * "41 54 "( 26 f (ξ)M' 2(' (pf (ξ) = x2 + ξ 2 2x − ξ.dΨ "!# ξ = ξm ' IB G;( ! 226"# .
"-−∞=0I -! !"(! 0& !; " !&% - ! "( & "& 26 .!! ( !ΔxΔpx = /2.IA9!& !" AK 5! ( ⎡⎤∞∞d(Ψ∗ Ψ)⎢⎥ξx d x = ξ ⎣ xΨ∗ Ψ|∞−Ψ∗ Ψ d x⎦ = −ξ.dx −∞−∞ΔxΔpx /2,IIi∂Ψ∂t= ĤΨ,Ĥ =p̂22m=−22mAΔ,-"#5 "IKΨ(r, t)|t=0 ≡ Ψ0 (r) = Ψ0 exp −r22δ 2+ip0 r.IQ(Q(, !( 5! " 6! !& !(! ( !#4 v = p0 /m - " 0!# ( 5- " ! "! !! (! !M x 9 5 ! '4 !* !"< MK Bj=2mi[Ψ∗ ∇Ψ − Ψ∇Ψ∗ ]r=0=p0m|Ψ0 |2 .Ψ0 (r) = f (p)eipr/d3 p)3(2π−ipr/f (p) = Ψ0 (r)eexp −∞,"( ;( '- 5 "#' ( ; "#'5 8#.! G'" *5& "& ! " "5 ! !" 8#M,Kr22δ 2+i(p0 − p)rK AdV =∞x2qxexp − 2 + iexp −dx=2δ−∞−∞y2∞2δ 2−∞z2dz =2δ 2 2 2q δ= (2π δ 2 )3/2 exp − 2 .2< ! ! 5( ! 3(p − p0 )2 δ 2prp2d pΨ(r, t) = Ψ0 (2π δ 2 )3/2 exp −−it.expi232dV.exp −dy2m(2π )=!! !" ! ! ( !" < M< ' !* ! '-! -! -"#( "( 26( !#'6( " * " eipr/ 226! f (p) !05 &!4 ' !# !* " !!- - !# -! (E "# p "(! ! * - !6 ('5 !; E = p2 2m?!! ! "( ' # " & "5"5 !k = p/ ! prEtpr= exp i − iexp i − i2mp22mtr22δ 2= Ψ0δ22π 23/2 2π 22t .d3 p(2π )3+i(p − p0 )2 δ 2prexp −+i −i2exp −p2= Ψ0dV.%*(1& 4 !" 5- "(! ( !!- ! ,"( 0 !5- "& 0'- q = p0 − p -! ! " "#22δ2+3/22π 2,"#&; 5- "('"#!!MΨ(r, t) = Ψ0(p0 − p)rδ2itr+p i +2mδ22π 2a=2mp0 δ 22−p20 δ 222d3 p = 2 2!p δexp −ap2 + 2pρ d3 p.exp − 0 22δ222+it2m, ρ=i2r+p0 δ 222!!5 (! -!"#3/2 3/2πt d3 p =p2D # 5 0'-( a = .!05 "-!# ; '!& 2 !!- !!# 4( 5 "( 226! f (p) !5 ! 05!# &51#4 ! ! KMf (p) = Ψ0 (r)e−ipr/ dV = Ψ0 exp −Ψ(r, t) = Ψ03/2 δ2p2< "# ( " !(! ( ' ! * .'"#5& ! '64 !!# prΨ(r, t) = f (p) exp i − i 2 2p δexp − 0 2 +2Ψ01 + it mδ 2exp −ρ2a=1 (r − ip0 δ 2 )22δ 2 1 + it mδ 2−p20 δ 222.
+ !# -! &( 26( "!(! -"# "4 I ! 4 A / !& 2 ; "(. <! 6" 0' &! " (! !& *(- !65M2(r − p0 t/m)222 δ.W ∼ |Ψ| = Ψ0 2 exp −2δ (t)δ (t)Q(K IQ(D # 05" 0 ! 0'- "( *!& ;5 "! ! tM 2 δ(t) = δ 1 + t mδ 2 , δ(0) = δ.N9 0' 6! ! ! ( !#4 v = p0 /m ;. !! ' ND - A *( ' ( c ! '"( !5 * - !6 * 6! ! !"#. - !6G ; G ! ! * - !6 !6"#( ('& !( !5* ' ! !"# ! !(( r = | r1 − r2 | U = U (r) %"( 26( !# ' ! ! .! 0* - !6M Ψ = Ψ(r1 , r2 ) C 5 " !! ! -! "-|Ψ(r1 , r2 )|2 dV1 dV2 !# (! !# ! -! - !6 A *! ( ".! 0J dV1 ! ! !- r1 - !6 I 3 "! 0JdV2 ! ! !- r2 ( ! !65* !(& 5 5 "' !# ! !( ! 05-5& M ∼ exp (−iEt/)C " 5 - !6 !# m1 m2 ! "#! !5 '5! (5Ĥ = T̂1 + T̂2 + U (r);A22T̂1 = −Δ1 , T̂2 = −Δ2 .2m12m2D # Δ1 Δ2 3 "" 5 !5* 226 1 !"(.! ( ! !! ! - !6 A I % '"#!! 5 ;" "41 4 cM−22m1Δ1 Ψ −22m2Δ2 Ψ + U (r)Ψ = EΨ.I<0 ! * 1 !"(! ( * ! !!"#5* - !6 ! 6! ! !"#& .! !! * 0 "!# ; "-MR=m1 r1 + m2 r2m1 + m2, r = r1 − r2 .K!5 ! 0'- (X, Y, Z) !5 ! 3 (x, y, z)<0' "−2 ∂ 2 Ψ2m1 ∂ x21−2 ∂ 2 Ψ2m2 ∂ x22 K Km1 x1 + m2 x2 "& - ! I < "# X =m1 + m2x = x1 − x2 ! ,05-5 " 0'( '5* "! ' !#∂Ψ∂ x1∂Ψ ∂X=∂X ∂ x1+∂Ψ ∂ xm1=∂ x ∂ x1∂Ψm1 + m2 ∂X+∂Ψ∂x.," * !4 '4M∂2Ψ∂∂Ψ∂X∂∂Ψ∂x=+=2∂ x1∂X∂ x1∂ x1∂x∂ x1=∂ x12m1∂2Ψ∂X 2m1 + m2+∂2Ψ2m1m1 + m2 ∂X∂ x+∂2Ψ∂ x2. "- "-4! ( '5 & x2 M∂Ψ∂ x2∂2Ψ∂ x22∂Ψ ∂X==∂X ∂ x2+∂Ψ ∂ x∂ x ∂ x22m2∂2Ψ−∂X 2m1 + m2m2=∂Ψm1 + m2 ∂X2m2∂2Ψm1 + m2 ∂X∂ x−+∂Ψ∂x,∂2Ψ∂ x2.< !"(( &5 5( "( !5* '5* "-−2 ∂ 2 Ψ2m1 ∂ x21−2 ∂ 2 Ψ2m2 ∂ x22=−2 ∂ 2 Ψ2M∂X 2− 5 "( ( 5 !5MM = m1 + m2 , μ =m1 m2m1 + m22 ∂ 2 Ψ2μ ∂ x2., "- 0'4! ( '5 ! y z % !5 * "41& 2 ( cM−22MΔR Ψ −22μΔr Ψ + U (r)Ψ = EΨ./!T̂R = −22MΔR !# ! !- & !5 6" ! !"#& !5 !!- 5 M *(1& ( 6! T̂r = −22μΔr 5! !- 4 4 ! !"# ( - !6 .!5 <"-5& '"#!! " !#4 "- !! !41!4 ' " - & *Q(KQ(@ !# ; ( '(Ψ(R, r) = Ψц.м.
(R)Ψотн (r),N ! 5& !"# 5! 6! !& 3! !"# - !6 < !"(( ! 5 "(-" "-; ( N & 2 ΔR Ψц.м.2 Δr Ψотн−+ −+ U (r) = E.O2MΨц.м.2μ Ψотн% "& - ! "-5 !(1 & !& !5* 0.* ' (! ! '"-5* 5* <! "( ! -!05 !5"(" # '"#5* '-(* ! 0*-!05 0 05" !(5 =! ! .(M2−ΔR Ψц.м. = Eц.м. Ψц.м.AB2M−22μΔr Ψотн + U (r)Ψотн = Eотн Ψотн .AA' * 5! 0 6! !-.& & Eц.м. ! 3 ! !"# & Eотн "- !6"#& '& !( U (r) /- 5"5 !0!# -!05Eц.м. + Eотн = E.D - A 7&! " !- ( "& .!-& & ("#& !6"#& (G ; ' ! 0 - "0& ("#& .!6"#& ( 1 !4! 4! ( 2"&En =π 2 22ma2n2 , n = 1, 2, 3, . . .
, 5& # !-! !(4 ! "(26( ! '" ("(! (!-& ! !"# 5U (x)(5 A-a /2a /2x0%50 ' -" ! -! " * ( (5 9E( !-41( 4 !.( !6!"#& E < 0 % .IIIIIIa! "& 26 !!- !!# cd2 ψdx2+2m2-U 0[E − U (x)] ψ = 0A ( ! -!5! ( ! (5 "- a 3 ; (5 % ".& ( U0 E1 # ! ! ( 7& "IY (x)!"# "( 0" !& % 0" ! d2 ψdx2+2m2-a /2[U0 − |E|] ψ = 0.K% 0" ! d2 ψdx2−2m|E|2a /2áx2 5 ) @A ,5 1 D @"A 60 D+ 1- , ψ = 0.%( 0'-(AI9 0' 5 0" # -! # "' 5& "( .!&;& !5 * - !6 3 ! 3 05" " !"# ! 5 "! "#'!# 4 - !6 ! 5 " -! 6! !5 - !6 !4 &.!4! ; "5 ! ( 0( - !6 =! !!' !# '"( * !& 05 !5 ; ! <5 54! !5 6" !5-!54! ' ! !(( !"K k2 =2m2q2 =[U0 − |E|],2m2|E|,; ( I K d2 ψdx2+ k2 ψ = 0, −a/2 < x < a/2,d2 ψdx22− q ψ = 0, x > a/2.?!! ! ;( !* & !#ψ = A2 cos kx, −a/2 < x < a/2,ψ = A3 e−qx , x > a/2.?; ;& x = a/2ψ|x=a/2−0 = ψ|x=a/2+0 ,ψ |x=a/2−0 = ψ |x=a/2+0NQ(K Q(ctg zcos z!A2 cos−kA2 sinka2ka2= A3 e−qa/2 ,gzO= −qA3 e−qa/2 .p /2/! 4 "! "( *( & Mctgka2=k1 + ctg (ka/2)! "& K "-ka222 kacos=±k=±.2ma2 U0 22mU0/0'-γ=22k2 + q 2AA z = ka/2 9 AB ! <"- 0 "!# 2- I ,.
!5 ;( 5"(4! ( ! " -! " AB "( *" 5"(!# ( ! ctg z > 0 I ! ! z ". !# '5 [0, π/2] [π, 3π/2] ! D( ; (AK " &! ; ( ABM22ma2z2 .A ," -! -! ' ! ! "05 (5 '-( ":-5 U0 ) ! γ 0! " AI '-( ! B 0 -. ! ?!! ! (! ( " (& γ z / * γ; ! ( =! '-! -! "# "& ( ! (!- & #7& ( " ( "& ( F-! -! z == ka/2 1 ! γ 1 "0 (5 U0 " < "# "5* z cos z ≈ 1 − z 2 2 AK !# z=1γz22@ ;!# ! ! "!"#5* 0"&% '; 0" "z (0) = 1/γ 9 "41 0". (1) 21111(1) "!# z == 2 1− 2 1 − 2 " z.A2γγγγ< !"(( ! 5 A -! "( ! γ ' AI"-E=−AKcos z = γ z.1−áà2 - " , @A ,) + +)) z ? ,) ctg z > 0AIma2 U0E = −U0 +z0<"- 0 !# "#'( !.kactg (ka/2)k!- ! ! cos= ±= ± ? -.22p /2zAB> 0.qK ma222U02 .AD! -! " 05 5 -" # 0" z (0) = 1/γ !"-" 05 ; E = 0D - A 7&! "#4 "0 2- . !-&K.& ("#& !6"#& (5 !& & ("(.! ( !- & #G ; G ! K.4 ( A F c"( - !65 !& ( ! Δψ +2m2[E − U (r)] ψ = 0.A/ !( ("(! ( 2- .
!-5 -! '"(!' !# Δψ =d2 ψdr 2+2 dψr dr=1 d2 (rψ)rdr 2 <! ( 264 u = rψ ; AMd2 udr 2+2m2[E − U (r)] u = 0.IQ(K NQ(% - ! ! 0" !(* ! ! "41& Mu +u +2m22m2[E + U0 ] u = 0,0 < r < R,2m2q2 = −[E + U0 ],2m29 ; & K ' 54! (U (r ) E."41 0'Mu = A sin kr + B cos kr,u = Ce−qr ,0Rr0 < r < R,r > R.," "#' "( ;MEб) |r=R−0 = u|r=R+0 ,IIγ=% ! 0 " '! " ! ! -! u = rψ < "# .!6"#( ( U (r) 4 -2 +- $ D) "( 26( ψ ; .),1 5 ( c A ! - ?!!.)! r → 0 ! ( u = rψ → 0? -! "( ;( 0" ! 0 < r < R '5! ( B == 0 ! -! !& 0" ! u = A sin kr <'( ; ! ;(; 0" ! r > R 1#4 "& 0 "--U 0A sin kR = Ce−qR ,kA cos kR = −qCe−qR .tg kR = − < 0.Nq<; ! "#'; # ! !sin kR = ± tg kR1 + tg2 kR= ±kk2 + q 2,OAA," 0 "!# ; 2- I ' I ; "!(41 !04 tg z < 0 I (."(! ( !"# " " (& γ z !!- "Mγ γmax =2πAI.tg zg êð zgzpp/2pp/2zàzá2 - " , @A ,)) ? B 0 tg z < 0? -! "( "-5 γ AB ! 4 * " ("(!- ( !*& (MU Umin =kAB.tg z < 0.в) u |r=R−0 = u |r=R+0 ./! 4 "!2mR2 U0sin z = ±γ z,sin z29 '- ! ( *4 & (а) u|r=0 = 0,Iz = kR,r > R.D # ;!* '-! 226 & r <--! ' # ! -! ! ( ! * ( (5 ! -! (' !(4 !- 4 !! !4! '-( E << 0 % 0'-(k2 = -!5 "( "- k q % 0'-(KE u = 0,K Oπ 2 28mR2.AK< '- U = Umin # *! ( '- E = 09 0' !*& ( # ! '!# !"# " "0 (5 !!- "Q(Q(D - A <"-!# !- ; '- 0 !- * .(* ! -!541 0!"#5& ! "!G ; % " '- 0 !- * (* 0.
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