832.动量算符和角动量算符 17/996 其解为 2n, 0,±1,±2 (3.2-6) tHin Py 0,±1, (3.2-7) tHin n2=0,±1,±2,… 3.2-8) L 由式(3.2-6)~(3.2-8)可知,本征值的间隔与L成反比,当L足够大 ● First●Prev●Next●Last● Go Back● Full Screen●cose●Quit
• First • Prev • Next • Last • Go Back • Full Screen • Close • Quit §3.2. ÄþÎÚÄþÎ 17/99 Ù) px = 2π~nx L , nx = 0, ±1, ±2, · · · , (3.2-6) py = 2π~ny L , ny = 0, ±1, ±2, · · · , (3.2-7) pz = 2π~nz L , nz = 0, ±1, ±2, · · · . (3.2-8) dª(3.2-6)*(3.2-8)§ m L ¤'§ L v
83.2.动量算符和角动量算符 18/99國 时,本征值的间隔可以任意地小,当L→∞时,本征谱由分立谱过 度到连续谱 现在,归一化常数、本征函数和归一化条件分别为 C=L-3/2 WpL32eP方京 (329) L/2 少2ypdr=1 L/2 箱归一化:把粒子限制在三维箱中并加上周期性边界条件的归一 化方法 动量本征函数(乘以时间因子exp(-E)就是自由粒子的波 函数.在其描述的状态中,粒子的动量有确定的值,该值就是动量 算符在此态中的本征值 ● First●Prev●Next●Last● Go Back● Full Screen●cose●Quit
• First • Prev • Next • Last • Go Back • Full Screen • Close • Quit §3.2. ÄþÎÚÄþÎ 18/99 § m ±?¿/§ L −→ ∞ § Ìd©áÌL ÝëYÌ© y3§8z~ê! ¼êÚ8z^©O C = L −3/2 , ψp = 1 L3/2 exp i ~ ~p · ~r, (3.2-9) Z L/2 −L/2 ψ ∗ pψpdτ = 1. 8zµrâf3n¥¿\þ±Ï5>.^8 z{© Äþ ¼ê ψp(~r) ¦±mÏf exp − i ~ Et Ò´gdâfÅ ¼ê©3Ù£ãG¥§âfÄþk(½ ~p§TÒ´Äþ Î3d¥ ©
3.2.动量算符和角动量算符 19/99 322角动量算符 在直角坐标系中,角动量算符L=产×的三个分量为 yPz-2Py =i zPx-rp (3.2-10) 2=-y=(-票是) L2+ 「(no-a2./a_a) (3.2-11) x 在讨论原子分子体系时,采用如图示的球坐标(r,6,p)是方便 的 ● First●Prev●Next●Last● Go Back● Full Screen●cose●Quit
• First • Prev • Next • Last • Go Back • Full Screen • Close • Quit §3.2. ÄþÎÚÄþÎ 19/99 3.2.2 ÄþÎ 3IX¥§ÄþÎ Lb = ˆ~r × ˆ~p n©þµ Lbx = ybpz − zbpy = ~ i y ∂ ∂z − z ∂ ∂y Lby = zbpx − xbpz = ~ i z ∂ ∂x − x ∂ ∂z Lbz = xbpy − ybpx = ~ i x ∂ ∂y − y ∂ ∂x (3.2-10) Lb2 = Lb2 x + Lb2 y + Lb2 z = −~ 2 " y ∂ ∂z − z ∂ ∂y 2 + z ∂ ∂x − x ∂ ∂z 2 + x ∂ ∂y − y ∂ ∂x 2 # (3.2-11) 3?Øf©fNX§æ^Xã«¥I (r, θ, ϕ) ´B ©
832.动量算符和角动量算符 20/99 图19球极坐标 ● First●Prev●Next●Last● Go Back● Full Screen●cose●Quit
• First • Prev • Next • Last • Go Back • Full Screen • Close • Quit §3.2. ÄþÎÚÄþÎ 20/99
32.动量算符和角动量算符 219回 r= rsin e cOS p y= rsin e sinφ z= rcos 0 (3.2-12 +y-+ coS tan p ar a sin 0 cos dr a sin esin p =a(x2+y2+z2)2 cos 6 ● First●Prev●Next●Last● Go Back● Full Screen●cose●Quit
• First • Prev • Next • Last • Go Back • Full Screen • Close • Quit §3.2. ÄþÎÚÄþÎ 21/99 x = rsin θ cos ϕ y = rsin θ sin ϕ z = r cos θ r 2 = x 2 + y 2 + z 2 cos θ = z r tan ϕ = y x (3.2-12) ∂r ∂x = ∂ ∂x x 2 + y 2 + z 2 1 2 = x r = sin θ cos ϕ ∂r ∂y = ∂ ∂y x 2 + y 2 + z 2 1 2 = y r = sin θ sin ϕ ∂r ∂z = ∂ ∂z x 2 + y 2 + z 2 1 2 = z r = cos θ