2? 1. I i I r2sinodod4xlr = 2 is radial probability distribution. II2 = 1R1 21Y1. 4 txi. 2dr

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1Y1 qualitative and quantitative procedures used for the l-d case can be Also, X(r) at r = is equivalent to LJ(r,9.) = for r.

1 e 4 tx iir the physical significance of the X(r) function. separate problem. Before proceeding to consider canonical examples, note immediately applied, but it is necessary to treat each value of L as a Consider. 1Y1 qualitative and quantitative procedures used for the l-d case can be Also, X(r) at r = is equivalent to LJ(r,9.) = for r. Hence the angles dr if pr This has exactly the form of a l-d problem with the effective potential ur jx_ex. n terms of the function X(r) rr(r), thee equation is THREE DMENSONAL RADAL PROBLEMS N SPHERCAl,. POLAR COORDNATES r? 1 These are equivalent to setting ft = 1, e =, p = energy in terms of Rydberg energy, Ry = distance in terms of Bohr radius, a = pe t is convenient to introduce atomic units: 7 d - e +lcl(l4.1))x EX. Hydrogen Atom so tx! is radial probability distribution. i rsinodod4xlr = and get regardless of angular orientation, we integrate t factor over all angles f we want the probability that the particle is between r and r dr, = 1R1 = R(r)Y(O,) U(r,t) = V(r) +. (L+ 1) energy given by d ;- + V(r)

1 w) T or Thus, The classical turning points are defined by examining the effective potential function The qualitative form of the radial wavefunctions can be readily inferred by..., ra!$b.

2 1 w) T or Thus, The classical turning points are defined by examining the effective potential function The qualitative form of the radial wavefunctions can be readily inferred by..., ra!$b. + + ( = - L(t+l)) = o Thus the problem becons RADAL PROBLEMS - functions and the wavefimctions. Roughly, we find: With these values for r, r, U(r r=r r r,9j, U(r,L). we can sketch the U(r,t) and hence r = i(l+1) and U(rm,L) = - m m = or -- = t(t.1) The minimum in U(r,) is given by = n[n ;. - (+1) ]. n r T = or U(r,L) = - 1 L(.+l) E = i(r,r) = 1; = at r = r,where r =. r r r<o U(r,L) = - + (L+l) r>o n 1,, 3, n Wuoo..j.nL Du4L4 E = - 1 iiusual case since no dependence ozj given by withxoatr=oandxoasr-.. Thebowidstateenergylevelsare

3 MWrLbt& UUML Ti = 1 not occur n = 1, not occur in m r = and tj(r ) = - 1 etc. 1.18, 6.8 for n = r(outer) = n -] r(inner) = n[n-vi?i] r(outer) = n[n+/n r(inner) = 1 El / / or s / F L= RADAL PROBLEMS - 3 the extremely flat character of U(r,t) at large r: tative form of the wavefunctions reflects the classical turning points and radial function has Ti L nodes (not counting that at r = ). The quali Note that the nodal properties are apparent from these sketches: the (n,.) levels all occur n = 1,, 3, in in r = and U(r ) - 3 for n = 4 18 for n = 3 8 for n = r = forn = 1 = 1 or is r = 6 and U(r) r(outer) = r(inner) = nfn- 4] - - F

RADAL PROBLEMS - 4 6 SP4E!UCALLY SYMMLrk!C SVSTkMS [ctap. 7 F- Pøweit eo. j:j. : S S 4 1 Fic. 7 6.

4 RADAL PROBLEMS SP4E!UCALLY SYMMLrk!C SVSTkMS [ctap. 7 F- Pøweit eo. j:j. : S S 4 1 Fic Th radial probability distribution function u for sccral values of the quantum numbers n, 1. (From E. U. Cum!on nci G. H. Shortley, The Thcory of Atomic Spcctrc, Carnbridgc University Prcss, Cambridge, 1953, by verrnissiou.)

5 3o bpctwe potkrr,4lcucvf$. JJEG-y LE.vLS

:1: ::: :: :!!ii!! tll [1.4.1 ik[ffl Nl :11 :1.1.. n: ni il;j :.1..1 ci Sc D U) l i jlt - :.

: h;, i;iii i: ;cihh n ii!!.1:./ V Sn 1OY Z S V JJflN 1 fl; rh, H -.1 :3 V s -a 44 US.1 i -S V.

6 :1: ::: :: :!!ii!! tll [1.4.1 ik[ffl Nl :11 :1.1.. n: ni il;j :.1..1 ci Sc D U) l i jlt - :.. ;: ii... 4 iii flu ti! 1111 i :;: ::::;:i; ::; ri lift hi 1H ih ii!;! Hi! H 1.1. :1 :! ill! lili 1111 C. 1:1 ii :1!.:,: i :1 lit ii ii ii:, uu l il ll ii P w ; fl i -._1 4 -, ill H H i ii H,:: :r._.i..1 iu T ii.. i i i iij ii.1,., ij.i UUJ 1!]! ±. 1 l _. : h;, i;iii i: ;cihh n ii!!.1:./ V Sn 1OY Z S V JJflN 1 fl; rh, H -.1 :3 V s -a 44 US.1 i -S V. *HH coo/az j S Sb L / V-9SC L Z ]H1 11 X 3 5-an-al 4sar.. oj apo9tflffl3fj 4)wn 14) :1. : J LLL. L lb :ib,!1i!. 3 t ii i [1.11 1i

energy unit Then sotropic Harmonic Oscillator s-..- Classical turning points: dr iir kr n.

7 r = [.(L+l)]1 4 and U(r,.) = [t(l+1)]1/ or Bound state energy levels given by Minimum in U(r,L): + = ( with x= Oat r = and x-o at Use for energy unit Then sotropic Harmonic Oscillator s-..- Classical turning points: dr iir kr n.;1)) = Lx RADAL PROBLEMS - 5 6j5V, r in - 3 L(L+1) o -t n+3-r - L(L.l)J and U(r,.) = n+3 r 1/ + 3) no dependence on. E = n+3, n =, 1,,... Again an unusual case because - r - (.+l)) = o Use for distance unit with w = (k/ln)w

in 7 RADAL PROBLEMS - 6 Sketch U(r,L) functions and wavefunctions, making use of what we know already from solution of the isotropic oscillator problem in Cartesian coordinates: 11 + - Energy n=4 or

8 in 7 RADAL PROBLEMS - 6 Sketch U(r,L) functions and wavefunctions, making use of what we know already from solution of the isotropic oscillator problem in Cartesian coordinates: Energy n=4 or 3s 1.n=5 or 4p / 1n or 3p or or units n= or s -. of 3..n= or is,n=1 or p r (inner) = o 1 r (inner) = E 4 3] 1 r (inner) = (outer) = (n+3) r (outer) = n4 /(3)) r (outer) = o inner/outer inner/outer inner/outer n = / / /.4.669/ /1.73 / /.6 1./.46 3 /3..81/.9 4 /3.33 r =,U = r =l.191,u =.83 r =l.57,u =4.9 n in in m in m n =,, 4,... n = 1, 3, 5,... n =, 4, 6, Dashed horizontal lines indicate location of energy levels allowed for a l-d oscillator but which are disallowed for the 3-D oscillator.

9 -, -. 1 Th fl9% n. 4,i to : io to flu NTlt,ttTEl b ;6 n Of t) i.! :: 1 :....,., 1 S JZ _i. 1.:. [i1tllt i ni i: ill.1.

10 V : V V ll [4iL iiilli. ii ii.1: W.1 V i V j:i. V V. ---r V...:. h: V : : : L z L L. V lox to TO ill: (:1.1. &Yov /o..jecs-( (e.yej E. i v).1 v :. ii:: liii,.1.1 ii ii ii: V : i : 1 V iii 1 i,, ti it i 1V. :1 : ill :1:.: : :1..

x RADAL PROBLEMS - 7 Note the relationships between the l-d problem and the = case of the 3-D problem: l-d Oscillator 3- Oscillator t= = 5 x n=4 n = 4 = n -

.. solutions, which all have maxima or minima there, whereas te n = 1, 3, 5,... solutions remain good since they have nodes at the midpoint.

> cases of the 3-D problem, there is no simple relation to the 1- problem.

11 x RADAL PROBLEMS - 7 Note the relationships between the l-d problem and the = case of the 3-D problem: l-d Oscillator 3- Oscillator t= = 5 x n=4 n = 4 = n - 1 x n = We see that inserting an infinite wall at the mid-point of the 1- potential eliminates the n =,, 4,... solutions, which all have maxima or minima there, whereas te n = 1, 3, 5,... solutions remain good since they have nodes at the midpoint. Hence the x > portions of the latter become the solutions for the 3-, 9. = problem, with = =,, 4,... (for n = 1,,...). For the. > cases of the 3-D problem, there is no simple relation to the 1- problem. However, there are of course simple relations with the solution of the 3- problem in Cartesian coordinates. These involve resolv ing the (n, n>,. degeneracies into the (n,l,m) states appropriate for spherical polar coordinates. Thus, as in the Kramer s treatment of spherical harmonics, one readily finds the following correspondences:

(1O D 1 () 3 egeneracy fl (units of - Tw) Momentum -

of Angular Cartesian RADAL PROBLEMS - 8 / Spherical Well,

n=l 13 give solutions corresponding to certain L values,

cartesian solutions for a particular n etc.

for energy imit ic / Use for distance imit a for a.

12 (1O D 1 () 3 egeneracy fl (units of - Tw) Momentum - Quanta States Energy No. of Angular Cartesian RADAL PROBLEMS - 8 / Spherical Well, nfinitely Deep n=3 3p 4f n= -1cw s 3d n= ls etc. n=l 13 give solutions corresponding to certain L values, as follows: Linear cothbinations of the degenerate cartesian solutions for a particular n etc. } states 1-3 states 7 (o- state (1) -5 states iia a Use for energy imit ic / Use for distance imit a for a. Consider V(r) for r < a / 9=l L= 3s 4d 4, n=5 Sf (3) (1) (1) 1 (3) (1) (1) (111) 3 (3) (1) (1) () (11) 6 () (11) () (11) 3 (l 1

13 Th E = forri. r dr + Then Thus have - = for < r < 1. Now require x= both RADAL PROBLEMS - 9 the following results: For 9. >, r (outer) = 1 always, whereas for r(inner) we have: Examine classical turning points:. in this problem. out that in fact there are no degeneracies among energy levels of different dealt with a cubical well and we are considering a spherical one. t turns For. > the energy levels will depend on the value of. as well as n. Note that here we cannot compare with the cartesian problem because that To make X For. >, minimum in tj(r,.) occurs at r = 1 and = t(l+l). Hence, find - = L(..l). f.(.+l) 1) or r (inner) U(r,j= for<r<l = ntr or E = nir in our reduced units 11 = 1,, 3, = at r = 1 requires X(r) = Nsjn, 1r (The cos r term is absent since need x = at r =.) forl=o at r and r = 1. For. = the solution is identical to the l-d case 7 E

14 spatial extent of the wavefunction is compressed. As 9 increases for a given n, the levels shift upwards because the 1zZ: n= L=1 RADAL PROBLEMS - 1 i:

15 ? 3L / 3 / s ( C. / to_-. L 1 Energy Levels and Effective Potential Curves for an nfinite Spherical Well -.-o qo i -. /z --._-.--- /

16 *L. llb. :.L n-i ii l V. F. :1;.. :.. 1 ttl in flbc k ft 1,-fl io x io to UL ct 1oLAJe e.vu) (A,utS D 1::. Li :...1 :1 F 1W.., ii :1. V,... i iJ :1 i 1iiii ii H!1... :1.. U.. i V ii liii :i1 fi1v11.! i:. iii tu i H V liii :1 i V 4f :1 1 ±1iH4 :it.tjjr! 41: ii. 1i :1: ii 1

Well is merely to emphasize qualitative features without getting involved in Oscillator n--l n ) Cr sotropic / problems, one for each value, the qualitative form of the wavefunctions, tedious details.

17 Well is merely to emphasize qualitative features without getting involved in Oscillator n--l n ) Cr sotropic / problems, one for each value, the qualitative form of the wavefunctions, tedious details. n every case, by resolving the problem into a set of order of levels, etc., can be deduced with practically no calculation. Comments RADAL PROBLEMS - 11 l.(z) is an associated 1 olojjl, j(z) a spherical Bessel function. r er L L+l -nfl t.l fr\ Z.l n -r/n H-Atom r e L r r e Problem General y as r - as r + used. Jo to be determined such that r x 4ndr = 1. Reduced units as defined above are 3 problems considered here, omitting in each case a normalization factor k = /determined by j(k)= r j,(kr) r Spherical. pfl - For reference, we list below the radial wavefunctions, y1,(r) for the spherical well problems is available in many texts - our aim in these notes The mathematical treatment of H-atom, 3-D isotropic oscillator and

$n=4 At small r, - const. r At large r, n= 3 n= n=5 = --_j +q L1=--r / n=o f5 L1=--r - ) (r L ) Cr r/ n= 1 n= 3 n= n-r-1\ n = N e r -nfl t+1.+1,r H ATOM RADAL WAVEFLJNCTONS n= 1 t+ 1 - const.$

18 n=4 At small r, - const. r At large r, n= 3 n= n=5 = --_j +q L1=--r / n=o f5 L1=--r - ) (r L ) Cr r/ n= 1 n= 3 n= n-r-1\ n = N e r -nfl t+1.+1,r H ATOM RADAL WAVEFLJNCTONS n= 1 t+ 1 - const. r e n l -r L=-j-- +r L1=--r = r 3 L 1 Lt=1 L=l L= L=4 t=5 Table of L = N e r 1 HARMONC OSCLLATOR WAVEFUNCTONS At small r, - const. r. At large r, const. r e n -n/n 81/6 (6 - r) 81/ (718r 4 4 A( r) = Table of NL (i.) ncluding the normalization factor N

19 C - C, ft C, CD - Cl, Ci ft ft -. p-s C - S 5-, 5 p-s C. 5 Vi rn -4 C-, Th.,i C-, cu Vi - C-,.1. C -i V V T1 l A A -.., -. CD l t S S CA CA P4,. t c,i O%J CA Vi.. V 3.. CD l C -. U, C l p-s %J + CẠ Cu + C, 1%. a o (11 D - l J k. a Vi Vi - U) -. - CD r -5 C (i 5-, Vi i U, C -C t N) -C S S - ii _% N) N) Ca CA t 3 - CD C, Ci ft S. p-s C CA - 3 Ci U) C -C

20 Energy Levels for a particle in an infinitely deep spherical well, in units of. ia fl T is 9.87 p d 33.9 s p 589 4d 8.4 3s 83.6 : 4s 157.7

~,. :'lr. H ~ j. l' ", ...,~l. 0 '" ~ bl '!; 1'1. :<! f'~.., I,," r: t,... r':l G. t r,. 1'1 [<, ."" f'" 1n. t.1 ~- n I'>' 1:1 , I. <1 ~'..

~,. :'lr. H ~ j. l' , ...,~l. 0 ' ~ bl '!; 1'1. :<! f'~.., I,, r: t,... r':l G. t r,. 1'1 [<, . f' 1n. t.1 ~- n I'>' 1:1 , I. <1 ~'.. ,, 'l t (.) :;,/.I I n ri' ' r l ' rt ( n :' (I : d! n t, :?rj I),.. fl.),. f!..,,., til, ID f-i... j I. 't' r' t II!:t () (l r El,, (fl lj J4 ([) f., () :. -,,.,.I :i l:'!, :I J.A.. t,.. p, - ' I I I