Physics 2D Lecture Slides Lecture 28: Mar 9th Vivek Sharma UCSD - PDF document

This is the first course at UCSD for which “Lecture on Demand” have been made available. 2000 Minutes of Streaming video served to hundreds of “demands” without interruption (24/7) Pl. take 10 minutes to fill out the Streaming Video Survey sent to you last week by the Physics Department. Your input alone will decide whether to extend this educational service to other classes at UCSD and to the next generation of UC students. Physics 2D Lecture Slides Lecture 28: Mar 9th Vivek Sharma UCSD Physics

The Coulomb Attractive Potential That Binds the electron and Nucleus (charge +Ze) into a Hydrogenic atom -e m e F V +e r 2 kZe = ( ) U r r The Hydrogen Atom In Its Full Quantum Mechanical Glory 1 1 ∝ = ⇒ ( ) x,y,z all mixe d up ! U r + + r 2 2 2 x y z As in case of particle in 3D box, we should use seperation of r variables (x,y,z ??) to derive 3 independent differential. eq ns. This approach will get very ugly since we have a "co njoined triplet" To simplify the situation, choose more appropriate variables → θ φ Cartesian coordinates (x,y,z) Spherical Polar (r , , ) coordinates 2 kZe = ( ) U r r

Spherical Polar Coordinate System Vol ume Element dV = θ φ θ ( sin ) ( )( ) dV r d rd dr θ θ φ 2 = r si n drd d dV The Hydrogen Atom In Its Full Quantum Mechanical Glory ∂ ∂ ∂ 2 2 2 ∇ = + + 2 Instead of writing Laplacian , r ∂ ∂ ∂ 2 2 2 x y z ∇ 2 write for spherical polar coordinates: ∂ ∂ ∂ ∂ ∂ ⎛ ⎞ ⎛ ⎞ 2 1 1 1 ∇ + θ + 2 2 = sin ⎜ r ⎟ ⎜ ⎟ ∂ ∂ θ ∂ θ ∂ θ θ ∂ φ 2 ⎝ ⎠ 2 ⎝ ⎠ 2 2 2 r r r r s n i r sin ψ ψ θ φ Thus the T.I.S.Eq. for (x,y,z) = (r, , ) be come s ∂ ∂ ψ θ φ ∂ ∂ ψ θ φ ⎛ ⎞ ⎛ ⎞ 1 (r, , ) 1 (r, , ) + θ + 2 ⎜ ⎟ ⎜ sin ⎟ r ∂ ∂ θ ∂ θ ∂ θ 2 ⎝ ⎠ 2 ⎝ ⎠ r sin r r r ∂ ψ θ φ 2 1 (r, , ) 2m ψ θ φ + (E-U(r)) (r, , ) = 0 θ ∂ φ 2 2 2 � 2 si n r 1 1 ∝ = with U r ( ) + + r 2 2 2 x y z

The Schrodinger Equation in Spherical Polar Coordinates (is bit of a mess!) The TISE is : ∂ ∂ ψ ∂ ∂ ψ ∂ ψ ⎛ ⎞ ⎛ ⎞ 2 1 1 1 2m + θ + ψ θ φ 2 ⎜ ⎟ ⎜ sin ⎟ + (E-U(r)) (r, , ) =0 r ∂ ∂ θ ∂ θ ∂ θ θ ∂ φ 2 ⎝ ⎠ 2 ⎝ ⎠ 2 2 2 � 2 r sin sin r r r r φ Try to free up second last term fro m all except θ ⇒ 2 2 T his requires multiplying thr uout by sin r ∂ ⎛ ∂ ψ ⎞ ∂ ⎛ ∂ ψ ⎞ ∂ ψ θ 2 2 2 2 2m sin ke r θ + θ θ + ψ 2 2 sin ⎜ ⎟ sin ⎜ sin ⎟ + (E+ ) =0 r ∂ ∂ ∂ θ ∂ θ ∂ φ ⎝ ⎠ ⎝ ⎠ 2 � 2 r r r ψ θ φ Θ θ Φ φ For Seperation of Variables, Write (r, , ) = R(r) . ( ) . ( ) ψ θ φ Θ θ Φ φ Plug it into the TISE above & divide thruout by (r, , )=R(r). ( ). ( ) ∂Ψ θ φ ∂ ( , , r ) R(r) = Θ θ Φ φ ( ). ( ) ∂ ∂ r r ∂Ψ θ φ ∂Θ θ ( , , ) r ( ) = Φ φ ⇒ Note that : ( ) ( ) when substituted in TISE R r ∂ θ ∂ θ ∂Ψ θ φ ∂Φ φ ( , , ) ( ) r = Θ θ ( ) ( ) R r ∂ θ ∂ φ Don’t Panic: Its simpler than you think ! θ ∂ ∂ θ ∂ ∂Θ ∂ Φ θ 2 ⎛ ⎞ ⎛ ⎞ 2 2 2 2 sin sin 1 2m sin ke R r + θ + 2 ⎜ ⎟ ⎜ sin ⎟ + ( E+ ) =0 r ∂ ∂ Θ ∂ θ ∂ θ Φ ∂ φ ⎝ ⎠ ⎝ ⎠ 2 � 2 R r r r φ Rearrange b y taking the term on RHS θ ∂ ∂ θ ∂ ∂Θ θ ∂ Φ ⎛ ⎞ ⎛ ⎞ 2 2 2 2 2 sin sin 2m sin ke 1 R r + θ 2 ⎜ ⎟ ⎜ s in ⎟ + (E+ ) =- r ∂ ∂ Θ ∂ θ ∂ θ Φ ∂ φ ⎝ ⎠ ⎝ ⎠ � 2 2 r R r r θ φ θ φ LHS i s f n. of r, & RHS is fn of only , for equality to be true for all r, , ⇒ 2 LHS= constant = RHS = m l

Deconstructing The Schrodinger Equation for Hydrogen θ Now go break up LHS to seperate the r & terms... .. θ ∂ ∂ θ ∂ ∂Θ θ 2 ⎛ ⎞ ⎛ ⎞ 2 2 2 sin sin 2m sin ke R r + θ 2 2 LHS: ⎜ ⎟ ⎜ si n ⎟ + (E+ ) =m r ∂ ∂ Θ ∂ θ ∂ θ ⎝ ⎠ ⎝ ⎠ � 2 l r R r r θ θ ⇒ 2 Divide Thruout by sin and arrange all terms with r aw a y from ∂ ∂ ∂ ∂Θ ⎛ ⎞ + ⎛ ⎞ 2 2 2 m 1 2m ke 1 R r − θ 2 (E+ )= l sin ⎜ r ⎟ ⎜ ⎟ ∂ ∂ θ Θ θ ∂ θ ∂ θ ⎝ ⎠ � 2 2 ⎝ ⎠ r sin sin R r r θ Same argument : LHS is fn of r, RHS is fn of ; θ ⇒ + For them to be equal for a ll r, LHS = const = RHS = ( 1) l l + What is the mysterious ( 1)? Just a number like 2(2+1) l l So What do we have after all the shuffling! Φ 2 d + Φ = 2 m 0.. ..... ............(1) φ l 2 d ⎡ ⎤ Θ ⎛ ⎞ + 2 1 m d d θ + − Θ θ = sin ( 1) l ( ) 0.....(2) ⎜ ⎟ ⎢ l l ⎥ θ θ θ θ ⎝ ⎠ 2 si n ⎣ sin ⎦ d d ⎡ ⎤ ∂ + ⎛ ⎞ + 2 2 1 2m ke ( 1) d R r l l = 2 (E+ )- ( ) 0....(3) ⎜ r ⎟ ⎢ ⎥ R r ∂ 2 ⎝ ⎠ � 2 2 ⎣ r ⎦ r d r r r These 3 "simple" diff. eqn describe the physics of the Hydrogen atom. All we need to do now is guess the solutions of the diff. equations Each of them, clearly, has a different functional form

And Now the Solutions of The S. Eqns for Hydrogen Atom Φ Φ + 2 d Φ = 2 The Azimuthal Diff. Equation : m 0 φ l 2 d φ Φ φ im Solution : ( ) = A e but need to check "Good Wavefunction Condition" l φ ⇒ Φ φ Φ φ + π Wave Function must be Single Valued for all ( )= ( 2 ) φ φ + π ⇒ Φ φ = ⇒ = ± ± ± im im ( 2 ) ( ) = A e A e m 0, 1, 2, 3....( Magneti c Q uantum # ) l l l Θ ⎡ ⎤ ⎛ ⎞ + 2 1 m d d θ + − Θ θ = The Polar Diff. Eq: ⎜ sin ⎟ ⎢ ( 1) l ⎥ ( ) 0 l l θ θ θ θ ⎝ ⎠ 2 sin ⎣ sin ⎦ d d Solutions : go by the name of "Associated Legendr e Functions" only exist when the integers and are related as follows l m l = ± ± ± ± = m 0, 1, 2, 3.... l ; l positive number l : Orbital Quantum Number l Wavefunction Along Azimuthal Angle φ and Polar Angle θ 1 = ⇒ Θ θ For 0, =0 ( ) = ; l m l 2 = ± ⇒ For 1, =0, 1 Three Possibilities for the Orbital part of wavefunction l m l 6 3 = = ⇒ Θ θ θ = = ± ⇒ Θ θ θ [ 1, 0 ] ( ) = cos [ 1, 1 ] ( ) = sin l m l m l l 2 2 10 = = ⇒ Θ θ θ − 2 [ l 2, m 0 ] ( ) = (3cos 1) l 4 .... and so on and so forth (see book for more Functions)

Radial Differential Equations and Its Solutions ⎡ ⎤ ∂ + ⎛ ⎞ + 2 2 1 2m ke ( 1) d R r l l = 2 The Radial Diff. Eqn : ⎜ ⎟ ⎢ (E+ ) - ⎥ ( ) 0 r R r ∂ 2 ⎝ ⎠ � 2 2 ⎣ r ⎦ r d r r r : Associated Laguerre Functions R(r ), Solutions exist only i f : Solu tio ns 1. E>0 or has negtive values given by ⎛ ⎞ � 2 2 ke 1 = = E=- 2a ⎜ ⎟ ; with a Bohr Radius 0 ⎝ 2 ⎠ 2 n mke 0 = − 2. And when n = integer such that 0,1,2,3,4,.......( 1) l n n = principal Quantum # or the "big daddy" quantum # The Hydrogen Wavefunction: ψ (r, θ , φ ) and Ψ (r, θ , φ ,t) To Summarize : The hydrogen atom is brought to you by the letters: ∞ n = 1,2,3,4,5,.... = − 0, 1,2,3,,4....( 1) Quantum # appear onl y in Trapped sys tems l n = ± ± ± ± m 0, 1, 2, 3,... l l The Spatial part of the Hydrog en Atom Wave Function is : ψ θ φ = Θ θ Φ φ = m ( , , ) ( ) . ( ) . ( ) Y r R r R l m nl lm nl l l l m Y are kn own as Sphe rical Harmonics. They define the angu lar stru cture l l in the Hydrogen-like atoms. iEt − Ψ θ ϕ = ψ θ φ � The Full wavefunction is (r, , , ) ( , , ) t r e

Radial Wave Functions For n=1,2,3 n R(r)= l m l 2 e -r/a 1 0 0 3/2 a 0 r 1 r -2a 2 0 0 (2- )e 0 3/2 a 2 2a 0 0 r 2 − 2 r r − + 3 a 3 0 0 (27 18 2 ) e 0 2 3/2 a 81 3a a 0 0 0 n=1 � K shell l=0 � s(harp) sub shell l=1 � p(rincipal) sub shell n=2 � L Shell l=2 � d(iffuse) sub shell n=3 � M shell l=3 � f(undamental) ss n=4 � N Shell l=4 � g sub shell …… …….. Symbolic Notation of Atomic States in Hydrogen → = = = = = ( 0) ( 1) ( 2) ( 3) ( 4 ) ..... l s l p l d l f l g l n ↓ 1 1 s 2 2 2 s p 3 3 3 3 s p d 4 4 4 4 4 s p d f 5 5 5 5 5 5 s p d f g Note that: •n =1 is a non-degenerate system ⎛ ⎞ 2 ke 1 •n>1 are all degenerate in l and m l. E=- 2a ⎜ ⎟ ⎝ 2 ⎠ n All states have same energy 0 But different angular configuration