Physics 2D Lecture Slides Lecture 28: Mar 9th

Transcription

1 This is the fist couse t UCSD fo which Lectue on Demnd hve been mde vibe. Minutes of Steming video seved to hundeds of demnds without inteuption (4/7) P. tke minutes to fi out the Steming Video Suvey sent to you st week by the Physics Deptment. You input one wi decide whethe to extend this eduction sevice to othe csses t UCSD nd to the next genetion of UC students. Physics D Lectue Sides Lectue 8: M 9th Vivek Shm UCSD Physics

2 The Couomb Attctive Potenti Tht Binds the eecton nd Nuceus (chge +Ze) into Hydogenic tom -e m e +e F V kze U () = The Hydogen Atom In Its Fu Quntum Mechnic Goy U ( ) = x,y,z mixed up! x + y + z As in cse of ptice in 3D box, we shoud use sepetion of vibes (x,y,z??) to deive 3 independent diffeenti. eqns. This ppoch wi get vey ugy since we hve "conjoined tipet" To simpify the sitution, choose moe ppopite vibes Ctesin coodintes (x,y,z) Spheic Po (, θφ, ) coodintes kze U () =

3 Spheic Po Coodinte System Voume Eement dv dv = ( sin θdφ) ( dθ)( d) = si nθddθdφ dv The Hydogen Atom In Its Fu Quntum Mechnic Goy Insted of witing Lpcin = + +, wite fo spheic po coodintes: x y z = sin + θ + sn i θ θ θ sin θ φ Thus the T.I.S.Eq. fo ψ(x,y,z) = ψ(, θ, φ) becomes ψ(, θ, φ) ψ(, θ, φ) sinθ + + sinθ θ θ ψ(, θ, φ) m + (E-U()) ψ (, θ, φ) = sin θ φ with U ( ) = x + y + z

4 The Schodinge Eqution in Spheic Po Coodintes (is bit of mess!) The TISE is : m ψ sin ψ + θ ψ + sinθ θ θ sin θ φ + (E-U()) ψ(, θ, φ) = Ty to fee up second st tem fom except φ This equies mutipying thuout by sin θ sin θ ψ ψ ψ m sin θ ke + sinθ sinθ + + (E+ ) ψ = θ θ φ Fo Sepetion of Vibes, Wite ψ(, θ, φ) = R(). Θ( θ). Φ( φ) Pug it into the TISE bove & divide thuout by ψ(, θ, φ)=r(). Θ( θ). Φ( φ) Ψ(, θφ, ) R() =Θ( θ). Φ( φ) Note tht : Ψ(, θφ, ) Θ( θ ) = R ( ) Φ( φ) θ θ when substituted in TISE Ψ(, θφ, ) Φ( φ) = R () Θ() θ θ φ Don t Pnic: Its simpe thn you think! sin θ R R sinθ Θ Θ θ θ Φ m sin θ ke E+ ) = Φ φ + sinθ + + ( Renge by tking the φ tem on RHS sin θ R sinθ Θ + R Θ θ θ m sin θ ke Φ Φ φ sinθ + (E+ ) =- LHS is fn. of, θ & RHS is fn of φ ony, fo equity to be tue fo, θ, φ LHS= constnt = RHS = m

5 Deconstucting The Schodinge Eqution fo Hydogen Now go bek up LHS to sepete the & θ tems..... sin θ R sinθ Θ m sin θ ke LHS: + + (E+ ) R Θ θ θ sinθ =m Divide Thuout by sin nd nge tems with w R R + m θ ke (E+ )= sin m y fom θ Θ sinθ θ Θsinθ θ θ Sme gument : LHS is fn of, RHS is fn of θ ; Fo them to be equ fo, θ LHS = const = RHS = ( + ) Wht is the mysteious ( + )? Just numbe ike (+) So Wht do we hve fte the shuffing! d Φ + = dφ m Φ.. d dθ m sin θ + ( + ) ( θ)...() Θ = sinθ dθ dθ sin θ d d R +...() (E+ )- R ( ) =...(3) m ke ( + ) These 3 "simpe" diff. eqn descibe the physics of the Hydogen tom. A we need to do now is guess the soutions of the diff. equtions Ech of them, cey, hs diffeent function fom

6 And Now the Soutions of The S. Eqns fo Hydogen Atom d Φ + Φ = dφ The Azimuth Diff. Eqution : m imφ Soution : Φ( φ) = A e but need to check "Good Wvefunction Condition" Wve Function must be Singe Vued fo φ Φ( φ)= Φ ( φ+ π) Φ( φ) = A e φ im = = ± ± ± im ( ) A e φ+ π m,,, 3...( Mgnetic Quntum # ) Φ d dθ m The Po Diff. Eq: sin θ + ( + ) ( θ) Θ = sinθ dθ dθ sin θ Soutions : go by the nme of "Associted Legende Functions" ony exist when the integes nd m e eted s foows m =, ±, ±, ± 3... ± ; = positive numbe : Obit Quntum Numbe Wvefunction Aong Azimuth Ange φ nd Po Ange θ Fo =, m = Θ( θ ) = ; Fo =, m =, ± Thee Possibiities fo the Obit pt of wvefunction 6 3 [ =, m = ] Θ( θ) = cos θ [ =, m = ± ] Θ( θ) = sin θ 4 [ =, m = ] Θ( θ) = (3cos θ )... nd so on nd so foth (see book fo moe Functions)

7 Rdi Diffeenti Equtions nd Its Soutions d R m ke ( + ) The Rdi Diff. Eqn: + (E+ )- R( ) d = Soutions : Associted Lguee Functions R(), Soutions exist ony if:. E> o hs negtive vues given by ke E=- ; with = = Boh Rdius n mke. And when n = intege such tht =,,,3, 4,...( n ) n = pincip Quntum # o the "big dddy" quntum # The Hydogen Wvefunction: ψ(,θ,φ) nd Ψ(,θ,φ,t) To Summize : The hydogen tom is bought to you by the ettes: n =,,3,4,5,... =,,,3,,4...( n ) Quntum # ppe ony in Tpped systems m =, ±, ±, ± 3,... ± The Spti pt ofthe Hydogen Atom Wve Function is: ψ(, θ, φ) = R ( ). Θ ( θ). Φ ( φ) = R m Y e known s Spheic Hmonics. They define the ngu stuctue in the Hydogen-ike toms. n m n m Y m ie t The Fu wvefunction is Ψ (, θϕ,, t) = ψ(, θ, φ) e

8 Rdi Wve Functions Fo n=,,3 n m R()= -/ 3/ e 3 (- )e 3/ 3/ - (7 8 + ) e 8 3 n= K she n= L She n=3 M she n=4 N She 3 = s(hp) sub she = p(incip) sub she = d(iffuse) sub she =3 f(undment) ss =4 g sub she.. Symboic Nottion of Atomic Sttes in Hydogen s ( = ) p ( = ) d ( = ) f( = 3) g( = 4)... n s s p 3 3 s 3 p 3d 4 4 s 4 p 4 d 4 f 5 5 s 5 p 5 d 5 f 5g Note tht: n = is non-degenete system n> e degenete in nd m. ke E=- A sttes hve sme enegy n But diffeent ngu configution

9 Enegy Sttes, Degenecy & Tnsitions Fcts About Gound Stte of H Atom n=, =, m = R = Θ θ = Φ φ = -/ () e ; () ; () 3/ π Ψ θφ = π -/ (,, ) e...ook t it cef. Spheicy symmetic no θφ, dependence (stuctue). Pobbiity Pe Unit Voume : Ψ (, θ, φ) = e π 3 Likeihood of finding the eecton is sme t θφ, nd depends ony on the di sepetion () between eecton & the nuceus. ke 3 Enegy of Gound Stte =- = 3.6eV Ove The Gound stte wvefunction of the hydogen tom is quite boing Not much chemisty o Bioogy coud deveop if thee ws ony the gound stte of the Hydogen Atom! We need stuctue, we need viety, we need some cuves! uy

10 Intepeting Obit Quntum Numbe () d dr m ke ( + ) Rdi pt of S.Eqn: + ( E+ )- R( ) d d = ke Fo H Atom: E = K + U = KRADIAL + K ORBITAL ; substitute this in E d dr m ( + ) K + RADIAL + KORBI TAL - R d d m () = Exmine the eqution, if we set K ORBITAL ( + ) = then m wht emins is diffeenti eqution in d dr m + [ KRAD IAL ] R( ) = which depends ony on dius of obit d d L Futhe, we so know t ht K ORBITAL = mvobit; L= p ; L =mv ob KORBIT = m L Putting it togthe: K mgnitude of Ang m m L p AL ( + ) ORBITAL =. Mom L = = + ( ) Since = positive intege=,,,3...(n-) ngu momentum L = ( + ) = discete vues L = ( + ) : QUANTIZATION OF Eect on's Angu Mom entu m Mgnetic Quntum Numbe m L = p (Right Hnd Rue) Cssicy, diection & Mgnitude of L wys we defined QM: Cn/Does L hve definite diection? Poof by Negtion: Suppose L ws pecisey known/defined (L z) ˆ Since L = p Eecton MUST be in x-y obit pne p z = ; pz z p z ; E =!!! m So, in Hydogen tom, L cn not hve pecise mesube vue Uncetinty Pincipe & Angu Momentum : L φ z

11 Mgnetic Quntum Numbe m Conside = L = ( + ) = 6 In Hydogen tom, L cn not hve pecise mesube vue Abitiy picking Z xis s efeence diection: L vecto spins ound Z xis (pecesses). The Z component of L : L = m; m =±, ±, ± 3... ± Z Note : since L < L (wys) since Z m < ( + ) It cn neve be tht L = m= ( + ) (beks Uncetinty Pincipe) Z So...the Eecton's dnce hs begun! L=, m =,±, ± : Pictoiy Eecton sweeps Conic pths of diffeent ϑ: Cos ϑ= L Z /L On vege, the ngu momentum Component in x nd y cnce out <L X > = <L Y > =

12 Whee is it ikey to be? Æ Rdi Pobbiity Densities Ψ (,θ, φ ) = Rn ( ). Θm (θ ). Φ m (φ ) = Rn Ym Pobbiity Density Function in 3D: P(,θ,φ ) = Ψ*Ψ = Ψ (,θ, φ ) = Rn. Ym Note : 3D Voume eement dv=.sin θ.d.dθ.dφ Pob. of finding ptice in tiny voume dv is P.dV = Rn. Ym..sin θ.d.dθ.dφ The Rdi pt of Pob. distibution: P()d π π dv P()d= Rn. d Θm (θ ) dθ Φ m (φ ) dφ When Θm (θ ) & Φ m (φ ) e uto-nomized then P()d= Rn.. d; in othe wods P()= Rn Nomiztion Condition: = R n d Expecttion Vues <f()>= f().p()d Gound Stte: Rdi Pobbiity Density P( )d = ψ ( ).4π d 4 P ( )d = 3 e Pobbiity of finding Eecton fo > 4 e d = 3 P > To sove, empoy chnge of vibe Define z= ; chnge imits of integtion P > = z z e dz (such integs ced Eo. Fn) =- [ z + z + ]e z = 5e = %!!

13 Most Pobbe & Avege Distnce of Eecton fom Nuceus Most Pobbe Distnce: 4 n= = m = P d = e 3 In the gound stte (,, ) ( ) Most pobbe distnce fom Nuceus Wht vue of is P() mx? dp 4 d =. e e 3 = + d d = + = = o =... which soution is coect? (see pst quiz) : Cn the eecton BE t the cente of Nuceus (=)? 4 ( = ) = =! Most Pobbe distnce = 3 (Boh guess P e Wht bout the AVERAGE oction <> of the eecton in Gound stte? <>= 4 P()d=. e d... chnge of vibe z= 3 = < >=... Use gene fom z =! = ( 4 3 z n z z e dz e dz n n n z= < >= = ed ight) )( n )...() 3 3!! Avege & most ikey distnce is not sme. Why? 4 Asnwe is in the fom of the di Pob. Density: Not symmetic Rdi Pobbiity Distibution P()= R() Becuse P()= R() No mtte wht R() is fo some n The pob. Of finding eecton inside nuceus =