Quantization and Special Functions
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1 Otocec Quatizatio ad Special Fuctios Christia B. Lag Ist. f. theoret. Physik Uiversität Graz Cotets st lecture Schrödiger equatio Eigevalue equatios -dimesioal problems Limitatios of Quatum Mechaics d lecture More dimesios Separatio of variables Hydrogeatom
2 Schrödiger s Equatio Erwi Schrödiger (887-96) i ( x, t) H ( x, t) t ψ = ψ time depedet Schrödiger equatio p i ψ( x, t) = + V( x) ψ( x, t) t m with p= i x this gives i ψ( x, t) = + V( x) ψ( x, t) t m x Statioary Schrödiger equatio Assumig factorizatio ψ ( x, t) ϕ( t) ψ( x) = oe gets i ϕ() ψ ( ) ( x) t () t t ϕ = ( x) m + V x x = E i i () t E () t () t exp( Et) t ϕ = ϕ ϕ + V( x) ψ( x) Eψ( x) = m x d ( x) ( E V( x)) ( x) mdx ψ = ψ Statioary Schrödiger equatio Diff. equatio + boudary coditios
3 Boudary coditios E < lim V( x) x Desity ρ(x)= ψ is localized Boud states, discrete eergy spectrum, Sturm-Liouville-problem E > lim V( x) x No localizatio Scatterig solutios, cotiuous eergy spectrum Self-adjoit operators Eigevalue problem for matrices: Aϕi = λϕ i i * Hermitia matrices: A= A ( A ) T ie.. a = a ij ji (Cf. Heiseberg) real eigevalues λ i eigevectors build a orthogoal system Eigevalue problem for operators: Self-adjoit operators: A= A where ( A f, g) ( f, Ag) real eigevalues λ i eigevectors = eigefuctios form a orthogoal basis Differetial operators (differetial equatios) with boudary coditios Sturm-Liouville problem (Cf. Schrödiger)
![Sturm-Liouville Problem: Boudary Values for Diffêretial Equatios [D=] Boud states (): a mole i a hole d mdx ψ ( x) = ( E V 0) ψ ( x) Boudary coditios: ψ ( x 0) = 0 ψ ( x a) = 0 me](/docs-images/96/128944507/images/4-1.jpg "( V)/ = k 0 = =3 = 0 a Oscillator equatio (Helmholtz eq.): bc.. ψ '' + k ψ = 0 Eigesolutios ad Eigevalues: c = ψ ( x) = exp( ± ikx) k= π / a ( = 0,,,.")
4 Sturm ad Liouville Jaques Charles Fracois Sturm ( ) Joseph Liouville (809-88) Collado, Arago, Ampere, Fourier; algebraic equatio; projective geometry, differetial geometry Cauchy, Mathieu, Dirichlet; fouder of Joural de Math. Pure ad Appliquées; Costitutig Assembly (848); astroomy, math. physics, pure math. Sturm-Liouville Problem: Boudary Values for Diffêretial Equatios [D=] Boud states (): a mole i a hole d mdx ψ ( x) = ( E V 0) ψ ( x) Boudary coditios: ψ ( x 0) = 0 ψ ( x a) = 0 me ( V)/ = k 0 = =3 = 0 a Oscillator equatio (Helmholtz eq.): bc.. ψ '' + k ψ = 0 Eigesolutios ad Eigevalues: c = ψ ( x) = exp( ± ikx) k= π / a ( = 0,,,...) ψ ( x) = csi( π x/ a) from orm a E V 0 ψ =Quatizatio! ( x) = dx ψ( x) =
5 [D=] Boud states (): harmoic oscillator d ψ( x) = ( E V( x)) ψ( x) mdx mω V( x) = x Boudary coditios: L (R,exp(-x )) =0 V(x) mω z = x, ψ = u( z)exp( z / ) E uz ( )'' zuz ( )' + uz ( ) = 0 ω E Hermite s D.E.: eigesolutios H (z), =0,,.., eigevalues: = E ( ) ω ω = = + =Quatizatio! Harmoic oscillator probability desity z / ψ ( z) = e H ( z)! π =0 ad H = = ad H = + 4z = ad H =z =3 ad H = z + 8z
6 Hermite s differetial equatio d d e y x e y x dx dx x ( ) + x ( ) = 0 Charles Hermite (8-90) yx ( )'' xyx ( )' + yx ( ) = 0 Eigevalues ( iteger) Eigevectors Hermite polyomials H (x) H 0=, H =x, H =4x -,... Orthogoality relatio: x dx e H ( x) H ( x) = δ π! m m (EP68); Bertrad, Jacobi, Liouville, Cauchy; umber theory, quadratic forms, matrices, 5th order algebraic eq. Problems ad limitatios of QM Reality problem Role of observer? Wave fuctio of uiverse Reality vs. probability (cf. crucial experimets)
7 Problems ad limitatios (cot d) Relativity Relativistic dispersio relatio 4 4 E =± p c + m c ± mc + + O 3 3 Particle creatio ad aihilatio Hilbert space is ormed Decay? p m Relativistic quatum field theory works, but has its ow problems p m c??? Cotets st lecture Schrödiger equatio Eigevalue equatios -dimesioal problems Limitatios of Quatum Mechaics d lecture More dimesios Separatio of variables Hydrogeatom
8 Laplace s differetial operator Laplace equatio ψ = 0 Pierre-Simo Laplace (749-87) I cartesia ccordiates: D = : d dx D = : d d + dx dy d d d D = 3: + + dx dy dz depeds o coordiate system D Alembert, Lagrage; Differetial calculus, math. physics/astroomy, theory of probability, stability of solar system ( méchaique céleste ) Laplace equatio: boudary coditios ψ ( x, yz, ) = 0 is a partial differetial equatio of elliptic type is a boudary value problem: Dirichlet: values of ψ o the boudary of the domai A ψ ( A)=... A Neuma: values of the derivative of ψ o the boudary: ψ. =... A A A
9 Laplace operator i D=3, spherical coordiates e.g. spherical coordiates r 0, 0 ϑ π, 0 ϕ < π x = r cosϕ siϑ y = r siϕ siϑ z = r cosϑ ψ (, r ϑϕ, ) r si ϑ ψ(, r ϑϕ, ) + + r r r r siϑ ϑ ϑ r si ϑ ϕ optimal for problem with spherically shaped boudaries [3D] Jack-i-a-box d d d + + ψ( x, yz, ) = ( kx + ky + kz ) ψ( xyz,, ) dx dy dz Asatz for separatio of variables depeds o geometry of the boudary - i.e. oe should choose optimal variables ψ ( x, yz, ) = X( xy ) ( yzz ) ( ) d X ( x ) = k x X ( x ) dx d ( ) ( ) Y y = k y Y y dy d ( ) ( ) Z z = k z Z z dz
10 Separatio of variables d X x = k x X x dx ( ) ( ) d Y y = k y Y y dy ( ) ( ) d Z z = k z Z z dz ( ) ( ) Eigevalue equatios (SL-systems) Geeral solutio: ψ ( x, yz, ) = cl mxl( xy ) ( yz ) m( z) lm,, with E = k m Eigevalues k x =l, k y =, k z =m; expasio coefficiets determied from boudary values. Hydroge atom e e V( r) = V( r) r r = is radially symmetric ( cetral potetial ) r V() r + E ψ(, r ϑ, ϕ) = 0 m ϑϕ e p Separatio of variables: ψ (, r ϑϕ, ) = a()( r b ϑϕ, ) Leads to separate equatios for the agular part ad the radial part with the solutios ar () = ul () r Laguerre equatio r b( ϑ, ϕ) = Y lm ( ϑ, ϕ) Geeralized Legedre equatio
11 Agular part Factorizatio gives for the agular part of the Laplace operator: ϑ ϑ ϕ siϑ siϑ + + αsi ϑ b( ϑ, ϕ) = 0 With b( ϑ, ϕ) = S( ϑ) T( ϕ) we get two differetial equatios d T T m T ( ϕ) = β ( ϕ) β =, ( ϕ) = e im dϕ ϕ d d S m S dϑ dϑ Substitute z=cos θ to get the geeralized Legedre equatio with the solutio si ϑ si ϑ ( ϑ ) + ( α si ϑ ) ( ϑ ) = 0 α = ll ( + ), S( ϑ) = P m ( z) l m im b( ϑϕ, ) = P ( z) e ϕ Y ( ϑ, ϕ) l lm Spherical harmoics Legedre s equatio Adrie-Marie Legedre (75-833) ( x ) y''( x) xy'( x) + l( l+ ) y( x) = 0 Eigesolutios regular i [-,] are the Legedre polyomials P l (x) P = 0 P ( x) = x P ( x) = (3x )... Legedre fuctios of st kid dx P ( x) P ( x) = δ l + l m lm Geeralized Legedre D.E. geeralized Legedre fuctios: d P x x P x dx m m m m/ l ( ) ( ) ( ) l( ) m Orthogoality relatio Laplace, d Alemebert; trajectories of projectiles, Berli, elliptic fuctios, umber theory, geometry, least squares method (died i poverty!)
12 Legedre s equatio (cot d) Geeralized Legedre D.E. [ ] ( x ) y''( x) m+ ) xy'( x) + l( l+ ) m( m+ ) y( x) = 0 geeralized Legedre fuctios: d P x x P x dx m m m m/ l ( ) ( ) ( ) l( ) m Radial part ur () ar () = r d m e l( l+ ) m + + () 0 E u r = dr r r effective potetial Leads to the geeralized Laguerre equatio ad the eigesolutio l+ r/ l+ r ur () = r e L l ( ) deote the geeralized Laguerre polyomials (highest order x -p ) r is measured here i uits of the Bohr Radius : 0.5A o me e
13 Laguerre s differetial equatio d x d x xe yx ( ) + e yx ( ) = 0 dx dx Edmod Nicolas Laguerre ( ) xy( x)'' + ( x) y( x)' + y( x) = 0 Eigevalues ( iteger) Eigevectors Laguerre polyomials L (x) L 0=, L =-x, L=-x+ x,... Orthogoality relatio: 0 x dx e L ( x) L ( x) = δ m m (EP46); Artillery officer; Approximatio theory, special fuctios Laguerre s differetial equatio (cot d) Geeralized Laguerre equatio : x yx ( )'' + ( p+ xyx ) ( )' + ( p) yx ( ) = 0 has as solutios the geeralized Laguerre polyomials: p p p d L p( x) = ( ) L( x) p (highest order x dx -p )
14 Hydroge quatum umbers: Eergy levels ψ lm (, r ϑϕ, ) rl ( ) e Y ( ϑ, ϕ) l l+ r r/ m l l =,,3,... l = 0,,... m= l, l+,..0,.. l, ( spi =± ) E me = 4 =3 = = l=0 l= l= V(r) # = (+3+5)=6 # = (+3)=8 E= ev # = Sample wave fuctio (30) for hydroge ψ lm (, r ϑϕ, ) rl ( ) e Y ( ϑ, ϕ) l l+ r r/ m l l ψ 30m (, r ϑϕ, ) rl( ) e Y( ϑϕ, ) 5 r r / ( lm,, ) = (3,,0) =-l-=0 radial odes l=: P =(3 cos θ-)/ m=0: o ϕ depedece From: B.Thaller, Visual Quatum Mechaics I/II (Telos) ad
15 Sample wave fuctio (433) for hydroge ( lm,, ) = (4,3,3) Period π/3 i ϕ! From: B.Thaller, Visual Quatum Mechaics I/II (Telos) ad Sample wave fuctio (933) for hydroge ( lm,, ) = (9,3,3) Period π/3 i ϕ! From: B.Thaller, Visual Quatum Mechaics I/II (Telos) ad
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