quantization and special functions - unigrazphysik.uni-graz.at/~cbl/teaching/specfunc/otocec.pdf ·...

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Quantization and Special Functions C. B. Lang, © 2003

Quantization and Special Functions

Christian B. Lang

Inst. f. theoret. PhysikUniversität Graz

Otocec 6.-9.10.2003


Contents

• 1st lecture

– Schrödinger equation– Eigenvalue equations– 1-dimensional problems– Limitations of Quantum

Mechanics

• 2nd lecture

– More dimensions– Separation of variables– Hydrogen atom

2


Schrödinger’s EquationErwin Schrödinger

(1887-1961)

( , ) ( , )i x t H x ttψ ψ∂

=∂

p ix

∂=

∂with this gives

2 2

2( , ) ( ) ( , )2

i x t V x x tt m xψ ψ

∂ ∂= − + ∂ ∂

2

( , ) ( ) ( , )2pi x t V x x t

t mψ ψ

∂= + ∂

“time dependent Schrödinger equation”


Stationary Schrödinger equation

2 2

2

1 ( ) ( )( ) 2

( )( )

V x xx m x

i tt t

ϕ ψϕ ψ

∂= − + ∂

∂∂

Assuming factorization ( , ) ( ) ( )x t t xψ ϕ ψ= one gets

E=

( ) ( ) ( ) exp( )ii t E t t E ttϕ ϕ ϕ∂

= → ∝ −∂

2 2

2 ( ) ( ( )) ( )2

d x E V x xm dx

ψ ψ= − −2 2

2 ( ) ( ) ( )2

V x x E xm x

ψ ψ ∂

− + = → ∂

“Stationary Schrödinger equation”Diff. equation + boundary conditions

3


Boundary conditions

| |lim ( )xE V x→∞<

| |lim ( )xE V x→∞>

Density ρ(x)=|ψ|2 is localized →Bound states, discrete energy spectrum,Sturm-Liouville-problem

No localization→Scattering solutions, continuous energy spectrum


Self-adjoint operators

• Eigenvalue problem for matrices:– Hermitian matrices:

• real eigenvalues λi

• eigenvectors build an orthogonal system

• Eigenvalue problem for operators:– Self-adjoint operators:

• real eigenvalues λi

• eigenvectors = eigenfunctions form an orthogonal basis

– Differential operators (differential equations) with boundary conditions → Sturm-Liouville problem

i i iAϕ λ ϕ=† *( ) . .T

ij jiA A A i e a a= ≡ =

† †where ( , ) ( , )A A A f g f Ag= ≡

(Cf. Heisenberg)

(Cf. Schrödinger)

4


Sturm and Liouville

Joseph Liouville (1809-1882)

Jaques Charles Francois Sturm(1803-1855)

Cauchy, Mathieu, Dirichlet; founder of Journal de Math. Pure and Appliquées; Constituting Assembly (1848);astronomy, math. physics, pure math.

Colladon, Arago, Ampere, Fourier; algebraic equation; projective geometry, differential geometry

Sturm-Liouville Problem: Boundary Values for Diffêrential Equations


[D=1] Bound states (1): a mole in a hole

( 0) 0( ) 0xx a

ψψ

≤ =≥ =

2'' 0kψ ψ+ =

Oscillator equation (Helmholtz eq.): . .( ) exp( )/ ( 0,1, 2,...)

( ) sin( / )

b cx ikxk n a n

x c n x a

ψπ

ψ π

= ± →= =

=

Boundary conditions:

2 2

02 ( ) ( ) ( )2

d x E V xm dx

ψ ψ= − −

2 202 ( ) /m E V k− =

0 a

n=1

n=2

n=3

from norm2ac =

2 2( ) | ( ) | 1x dx xψ ψ= =∫

=Quantization!

Eigensolutions and Eigenvalues:

20nE V n− ∝

5


[D=1] Bound states (2): harmonic oscillator

Boundary conditions: L2(R,exp(-x2))

2 2

2

22

( ) ( ( )) ( )2

( )2

d x E V x xm dx

mV x x

ψ ψ

ω

= − −

=

2, ( )exp( / 2)

2( ) '' 2 ( ) ' 1 ( ) 0

mz x u z z

Eu z zu z u z

ω ψ

ω

= = − →

− + − =

Hermite‘s D.E.:eigensolutions Hn(z), n=0,1,2.., eigenvalues:

12

2 1 2 ( )nn

E n E nωω

− = → = +

n=0

n=1

V(x)

=Quantization!


Harmonic oscillator probability density

n=0 and Hn=1

-4 -2 2 4

0.1

0.2

0.30.4

0.5

n=1 and Hn=2z

-4 -2 2 4

0.1

0.2

0.3

0.4

n=2 and Hn=−2+4z2

-4 -2 2 4

0.050.10.150.20.250.30.35

n=3 and Hn=−12z+8z3

-4 -2 2 4

0.050.10.150.20.250.30.35

2 / 212 !

( ) ( )nz

n nnz e H z

πψ −=

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Hermite’s differential equation

2 2

( ) 2 ( ) 0

( ) '' 2 ( ) ' 2 ( ) 0

x xd de y x ne y xdx dx

y x x y x n y x

− − + = →

− + =

Eigenvalues 2n (n integer)Eigenvectors Hermite polynomials Hn(x)

Orthogonality relation:

2

( ) ( ) 2 !x nn m nmdx e H x H x nδ π

∞−

−∞

=∫

20 1 2H =1, H =2x, H =4x -2, ....

Charles Hermite(1822-1901)

(EP68); Bertrand, Jacobi, Liouville, Cauchy;number theory, quadratic forms, matrices, 5th order algebraic eq.


Problems and limitations of QM

• Reality problem– Role of observer?– Wave function of universe– Reality vs. probability (cf. crucial experiments)

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• Relativity– Relativistic dispersion relation

• Particle creation and annihilation– Hilbert space is normed– Decay?

Problems and limitations (cont’d)

⇒ Relativistic quantum field theory …works, but has its own problems…

2 42 2 2 4 2

3 32p pE p c m c mc Om m c

= ± + ≈ ± + +

?????????


Contents

• 1st lecture

– Schrödinger equation– Eigenvalue equations– 1-dimensional problems– Limitations of Quantum

Mechanics

• 2nd lecture

– More dimensions– Separation of variables– Hydrogen atom

8


Laplace’s differential operator

2

2

2 2

2 2

2 2 2

2 2 2

1:

2 :

3:

dDdxd dDdx dyd d dDdx dy dz

= ∆ ≡

= ∆ ≡ +

= ∆ ≡ + +

Pierre-Simon Laplace(1749-1827)

D‘Alembert, Lagrange;Differential calculus, math. physics/astronomy, theory of probability, stability of solar system („méchanique céleste“)

In cartesian ccordinates:

0ψ∆ =

Laplace equation

…depends on coordinate system…


• Dirichlet: values of ψ on the boundary of thedomain A

• Neumann: values of the derivative of ψ on theboundary:

Laplace equation: boundary conditions

…is a partial differential equation of „elliptic type“……is a boundary value problem:

( , , ) 0x y zψ∆ =

( A)=...ψ ∂

A.n| =...ψ ∂∂

A

A∂

A

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Laplace operator in D=3, spherical coordinates

e.g. spherical coordinates

cos sinsin sincos

x ry rz r

ϕ ϑϕ ϑϑ

===

0, 0 , 0 2r π πϑ ϕ≥ ≤ ≤ ≤ <

22

2 2 2 2 2

1 1 1( , , ) sin ( , , )sin sin

r r rr r r r r

ψ ϑ ϕ ϑ ψ ϑ ϕϑ ϑ ϑ ϑ ϕ

∂ ∂ ∂ ∂ ∂ ∆ ≡ + + ∂ ∂ ∂ ∂ ∂

…optimal for problem with spherically shaped boundaries…


[3D] Jack-in-a-box

( )2 2 2

2 2 22 2 2 ( , , ) ( , , )x y z

d d d x y z k k k x y zdx dy dz

ψ ψ

+ + = − + +

Ansatz for separation of variables depends on geometry of the boundary -i.e. one should choose optimal variables

( , , ) ( ) ( ) ( )x y z X x Y y Z zψ = 22

2

22

2

22

2

( ) ( )

( ) ( )

( ) ( )

x

y

z

d X x k X xdxd Y y k Y ydyd Z z k Z zdz

= −

= −

= −

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Separation of variables

Eigenvalue equations(SL-systems)

General solution:

n, ,

( , , ) ( ) ( ) ( )l m l n ml n m

x y z c X x Y y Z zψ = ∑

Eigenvalues kx=l, ky=n, kz=m; expansion coefficients determined from boundary values.

22

2

22

2

22

2

( ) ( )

( ) ( )

( ) ( )

x

y

z

d X x k X xdxd Y y k Y ydyd Z z k Z zdz

= −

= −

= −

22

2E k

m=with


Hydrogen atom

2 2

( ) ( )| |e eV r V rr r

= − ≡ − =

is radially symmetric („central potential“)e

p

Separation of variables:

2

( ) ( , , ) 02 r V r E r

m ϑϕ ψ ϑ ϕ

∆ − + =

( , , ) ( ) ( , )r a r bψ ϑ ϕ ϑ ϕ=

Leads to separate equations for the angular part and the radial part withthe solutions

1( ) ( )la r u rr

=

( , ) ( , )lmb Yϑ ϕ ϑ ϕ=

Laguerre equation

Generalized Legendre equation

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Angular partFactorization gives for the angular part of the Laplace operator:

22

2sin sin sin ( , ) 0bϑ ϑ α ϑ ϑ ϕϑ ϑ ϕ

∂ ∂ ∂+ + = ∂ ∂ ∂

With ( , ) ( ) ( )b S Tϑ ϕ ϑ ϕ= we get two differential equations

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2 ( ) ( ) , ( ) eimd T T m Td

ϕϕ β ϕ β ϕϕ

= − → = =

2 2sin sin ( ) ( sin ) ( ) 0d d S m Sd d

ϑ ϑ ϑ α ϑ ϑϑ ϑ

+ − =

Substitute z=cos θ to get the generalized Legendre equation with the solution

( 1), ( ) ( )ml l S P zlα ϑ= + =

( , ) ( ) ( , )m iml lmb P z e Yϕϑ ϕ ϑ ϕ= ∝

„Sphericalharmonics“


Legendre’s equation Adrien-Marie Legendre(1752-1833)

Laplace, d‘Alemebert;trajectories of projectiles, Berlin, elliptic functions, number theory, geometry, least squares method(died in poverty!)

2(1 ) ''( ) 2 '( ) ( 1) ( ) 0x y x xy x l l y x− − + + =

Eigensolutions regular in [-1,1] are theLegendre polynomials Pl(x)

0

121

2 2

1( )

( ) (3 1)...

PP x x

P x x

==

= −

„Legendre functions of 1st kind“

2 / 2( ) ( 1) (1 ) ( )m

m m ml lm

dP x x P xdx

≡ − −

„Generalized Legendre D.E.→generalized Legendre functions:

112

1

1( ) ( )2 1l m lmdx P x P xl

δ−

=+∫ Orthogonality

relation

12


Legendre’s equation (cont’d)

[ ]2(1 ) ''( ) 2 1) '( ) ( 1) ( 1) ( ) 0x y x m xy x l l m m y x− − + + + − + =

2 / 2( ) ( 1) (1 ) ( )m

m m ml lm

dP x x P xdx

≡ − −

→ generalized Legendre functions:

„Generalized Legendre D.E.“


Radial part

2 2

2 2 2 2

2 ( 1) 2 ( ) 0d m e l l m E u rdr r r

++ − + =

„effective potential“

1 / 2 1 21( ) ( )l r n l r

n l nu r r e L+ − +− −=

( )( ) u ra rr

=

Leads to the „generalized Laguerre equation“ and the eigensolution

r is measured here in units of the „Bohr Radius“:

2

2 0.5Ao

em e≈

denote the generalizedLaguerre polynomials(highest order xn-p)

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Laguerre’s differential equation

( ) ( ) 0

( ) '' (1 ) ( ) ' ( ) 0

x xd dxe y x ne y xdx dx

x y x x y x n y x

− − + = →

+ − + =

Eigenvalues 2n (n integer)Eigenvectors Laguerre polynomials Ln(x)

Orthogonality relation:

0

( ) ( )xn m nmdx e L x L x δ

∞− =∫

210 1 2 2L =1, L =1-x, =1-2x+ x , ....L

Edmond Nicolas Laguerre(1834-1886)

(EP46); Artillery officer;Approximation theory, specialfunctions


Laguerre’s differential equation (cont’d)

( ) '' ( 1 ) ( ) ' ( ) ( ) 0x y x p x y x n p y x+ + − + − =

“Generalized Laguerre equation”:

( ) ( 1) ( )p

p pn p np

dL x L xdx− = − (highest order xn-p)

has as solutions the generalized Laguerre polynomials:

14


Hydrogen quantum numbers: Energy levels

2 1 /21

12

( , , )

( ) ( , )

1,2,3,...0,1,... 1

, 1,..0,.. 1,1( )

nlm

l l r n mrn l ln

r

r L e Y

nl nm l l lspin

ψ ϑ ϕ

ϑ ϕ+ −− −

∝

== −= − − + −

= ±

n=1

n=2

n=3

l=0l=1

l=2

E= - 13.6 eV

# =2 (1+3+5)=16

# =2 (1+3)=8

# =2

4

2 21

2nmeE

n= −

V(r)


Sample wave function (320) for hydrogen

From: B.Thaller, Visual Quantum Mechanics I/II (Telos)and http://www.uni-graz.at/imawww/vqm/

2 1 /21

3202 5 / 3 02

0 23

( , , )

( ) ( , )

( , , )

( ) ( , )

nlm

l l r n mrn l ln

mrr

r

r L e Y

r

r L e Y

ψ ϑ ϕ

ϑ ϕ

ψ ϑ ϕ

ϑ ϕ

+ −− −

−

∝

∝

( , , ) (3,2,0)n l m =

n=-l-1=0 radial nodesl=2: P2=(3 cos2θ-1)/2m=0: no ϕ dependence

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( , , ) (4,3,3)n l m =

Period 2π/3 in ϕ !




( , , ) (9,3,3)n l m =

Period 2π/3 in ϕ !


quantization and special functions - unigrazphysik.uni-graz.at/~cbl/teaching/specfunc/otocec.pdf ·...

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