Reduced Form of TISE for H-Atom: Separation of Variables Movement - - PowerPoint PPT Presentation

Reduced Form of TISE for H-Atom: Separation of Variables

Reduced Mass:

e e N N e N CM e e N N e N CM e e N N e N CM e N

m x m x x x x x M m y m y y y y y M m z m z z z z z M m m M µ + = − = + = − = + = − = =

2 2 2

2 2

2 2

in terms of CM and electronic coordinates ( , ) ( ) ( )

, , , , , , , ,

N N N

Total e e e N N N e e e e N N N N

CM e e e e

E M Ze r

Separate H

and E

x y z x y z x y z x y z

µ = − =

Ψ = Ψ

• Ψ

   ÷  

∇ Ψ Ψ ∇ Ψ Ψ

− −

h h

Free Particle: movement of The whole atom: You solved it! Relative motion of the electron and With respect to the Nucleus

2 2 2 2 2 2 2 2 2 2 2

( , , ) ( , , ) ( , , ) ( , , ) ( , , ) 2

e e e e i i i

e e

Ze x y z x y z x y z x y z x y z x y z x y z

E

µ

=

  ∂ ∂ ∂ − Ψ + Ψ + Ψ − Ψ  ÷ ∂ ∂ ∂ + +  

Ψ

h

Problem: 2nd order PDE with 3 variables - can not be separated!

Relative motion of electron wrt nucleus:

: . Movement of electronmuch faster than heavy nucleus Separate translational motion relative frame ⇒

Spherical Polar Coordinates

Used for spherically symmetric systems Conversion from Cartesian coordinates

Hamiltonian: Spherical Polar Coordinates

Solve this PDE need to separate variables r, θ, φ: POSSIBLE

Looks can be deceiving! Looks can be deceiving!

TISE for H-Atom in spherical-polar coordinates

µ µ µ µ

s ( ) ( ) ( ) : ( , , ) ( ) ( ) ( ) Special olution if H H r H H r R r θ φ θ φ θ φ = + + Ψ = Θ Φ

[ ] [ ] [ ]

2 2 2 2 2 2 2

1 ( ) ( ). ( ) sin ( ) ( ). ( ) sin 1 1 ( ) ( ). ( ) sin 2 d d d d r R r R r dr dr d d r d R r d Ze E R r θ φ θ θ φ θ θ θ φ θ θ φ µ       Θ Φ + Θ Φ  ÷  ÷           + Φ Θ       + +  ÷   h ( ) ( ) ( ) 0 r θ φ Θ Φ

=

2 2 2 2 2 2 2

1 1 1 ( , , ) sin ( , , ) ( , , ) sin sin 2 ( , , ) 0 r r r r r r r Ze E r r θ φ θ θ φ θ φ θ θ θ θ φ µ θ φ   ∂ ∂ ∂ ∂ ∂     Ψ + Ψ + Ψ  ÷  ÷   ∂ ∂ ∂ ∂ ∂         + + Ψ  ÷  

=

h

2nd order Partial Differential Equation with three variables

2 2 2 2 2 2 2 2

sin sin 2 1 sin sin d dR d d Ze d r E r R dr dr d d r d θ θ µ θ θ θ θ φ   Θ Φ     + + + = −  ÷  ÷  ÷ Θ Φ       h

2

( , ) ( ) ( , ) ( ) . ( ) LHS F r G RHS F r G Const m say θ φ θ φ = = = ⇒ = = =

2 2 2 2 2 2 2

1 2 sin sin sin d d R d d R d Ze r E r dr dr d d d r

R R

µ θ θ θ θ θ φ         Φ Θ ΘΦ + + + + =    ÷  ÷  ÷        

Θ Φ ΘΦ

h 2 2 2

1 m φ ∂ Φ ∴ + = Φ ∂

2 2

sin : . . r Multiply by and rearrange R θ Θ Φ

Separation of Variables r, θ, φ

( , , ) ( ). ( ). ( )

. .

r R r

R

θ φ θ φ Ψ = Θ Φ =

ΘΦ

Solve 2 nd order DE to

• btain functional form of

Φ(φ)

: ( ) Solution

im

Ae

φ

φ

±

Φ =

Solving φ-part is relatively simple!

Another Quantum Number “popped out” out of Boundary Conditions: Quantization of Angular Momentum! Magnetic Quatnum Number: Can take any integral value (including zero); Restricted by another quantum number (m<l). Loosely relates to direction

• f orbital angular momentum; splitting of energy levels in magnetic field.

Boundary Condition

2 2 2

1 ( ) ( ) m φ φ φ ∂ ∴ Φ + = Φ ∂

2 2 2

& im m φ φ ∂Φ = ± Φ ∂ ∂ Φ = − Φ ∂ Q

Solving R(r) and Θ(θ) part not so simple, but can be done (in ~5-6 lectures!)

2 2 2 2 2 2 2

2

sin sin 2 sin sin d sin : d dR d r Ze r E R dr dr d r Simply ivide by and rearrange

m

θ θ µ θ θ θ θ θ   ∂ Θ     + + + =  ÷  ÷  ÷ Θ ∂      

−

h

Need mathematical skills to solve the Differential Equations for R(r) and Θ(θ) - it will take several lectures to do so….

2 2 2 2 2 2

1 2 1 sin ( .) sin sin d dR r Ze m d d r E const R dr dr r d d µ θ β θ θ θ θ   Θ     + + = − =  ÷  ÷  ÷ Θ       h

Only r : Can be solved to obtain R(r) part Only θ: Can be solved to obtain Θ(θ) part Boundary conditions applied!

Θ(θ)Φ(φ) are Spherical Harmonics Yl

m(θ,φ)

l=Azimuthal Quantum Number

• r orbital quantum number l≤n-1

Easier to solve if written differently: Rigid-Rotor  already solved the angular (θ, φ) part: Related to Angular Momentum! L2 Square of angular momentum: Eigenfunctions “Spherical Harmonics”

·

µ

2 2 2 2

1 1 sin ( , ) ( , ) sin sin L θ θ φ θ φ θ θ θ θ φ   ∂ ∂ ∂ + Ψ = Ψ   ∂ ∂ ∂  

n= Principal Quantum Number=1,2,3,…

Radial Wavefunction depends on n and l:

Solve for R(r): Quantized Energies

Only n dependence of E: V term in H is needed for providing energies as

• eigenvalues. Angular parts does not have it!!!

Essentially same as Bohr’s Equation, with slight changes

Energies: En:

Particle in 3D-Box:Three Quantum Numbers

3 Quantum Numbers needed to Describe the system completely

Normalization Conditions (each dimension)

How to obtain normalized Ψn,l,m(r,θ,π)?

Loose Meaning of Quantum Numbers

L=0  s-orbital L=1  p-orbital L=2  d-orbital L=3  f-orbital

Radial Solutions depend on n and l (l=n-1)

Additional restrictions on l: n ≥ l+1 arise when DE is solved

2 1 2

( , ). . .

l l nl n l Z na

Zr na

r

R A n l r L

e

   ÷  ÷ +   +

−

  =  ÷  

f(r,n,l,Z) g(r,Z) Exponential Decay g(r)

Note: Rnl0 as rinfinity

Complete Wavefunction of H-Atom Lets not get intimidated: Please do not even try to remember

H-Atom Complete Ψ(r,θ,φ) for n=1,2

σr/a0

F(r) only Linear combination Of two solutions is Also a solution 1s 2s 2pz 2px,

y

F(r,θ) F(r,θ,φ) F(r) only