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4E : The Quantum Universe Lecture 26, May 17 Vivek Sharma - - PowerPoint PPT Presentation
4E : The Quantum Universe Lecture 26, May 17 Vivek Sharma - - PowerPoint PPT Presentation
4E : The Quantum Universe Lecture 26, May 17 Vivek Sharma modphys@hepmail.ucsd.edu Radial Probability Distribution P(r)= r 2 R(r) Because P(r)=r 2 R(r); No matter what R(r) is for some n, The prob. Of finding electron inside the nucleus = 0 !!
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Normalized Spherical Harmonics & Structure in H Atom
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Excited States (n>1) of Hydrogen Atom : Birth of Chemistry !
211 210 21- 200 n 1 211 21 1 1
Features of Wavefunction in & : Consider Spherically Symmetric (last slide) Excited , , States (3 & each with same E ) : are all states 1 Z =R Y 2, 2 = p a n l ψ ψ ψ θ φ ψ ψ π = = ⇒ = ⎛ ⎛ ⎞ ⎜ ⎜ ⎟ ⎝ ⎠ ⎝
21 1
2 * 2 211 211 210 21 1 l 1 3/ 2
sin | | | | sin Max at = ,min at =0; Symm in 2 W (r) hat about (n=2, =1, Y ( , ); 1 3 Y ( , ) cos ; 2 Function is max at =0, min a m 2 . 8 t ) = . i
Zr a
Z e R r e a
φ
θ π ψ ψ ψ θ θ ψ θ φ θ φ θ π θ θ π θ φ
−
= ∝ = ⎞ ⎛ ⎞ ⎛ ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ = ⎝ ∝ ⎠ ⎠
- z
We call this 2p state because of its extent in z
2pz
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Excited States (n>1) of Hydrogen Atom : Birth of Chemistry !
2 1 2 2
Remember Principle of Linear Superposition for the TISE which is basically a simple differential equat Principle of Linear Superposition If are sol. ion:
- f TISE
then a "des
- 2m
igne a U nd E ψ ψ ψ ψ ψ ∇ + ⇒ =
- '
1 2 2 2 ' ' ' '
To check this, just substitute in pla r" wavefunction made of linear sum i ce of & convince yourself that s also a
- sol. of the diff. equ
- ation !
2m The a b U E ψ ψ ψ ψ ψ ψ ψ ψ = + ∇ + =
- diversity in Chemistry and Biology DEPENDS
- n this superposition rule
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Designer Wave Functions: Solutions of S. Eq !
[ ] [ ]
x y
2p 211 21 1 2p 211 21 1
Linear Superposition Principle means allows me to "cook up" wavefunctions 1 ......has electron "cloud" oriented along x axis 2 1 ......has electron "cloud" oriented along 2 ψ ψ ψ ψ ψ ψ
− −
= + = −
200 210 211 21 1
2 ,2 ,2 ,2 Similarly for n=3 states ...and so on ...can get very complicated structure in & .......whic y axis So from 4 solutio h I can then mix & match ns to make electron , , , s "
x y z
s p p p θ φ ψ ψ ψ ψ
− →
most likely" to be where I want them to be !
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Designer Wave Functions: Solutions of S. Eq !
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Cross Sectional View of Hydrogen Atom prob. densities in r,θ,φ Birth of Chemistry (Can make Fancy Bonds Overlapping electron “clouds”)
Z Y
What’s the electron “cloud” : Its the Probability Density in r, θ,φ space space!
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More Radial Probabilities P(r) Vs. r/a0
Net Prob. densities for n=2 states
spherically symmetric dumbbell Doughnut (toroid)
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Radiative Transitions: An Impromptu Discussion
- Concepts
– Write and use the wavefunctions which arise as solutions of the time-dependent Schrodinger equation – Use concept of linear superposition of time-dependent wavefunctions – Calculate expectation value for <x>, when is it time-dependent?
- See slides (posted) from Lecture 27 ( See also Modern