Lec19_on Hydrogen Atom

8/3/2019 Lec19_on Hydrogen Atom

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Lecture 19: The Hydrogen Atom

Reading: Zumdahl 12.7-12.9 Outline The wavefunction for the H atom

Know what wave functions look like for a particletrapped in a box; now we need to know what theylook like for an electron attracted to a nucleus; and

the energy of each wave function. Quantum numbers and nomenclature Orbital (i.e. wavefunction) shapes and energies

Problems (Chapter 12, 5th Ed.) 48, 49, 50, 52, 54, 55, 56, 57, 60


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H-atom wavefunctions Recap: The Hamiltonian is a sum of kinetic (KE,

or T) and potential (PE, or V) energy.

The hydrogen atom potential energy is given by:

e-

P+r

2 ( ) eV V r

r

= =

12

H T V E T V V

= += + =

r0

V (

P o

t e n

t i a l E

. )

The bar means average

over the position of theelectron.


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The Coulombic PE (V) can be generalized

e-

P+r

( )

( )2

2

2

2

( ) 4

4

where v2

o

A o

ZeV r r

F ee e N

pT p mm

=

= =

= = Z

Z = atomic number (= 1 for hydrogen) r is the distance between the electron and the nucleus Only one electron allowed (for now).


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H-atom Coordinates Frame The radial dependence of the potential suggests

that we should from Cartesian coordinates to spherical polar coordinates.

p+

e-

r = interparticle distance(0 r )

= angle from z toxy plane(0 )

= rotation in xy plane(0 2)

Major (azimuthal) angle

Minor angle


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H-atom Allowed EnergiesWhen we solve the Schrodinger equation using the Coulomb potential, we find that the bound-state energy levels arequantized or discrete:

2 4 2

182 2 2 2

02.178 108n Z me Z E x J n h n

= =

n (an integer counter) is the principal quantum number ,and ranges from 1 to infinity. n=1 is the lowest energy(level) or ground state for an electron bound to ahydrogen-like nucleus.This is the same formula Bohr gave us.Compare and contrast these energy levels with those of the particle in a box.


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Solve the Wave Equation for the Electron bound to th Nucleus

Set up the Schrdinger equation (SE) for the wave functionin terms of x,y and z coordinates, then rewrite in polar coordinates (because V depends only on r).

Solve the SE the same way Schrdinger did: Look theanswer up in a math book (Courant and Hilbert, in hiscase).

The solution gives a set of wave functions, and the energyof each wave function. The wave functions (and energies) are distinct and

countable (although in principle there are an infinitenumber of wavefunctions).

The wavefunctions are now called orbitals as they describethe probability of the electron in the vicinity of the nucleus.They are not orbits but regions of space wherein theelectron orbits, hence orbitals.


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Form of WaveFunctions (for Orbitals) Like the particle in a box the wave function

depends on the coordinate and a quantum number (like x and n).

There are three coordinates so the wave function isa product of a part that Depends only on r and has n (and l) with it Depends only on theta and has l (and m) with it Depends only on phi (and has m with it)

The total wave function has the form:( ) ( ) ) ( ), , , ,, ,n l m n l l m mr R r =

sin cossin sincos

x r

y r

z r

===

Relation between Cartesian and polar coordinates.


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Orbitals Orbitals are a description of where the

electron resides (like a house) Quantum numbers are like the address of

the house. The orbital does exist even without the

electron (so an empty orbital is called avirtual orbital).


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Energy levelsThe energy expression for the QM result is the same as

Bohrs, because the Virial Relation (which is also true for planets going around the sun) is also true for QuantumMechanics and embodies the balance between potentialand kinetic energy.

2 2

2 2

22

2 22 2 2

2

>0 02 4

2 This is the Virial Relation1 1 12 2 4

41 14 4 2 4 2 2

2

o

o Bohr

o o

p ZeKE T PE V m r

V T

V V E T V V

V T

Zer V m Ze m Ze

E pT rp nh

m

= = = =

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Lec19_on Hydrogen Atom