handout lecture18

7/29/2019 Handout Lecture18

1/22

Lecture 18: The Hydrogen Atom Reading: Zuhdahl 12.7-12.9

Outline The wavefunction for the H atom Quantum numbers and nomenclature

Orbital shapes and energies


2/22

H-atom wavefunctions

Recall from the previous lecture that theHamiltonian is composite of kinetic (KE) andpotential (PE) energy.

The hydrogen atom potential energy is given by:

e-

P+r

r0

V(r) =-e

2

r


3/22

H-atom wavefunctions (cont.) The Coulombic potential can be generalized:

e-

P+r

V(r) =-Ze

2

r Z

Z = atomic number (= 1 for hydrogen)


4/22

H-atom wavefunctions (cont.) The radial dependence of the potential suggests

that we should from Cartesian coordinates to sphericalpolar coordinates.

p+

e-

r = interparticle distance(0 r )

q = angle from xy plane

(p/2 q - p/2)

f = rotation in xy plane(0 f 2p)


5/22

H-atom wavefunctions (cont.) If we solve the Schrodinger equation using this

potential, we find that the energy levels arequantized:

En= -

Z2

n2

me4

8e0

2h

2

= -2.178x10

-18J

Z2

n2

n is the principle quantum number, and ranges

from 1 to infinity.


6/22

H-atom wavefunctions (cont.) In solving the Schrodinger Equation, two other

quantum numbers become evident:

l, the orbital angular momentum quantum number.Ranges in value from 0 to (n-1).

ml, the z component of orbital angular momentum.

Ranges in value from -l to 0 to l.


7/22

H-atom wavefunctions (cont.) In solving the Schrodinger Equation, two other

quantum numbers become evident:

l, the orbital angular momentum quantum number.Ranges in value from 0 to (n-1).

m, the z component of orbital angular momentum.

Ranges in value from -l to 0 to l.

We can then characterize the wavefunctions based onthe quantum numbers (n, l, m).


8/22

Orbital Shapes Lets take a look at the lowest energy orbital, the

1s orbital (n = 1, l = 0, m = 0)

y1s =1

p

Z

ao

32

e

-Z

a0

r

=1

p

Z

ao

32

e-s

a0 is referred to as the Bohr radius, and = 0.529

En= -2.178x10-18J

Z2

n2

= -2.178x10

-18J

1

1


9/22

Orbital Shapes (cont.) Note that the 1s wavefunction has no angular

dependence (i.e., Q and F do not appear).

y1s = 1

pZa

o

32

e

-Z

a0

r

= 1p

Za

o

32

e-s

y*yProbability =

Probability is spherical


10/22

Orbital Shapes (cont.) Naming orbitals is done as follows

n is simply referred to by the quantum number l (0 to (n-1)) is given a letter value as follows:

0 = s 1 = p 2 = d 3 = f

- ml (-l0l) is usually dropped


11/22

Orbital Shapes (cont.)

Table 12.3: Quantum Numbers and Orbitals

n l Orbital ml # of Orb.

1 0 1s 0 12 0 2s 0 1

1 2p -1, 0, 1 33 0 3s 0 1

1 3p -1, 0, 1 32 3d -2, -1, 0, 1, 2 5


12/22


Example: Write down the orbitals associated with n = 4.

Ans: n = 4l = 0 to (n-1)= 0, 1, 2, and 3= 4s, 4p, 4d, and 4f

4s (1 ml sublevel)4p (3 ml sublevels)4d (5 m

l

sublevels4f (7 ml sublevels)


13/22

Orbital Shapes (cont.)s (l = 0) orbitals

r dependence only

as n increases, orbitalsdemonstrate n-1 nodes.


14/22

Orbital Shapes (cont.)2p (l = 1) orbitals

not spherical, but lobed.

labeled with respect to orientation along x, y, and z.

y2pz

=1

4 2p

Z

ao

32

se-s

2 cosq


15/22

Orbital Shapes (cont.)3p orbitals

more nodes as compared to 2p (expected.).

still can be represented by a dumbbell contour.

y3pz

=2

81 p

Z

ao

32

6s-s2( )e-s

3 cosq


16/22


3d (l = 2) orbitals

labeled as dxz, dyz, dxy, dx2-y2 and dz2.


17/22


3d (l = 2) orbitals

dxy dx2-y2


18/22


3d (l = 2) orbitals

dz2


19/22


4f (l = 3) orbitals

exceedingly complex probability distributions.


20/22

Orbital Energies

energy increases as 1/n2

orbitals of same n, but differentl are considered to be of equalenergy (degenerage).

the ground or lowest energyorbital is the 1s.


21/22

Spin Further experiments

demonstrated the need

for one more quantumnumber.

Specifically, someparticles (electrons inparticular)demonstrated inherentangular momentum.


22/22

Spin (cont.) The new quantum

number is ms

(analagous to ml).

For the electron, mshas two values:

+1/2 and -1/2

ms = 1/2

ms = -1/2

handout lecture18

Documents