3313 Andrew Brandt 1

Physics 3313 - Lecture 16

4/1/2009

Wednesday April 1, 2009Dr. Andrew Brandt

1. Hydrogen Atom Wave Function2. Angular Momentum3. Orbital and Magnetic Quantum Numbers4. Angular Momentum Operator5. TEST moved to 4/27

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Hydrogen Atom Wave Function

3/30/2009

• http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydwf.html#c1

• different orbital angular momentum states identified by a letter in orbital notation

, ,nlm nl lm mr R r

l value 0 1 2 3 4

Orbital s p d f g

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Angular Momentum

• Radial equation (above) should only be concerned with radial motion (towards and away from nucleus), but energy could have an orbital term

• If then this term would cancel out

• since L=r x p =mvr, so

4/1/2009

22

2 2 20

11 20

4

l ld dR m er E R

r dr dr r r

radial orbitalE KE KE U

2

2

1

2orbital

l lKE

mr

22

2

1

2 2orbital

LKE mv

mr

( 1)L l l

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Orbital (l) and Magnetic (ml) Quantum Numbers

• l is related to orbital angular momentum; angular momentum is quantized and conserved, but since h is so small, often don’t notice quantization

• Electron orbiting nucleus is a small current loop and has a magnetic field, so an electron with angular momentum interacts with an external magnetic field

• The magnetic quantum number ml specifies the direction of L (which is a vector—right hand rule) and gives the component of L in the direction of the magnetic field Lz

• Five ml values for l=2 correspond to five different orientations of angular momentum vector.

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( 1)L l l

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Angular Momentum• L cannot be aligned parallel with an external magnetic

field (B) because Lz is always smaller than L (except when l=0)

• In the absence of an external field the choice of the z axis is arbitrary (measure projection as in any direction)

• Why only Lz quantized? What about Lx and Ly?

• Suppose L were in z direction, then electron would be confined to x-y plane; this implies z position is known and pz is infinitely uncertain, which is not true if part of a hydrogen atom

• Therefore average values of Lx =Ly =0 and it is only necessary to specify L and Lz4/1/2009

( 1) ll l m

lm

( 1)l l l

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Precession of Angular Momentum

• The direction of L is thus continually changing as it precesses around the z axis

(note average values of Lx =Ly =0 )

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Angular Momentum Operator

• Consider angular momentum definition: so

• We can define the angular momentum operator in cartesian and spherical coordinates:

• with

gives similarly

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L r p ������������� �

x z yL yp zp Y x zL zp xp z y xL xp yp

xp ix

ˆz

d dL i x y i

dy dx

ˆ

z nlmL i

nl lm mi R r

ˆ ( )( )l lzL i im m

yp iy

zp i

z

limAe

2 2ˆ ( 1)L l l

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QM Modifications to Bohr Model

• In Bohr model electron has circular orbit around nucleus with

and =90o and changes with time

• QM mods:

1) No definite r, , and , but only probabilities due to wave nature of electron

2) |2| independent of time and varies from place to place, so can’t think of electron as orbiting

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20nr n a

2 2 2 2| | | | | | | |R 2 * 2| | ( )l lim imAe Ae A

probability constant independent of azimuthal angle (spherical symmetry)

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Radial Wave Function

• Radial part of wave function varies with r and differs for different n,l combinations

• R is maximum at r=0 (inside nucleus) for s states but approaches 0 at r=0 for l>0

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Probabilities• Probability of finding electron in

hydrogen atom in a spherical shell between r and r+dr is given by

• with

• since angular wave functions are normalized

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222 22

0 0sinP r dr r R dr d d

22r R dr

2

,| | dV

2 2 2 2| | | | | | | |R

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Probability Distributions

4/1/2009