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**5**

Slide 38

In 3D: If the amplitude of the wavefunction of a particle at some point (x,y,z) is Y(x,y,z) then the probability of finding the particle between x and x+dx, y+dy and z+dz, ie, in a volume dV = dx dy dz is given by

Y(x,y,z) dV

dx

dy

dz

x

z

y

It follows therefore that

2

2

Y(x,y,z)

is a probability density.

In spherical coordinates

2

Y(r,q,f)

Slide 39

More important to know probability of finding electron at a given distance from nucleus!

dx

dy

dz

x

z

y

dV=dxdydz

Cartesian coordinates

not very useful to describe orbitals!

… in a shell of

dA = r2sinqdfdq

dA=dxdy

dV = r2sinqdfdqdr

Surface element

Volume element

Slide 40

Slide 41

For spherically symmetric orbitals:

the radial distribution function is defined as

2

P(r)= 4pr2 Y(r)

and P(r)dr is the probability of finding the electron in a shell of radius r and thickness dr

Slide 42

For spherically symmetrical orbitals

P(r)

Slide 43

Important

Born

interpretation

prob = Y2dt

Hence plot

P(r) = r2R(r)2

P(r)=4pr2Y2

(for spherical symmetry)

Pn(r) has

nodes

Slide 44

So what do we learn?

R(r)

R(r)2

P(r)

r

r

r

The 2p orbital is on average closer to the nucleus, but note the 2s orbital’s high probability of being

Slide 45

3d vs 4s

P(r)

Slide 46

The Ylm(,) angular part

of the solution form a set

of functions called the

n.b. These are generally

imaginary functions -we use

real linear combinations to

picture them.

Slide 47

Shapes arise from combination of radial R(r) and angular parts Ylm(,) of the wavefunction.

Usually represented as boundary surfaces which include of the probability density. The electron does not undergo planetary style circular orbits.

- Atomic Structure and Periodic Trends
- Why study atomic electronic structure?
- The Periodic Table
- The Hydrogen Atom
- Energy Levels?
- The Rydberg Formula
- Bohr Theory (old quantum)
- The problem with Bohr Theory
- Quantum mechanical Principles and the Solution of the Schrödinger Equation
- The Results of Quantum Mechanics
- Spherical Polar Coordinates
- The quantum numbers;
- The Radial Wavefunctions
- Revisit: The Born Interpretation
- Radial Wavefunctions and the Born Interpretation
- The Surface area of a sphere is hence:
- Construction of the radial distribution function
- Radial distribution function P(r)
- The Angular Wavefunction
- The Shapes of Wavefunctions (Orbitals)
- Electron densities representations
- The energies of orbitals
- The Ionization Energy
- Other Atoms
- Periodic Trends
- Space for extra Notes
- More on Ionization Energies
- More Periodic Trends
- Appendices

- Radiation Safety and Operations
- Motion
- Newton’s third law of motion
- Health Physics
- Practical Applications of Solar Energy
- Space Radiation
- Sensory and Motor Mechanisms

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