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5. The Quantum Mechanical Model of the Atom
W . Heisenberg 

E. Schrodinger  Wave mechanics
 or Quantum mechanics
L. DeBroglie 
By analogy with a guitar string,
fastened at both ends:
Particles take on wave properties:
view the electron as a standing wave!
• If we assume a circular trajectory, we should have:
Circumference  a whole number of wavelengths
Cancellation by destructive
interference
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2 r  n 
h
But  
mv

h

n
  m vr  n
2
( De Broglie ) 

Bohr assumption!
 Schrödinger (1925): Wave equation to calculate the total
energy of the atom.
Idea: particle is distributed in space just like
the amplitude of a wave. A high intensity
is equivalent to a high probability of finding
the particle!
ˆ Ψ  EΨ
H
Erwin Schrödinger
Schrödinger wave equation (SWE)
Nobel 1933
Operators
• In quantum mechanics, every measurable quantity
has its own operator.
• All the operators operate on the wave function .
 position 𝑥ො
෡=a
𝑨
3D position 𝑟Ԧመ
scalar
operator
Velocity 𝑣෠Ԧ
Wave
Linear momentum 𝑝෠Ԧ
function
෠
Angular momentum 𝐿
ˆ Ψ  EΨ
H
෠
Kinetic energy 𝑇
Schrödinger wave equation
(SWE)
෠
Potential energy 𝑉
෡ = 𝑇෠ + 𝑉෠ Hamiltonian operator
Total energy 𝐻
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Ψ 

Ĥ 

E 
wave function
  (x, y, z)
Hamiltonian operator
Total energy
 Tˆ  Vˆ
 K .E  P.E
of the moving electron
=
of attraction between
e and the nucleus
 SWE can be solved exactly for the H-atom
 Many solutions for  are found. Each solution
represents an atomic orbital.
 Atomic Orbital
Definition: an atomic orbital is a region in space where there is a
finite probability of finding an electron. It can be obtained from
the values of the wave function at different points in space.
 Not a Bohr orbit.
 Does not give the detailed pathway of the electron (exact trajectory
is not known). Only a shape of the electron density region can
be known.
 The lowest energy orbital (or ground state orbital) of the H
atom  1s orbital.
 Heisenberg (1927):
There is a limitation to just how precisely we can know both the
position and the momentum of a given particle.
Heisenberg’s UNCERTAINTY PRINCIPLE.
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It is impossible to know simultaneously both the position
and the momentum of a given elementary particle (such as
an electron).
Mathematically,



x  (mv) 
h
4
x  uncertainty in position
(mv)  uncertainty in momentum
Notes:
Werner Heisenberg
Nobel 1932
 No well-defined orbit for the e as in the Bohr model.
 Limitation (uncertainty gap) becomes small for large
particles.
 Physical significance of the wave function
(Born interpretation)
2  probability of finding an electron near a particular
point in space.
2 (x, y, z) d  probability in a volume element d
2 (x, y, z)  probability density or distribution. (at a
given point).
 2 ( x1 , y1 , z1 )
N1


 2 ( x2 , y 2 , z 2 ) N 2
ratio of the probability densities
Vagueness consistent with the concept of the Heisenberg
Uncertainty principle.
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 Representation of a probability distribution or atomic
orbital
1s orbital of the H atom:
0.06
0.87
 (r, , )
 (x, y, z)
0.32
0.15
etc…

1s  C ea r


Figures from Atkins, Physical Chemistry, 6th Ed.
 (r, , )  R (r) Y (, )
Angular
part
Radial
part
Every solution of SWE (), represents a different region in space
or ATOMIC ORBITAL
Orbital
Wave function
3/ 2
1s
1 Z
 
 1s
  a0 
2s
 2 s
1
4 2
2pz
 2p 
1
4 2
Z
 
 a0 
2px
 2p 
1
4 2
Z
 
 a0 
2py
 2p 
1
4 2
Z
 
 a0 
.
z
x
y
Z
 
 a0 
e  Zr / a0
3/ 2

Z r   Zr / 2 a0
 2 
e
a0 

3/ 2
3/ 2
 Z r   Zr / 2 a0

 e
cos 
 a0 
 Z r   Zr / 2 a0

 e
sin  cos 
 a0 
3/ 2
 Z r   Zr / 2 a0

 e
sin  sin 
 a0 
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