Lecture 16: Intro. to Quantum Mechanics

1
Lecture 16 Intro. to Quantum Mechanics

• Reading Zumdahl 12.5, 12.6
• Outline
• Basic concepts.
• A model system particle in a box.
• Other confining potentials.

2
Quantum Concepts

• The Bohr model was capable of describing the
  discrete or quantized emission spectrum of H.

• But the failure of the model for multielectron
  systems combined with other issues (the
  ultraviolet catastrophe, workfunctions of metals,
  etc.) suggested that a new description of atomic
  matter was needed.

3
Quantum Concepts

• This new description was known as wave mechanics
  or quantum mechanics.

• Recall, photons and electrons readily demonstrate
  wave-particle duality.

• The idea behind wave mechanics was that the
  existence of the electron in fixed energy levels
  could be though of as a standing wave.

4
Exercise

• What is the wavelength of an electron (mass 9.11
  x 10-31 kg) traveling at a speed of 1.0 x107 m/s?

l h / p h/mv
l 6.626x10-34 Js /(9.11x10-31kg)(1x107m/s)
l 6.626x10-34 Kgm2/s /(9.11x10-31kg)(1.x107m/s)
l 7.3x10-11 m
5
Exercise

• What is the wavelength of a baseball (mass 0.1
  kg) traveling at a speed of 35 m/s?

l h / p h/mv
l 6.626x10-34 Js /(0.1kg)(35m/s)
l 6.626x10-34 Kgm2/s /(0.1kg)(35m/s)
l 1.9x10-34 m
6
Uncertainty Principle

• Another limitation of the Bohr model was that it
  assumed we could know both the position and
  momentum of an electron exactly.

• Werner Heisenberg development of quantum
  mechanics leads him to the observation that there
  is a fundamental limit to how well one can know
  both the position and momentum of a particle.

Uncertainty in position
Uncertainty in momentum
7
Example

• Example
• What is the uncertainty in velocity for an
  electron in a 1Å radius orbital in which the
  positional uncertainty is 1 of the radius.

Dx (1 Å)(0.01) 1 x 10-12 m
huge
8
Example

• Example (youre quantum as well)
• What is the uncertainty in position for a 80
  kg student walking across campus at 1.3 m/s with
  an uncertainty in velocity of 1.

Dp m Dv (80kg)(0.013 m/s) 1.04 kg.m/s
Very smallwe know where you are.
9
De Broglies wavelength

• He provided a relationship between the electron
  properties and their wavelength which
  experimentally demonstrated by diffraction
  experiments

l h / p h/mv
10
Quantum Concepts (cont.)

• What is a standing wave?

A standing wave is a motion in which
translation of the wave does not occur.
In the guitar string analogy
(illustrated), note that standing waves
involve nodes in which no motion of the
string occurs.
Note also that integer and half- integer
values of the wavelength correspond to
standing waves.
11
Quantum Concepts (cont.)

• Louis de Broglie suggests that for the e- orbits
  envisioned by Bohr, only certain orbits are
  allowed since they satisfy the standing wave
  condition.

not allowed
12
Schrodinger Equation

• Erwin Schrodinger develops a mathematical
  formalism that incorporates the wave nature of
  matter

Kinetic Energy
The Hamiltonian
d2/dx2
x
The Wavefunction
E energy
13
Wavefunction

• What is a wavefunction?

a probability amplitude

• Consider a wave

Intensity

• Probability of finding a particle in space

Probability

• With the wavefunction, we can describe spatial
  distributions.

14
Potential Energy and Quantization

• Consider a particle free to move in 1
  dimension

The Free Particle
Potential E 0
The Schrodinger Eq. becomes
0
Energy ranges from 0 to infinity.not
quantized.
15
Potentials and Quantization (cont.)

• What if the position of the particle is
  constrained by a potential

Particle in a Box
Potential E
0 for 0 x L
? all other x
Now, position of particle is limited to the
dimension of the box.
16
Potentials and Quantization (cont.)

• What do the wavefunctions look like?

n 1, 2, .
Like a standing wave
y
yy
17
Potentials and Quantization (cont.)

• What does the energy look like?

n 1, 2,
Energy is quantized
E
y
yy
18
Potentials and Quantization (cont.)

• Consider the following dye molecule, the
  length of which can be considered the length of
  the box an electron is limited to

L 8 Å
What wavelength of light corresponds to DE from
n1 to n2?
(should be 680 nm)
19
Potentials and Quantization (cont.)

• One effect of a constraining potential is
  that the energy of the system becomes quantized.

Back to the hydrogen atom
constraining potential
20
Potentials and Quantization (cont.)

• Also in the case of the hydrogen atom, energy
  becomes quantized due to the presence of a
  constraining potential.

Schrodinger Equation
Recovers the Bohr behavior