Lecture 16: Intro. to Quantum Mechanics

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.

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.

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”.

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?

= h / p = h/mv

= 6.626x10-34 Js /(9.11x10-31kg)(1x107m/s)

= 6.626x10-34 Kgm2/s /(9.11x10-31kg)(1.x107m/s)

= 7.3x10-11 m

Exercise• What is the wavelength of a baseball (mass 0.1 kg)

traveling at a speed of 35 m/s?

= h / p = h/mv

= 6.626x10-34 Js /(0.1kg)(35m/s)

= 6.626x10-34 Kgm2/s /(0.1kg)(35m/s)

= 1.9x10-34 m

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.

x p h

4Uncertainty in position Uncertainty in momentum

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.

x = (1 Å)(0.01) = 1 x 10-12 m

p h

4x

6.626x10 34 J.s 4 1x10 12 m

5.27x10 23 kg.m /s

v pm

5.27x10 23 kg.m /s

9.11x10 31kg5.7x107 m

s huge

Example• Example (you’re 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%.

p = m v = (80kg)(0.013 m/s) = 1.04 kg.m/s

x h

4p

6.626x10 34 J.s 4 1.04kg.m /s

5.07x10 35 m

Very small……we know where you are.

De Broglie’s wavelength• He provided a relationship between the electron properties and

their ‘wavelength’ which experimentally demonstrated by diffraction experiments

= h / p = h/mv

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.

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

Schrodinger Equation• Erwin Schrodinger develops a mathematical

formalism that incorporates the wave nature of matter:

ˆ H E

ˆ H The Hamiltonian:

ˆ p 2

2m (PE)

Kinetic Energy

The Wavefunction:x

E = energy

d2/dx2

Wavefunction• What is a wavefunction?

= a probability amplitude

• Consider a wave:

y Ae i 2 t

y 2 Ae i 2 t Ae i 2 t A2Intensity =

• Probability of finding a particle in space:

*Probability = • With the wavefunction, we can describe spatial distributions.

Potential Energy and Quantization• Consider a particle free to move in 1 dimension:

x

p “The Free Particle”

Potential E = 0

• The Schrodinger Eq. becomes:

ˆ H ˆ p 2

2m PE

(p)

ˆ p 2

2m(p)

p2

2m(p)

12

mv 2(p) E(p)

• Energy ranges from 0 to infinity….not quantized.

0

Potentials and Quantization (cont.)• What if the position of the particle is constrained

by a potential:“Particle in a Box”

Potential E

x0

inf.

0 L = 0 for 0 ≤ x ≤ L

= all other x

• Now, position of particle is limited to the dimension of the box.

Potentials and Quantization (cont.)• What do the wavefunctions look like?

x 2L

sin nxL

Like a standing wave

n = 1, 2, ….

Potentials and Quantization (cont.)• What does the energy look like?

Energy is quantized

E

E n2h2

8mL2

n = 1, 2, …

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:

E h2

8mL2 n final2 ninitial

2 h2

8m(8Å)2 22 1 2.8x10 19 J

What wavelength of light corresponds to E from n=1 to n=2?

L = 8 Å

700nm(should be 680 nm)

N

N

+

Potentials and Quantization (cont.)• One effect of a “constraining potential” is that the

energy of the system becomes quantized.

• Back to the hydrogen atom:

e-

P+r

V (r) e2

r

constraining potential

r0

Potentials and Quantization (cont.)• Also in the case of the hydrogen atom, energy becomes

quantized due to the presence of a constraining potential.

r0

Schrodinger Equation

V (r) e2

r

0

Recovers the “Bohr” behavior