1. Application of quantum in chemistry

2. The Particle in A Box

3. The ‘Classical’ Case

5. The ‘Quantum’ Case

6. The absolutely small particle in the nanometer size box is a quantum particle, and it must obey the Uncertainty Principle, that is, ΔxΔp= h/4π. If V=0 and x= L/2, we know both x and p. The result would be ΔxΔp 0, the same as the classical racquetball.
This is impossible for a quantum system. Therefore, V cannot be zero. The particle cannot be standing still at a specific point. If V cannot be zero, then Ek can never be zero. The Uncertainty Principle tells us that the lowest energy that a quantum racquetball can have cannot be zero. Our quantum racquetball can never stand still.

7. Energies of a Quantum Particle in a Box

8. Wave functions must be zero at the walls

9. Nodes are the points where the wave function crosses zero

10. Energies are quantized

11. A Discreet set of energy levels

14. Why are Cherries Red and Blueberries blue ?

15. The Colour of Fruit This energy corresponds to Deep Red Colour
If L=0.7 nm, =540 nm
If L=0.6 nm, =397 nm
Green Colour
Blue Colour

16. Particle in a box
Step 1: Define the potential energy Step 2: Solve the Schrodinger equation Step 3: Define the wave function Step 4: Determine the allowed energies Step 5: Interpret its meaning

17. Particle in 1-dimensional box
Infinite walls
Time Independent Schrödinger Equation
V(x)=0
V(x)=∞ L x
Region I
Region II
Region III KE PE TE
Applying boundary conditions:
Region I and III:

18. Functions with this property sin and cos.
Second derivative of a function equals a negative constant times the same function.
Functions with this property sin and cos.
Copyright – Michael D. Fayer, 2007

19. b) x=L ψ=0 Region II: Thus, wave function: But what is ‘A’? a) x=0 ψ=0
This is similar to the general differential equation:
But what is ‘A’?
Applying boundary conditions:
a) x=0 ψ=0

20. Normalizing wave function:
Calculating Energy Levels:
Thus normalized wave function is:
Thus Energy is:

21. Particle in a 1-Dimensional Box
Difference b/w adjacent
energy levels:
2) Non-zero zero point energy
3) Probability density is structured
with regions of space demon-
-strating enhanced probability.
At very high n values, spectrum
becomes continous-
convergence with CM
(Bohr’s correspondance
principle) + + + - + + E y
y*y

22. Particle in a 3-D box

23. Question: An electron is in 1D box of 1nm length
Question: An electron is in 1D box of 1nm length. What is the probability of locating the electron between x=0 and x=0.2nm in its lowest energy state?

24. Question: An electron is in 1D box of 1nm length
Question: An electron is in 1D box of 1nm length. What is the probability of locating the electron between x=0 and x=0.2nm in its lowest energy state?
Solution:

25. Question: What are the most likely locations of a particle in a box of length L in the state n=3

26. Example: What are the most likely locations of a particle in a box of length L in the state n=3

27. Expectation value of position and its uncertainty

28. Expectation values
Position
Uncertainity

29. Expectation value of Momentum
And square of momentum

30. Momentum

31. Estimating pigment length
Assumptions:

34. Wavelength of transition for Anthracene
Particle in a Box Simple model of molecular energy levels.
Anthracene L
p electrons – consider “free” in box of length L. Ignore all coulomb interactions.

35. Pigments and Quantum mechanics
High degree of conjugation!!
Electrons have wave properties and they don’t jump off the pigments when they reach its ends.
These electrons resonances determine which frequencies of light and thus which colors, are absorbed & emitted from pigments

36. Electron resonances in a cyclic conjugated molecule
A crude quantum model for such molecules assumes that electrons move freely in a ring.
Resonance condition:
R: radius of molecule, λ wavelength of electron

37. Energy is once again quantized
Energy is once again quantized. It depends on variable n which posseses discrete values only