Problems that you should be able to solve
1) Characterize the difference(s) between particles and waves.
2) Name two experiments/observations that violate classical mechanics.
3) Calculate the de Broglie wavelength of:
- The mass of 1.0 g traveling at 1.0 m/s
- A 100 kg person traveling at 8 km/s
4) Name one experimental technique that makes use of the de Broglie wavelength for particles.
5) Werner Heisenberg proposed in 1927 a principle that is one of the fundamentals of quantum mechanics and of central importance for the philosophical questions arising from the new theory. The speed of a certain proton is 350 km/s. If the uncertainly in its momentum is 0.01%, what uncertainty in its location must be tolerated?
6) Position and momentum are related by the Heisenburg uncertainty principle, what other two variables are also related this principle.
7) The Schrödinger equation is of paramount importance for solving quantum mechanical problem and represents the total energy of the system. Identify the term that represents the kinetic energy in this equation.
8) A wavefunction is a solution to the Schrödinger equation; what three properties must the wavefunction satisfy.
9) Calculate the probability that an electron will be found between x = 0.1 and x = 0.2 nm in a box of length L=10 nm, when its wavefunction is .
10) What is the probability of finding the electron in the center of a 1-D box with infinite potential at the walls when its wavefunction is in the n=-1 and n=2 states? What is the probability of finding the particle between x=0 and x=L in this box?
11) How is the square of the wavefunction, , similar to the Maxwell distribution for the speed of gas particles?
12) What is the lowest energy for the particle in a box? How does this relate to the Heisenberg Uncertainty principle?
13) How are the orbitals for electrons in a hydrogen atom related to the wavefunction which is the solution to the Schrödinger equation?
14) What are the quantum numbers for the highest energy electron in a boron atom? What quantum numbers determine the energy of this electron? Does this differ for the energies of electrons in 1) hydrogen and 2) lithium?
15) How many orbitals to the hydrogen atom have the quantum numbers of n=2 and l=1? Draw the angular parts to each orbital?
16) What is the range for each quantum numbers in the solution(s) to the hydrogen atom?
17) Hybridization is the mixing of atomic orbitals on an atom prior to the generation of a molecular bond. The hybridized orbitals then can generated typically (for the examples given in class) sigma bonds. What is the difference between ? and ? bonds? How do atoms generate ? bonds?
18) Order the following diatomics with respect to stronger bonding: N2, H2-, O2, He2.
19) Draw out the molecular orbital diagram (mixing of the atomic orbitals to generate molecular orbitals) for N2 and O2. Calculate the bond orders for each diatomic. What is the multiplicity of each diatomic? Which molecule is more likely to emit fluorescence? Which will interact with an external magnetic field?
20) Sketch the spatial distribution (hence structure) of the electron density for the valence orbitals in one of the carbon atoms in each of the following molecules:
24) Spectroscopy involves the coupling of electromagnetic radiation between wavefunctions for a specific system. Why do spectra typically do not look like “sticks” associated with the difference in energy between wavefunctions?
25) Not all transitions are allowed in spectroscopy. For the electronic transitions in a hydrogen atom, what two selection rules are required to be satisfied before the transition can be observed?
32) Explain why
- Molecules absorb visible (and UV) photons at selected frequencies to give excited electronic states and then usually emit photons at lower frequencies, a phenomenon called fluorescence.
- Atom in contrast absorb photons at selected frequencies to get excited electronic state and then emit photons at the same frequencies.
33) What main difference between fluorescence and phosphorescence?
35) The wavefuction: is not normalized for the range x=0 to x=L. Write the equation to normalize this wavefunction. Why do wavefunctions need to be normalized?
36) Explain how a Scanning Tunneling Microscope works including tunneling, electrons, distance dependence and the inherent resolution of the technique.