Set#6

```CHEM344 HW#6
Due: Fri, Mar [email protected] BEFORE CLASS!
HW to be handed in:
Atkins
Chapter 7:
Exercises: 7.11(b), 7.12(b), 7.13(b),
Problems: 7.4,
Chapter 8:
Exercises: 8.6(b), 8.8(b), 8.19(b),
Problems: 8.10, 8.14(a only), 8.32,
Extra (do not hand in):
Chapter 7:
Problems: 7.8,
Chapter 8:
Exercises: 8.1(b), 8.3(b), 8.4(b), 8.7(b), 8.18(b),
Problems: 8.28,
1. [7.11(b)] An unnormalized wavefunction for an electron in a carbon nanotube of length L is
sin(2πx/L). Normalize this wavefunction.
2. [7.12(b)] For the system described in Exercise 7.11b, what is the probability of finding the
electron in the range dx at x = L/2?
3. [7.13(b)] For the system described in Exercise 7.11b, what is the probability of finding the
electron between x = L/4 and x = L/2?
4. [7.4] The ground-state wavefunction for a particle confined to a one-dimensional
⁄
box of length L is
( )
( )
Suppose the box is 10.0 nm long. Calculate the probability that the particle is (a) between x =
4.95 nm and 5.05 nm, (b) between x = 1.95 nm and 2.05 nm, (c) between x = 9.90 nm and
10.00 nm, (d) in the right half of the box, (e) in the central third of the box.
5. [8.6(b)] What are the most likely locations of a particle in a box of length L in the state n = 5?
6. [8.8(b)] A nitrogen molecule is confined in a cubic box of volume 1.00 m3. Assuming that the
molecule has an energy equal to 3/2kT at T = 300 K, what is the value of n = (nx2 + ny2 + nz2)1/2
for this molecule? What is the energy separation between the levels n and n + 1? What is its de
Broglie wavelength?
Note: this sort of glosses over the 3-D aspect of the box, but you could
have gotten same answer for nx by solving for Ex=½kT
7. [8.19(b)] Calculate the value of |ml | for the system described in the preceding exercise
corresponding to a rotational energy equal to the classical average energy at 25°C (which is
equal to 1/2kT).
Note: from 8.19(a) the system is a proton constrained to rotate in a circle of radius 100 pm
around a fixed point.
Note: moved to Problem Set #7
8. [8.10] The wavefunction inside a long barrier of height V is ψ = Ne−κ x. Calculate (a) the
probability that the particle is inside the barrier and (b) the average penetration depth of the
particle into the barrier.
9. [8.14 (a) only] Determine the values of Δx = (<x2> − <x>2)1/2 and Δp = (<p2> − <p>2)1/2 for (a)
a particle in a box of length L and (b) a harmonic oscillator. Discuss these quantities with
reference to the uncertainty principle.
10. [8.32] Many biological electron transfer reactions, such as those associated with biological
energy conversion, may be visualized as arising from electron tunnelling between protein-bound
co-factors, such as cytochromes, quinones, flavins, and chlorophylls. This tunnelling occurs
over distances that are often greater than 1.0 nm, with sections of protein separating electron
donor from acceptor. For a specific combination of donor and acceptor, the rate of electron
tunnelling is proportional to the transmission probability, withκ ≈ 7 nm−1 (eqn 8.19). By what
factor does the rate of electron tunneling between two co-factors increase as the distance
between them changes from 2.0 nm to 1.0 nm?
Extra (do not hand in):
11. [7.8] The normalized wavefunctions for a particle confined to move on a circle are ψ(φ) =
(1/2π)1/2e−imφ, where m = 0, ±1, ±2, ±3, . . . and 0 ≤ φ ≤ 2π. Determine <φ>.
12. [8.1(b)] Calculate the energy separations in joules, kilojoules per mole, electronvolts, and
reciprocal centimetres between the levels (a) n = 3 and n = 1, (b) n = 7 and n = 6 of an electron
in a box of length 1.50 nm.
13. [8.3(b)] Calculate the expectation values of p and p2 for a particle in the state n = 2 in a
square-well potential.
14. [8.4(b)] Calculate the expectation values of x and x2 for a particle in the state n = 2 in a
square-well potential.
15. [8.7(b)] Calculate the percentage change in a given energy level of a particle in a cubic box
when the length of the edge of the cube is decreased by 10 percent in each direction.
16. [8.18(b)] Confirm that wavefunctions for a particle in a ring with different values of the
quantum number ml are mutually orthogonal.
17. [8.28] Starting from the operator ̂ = xpy − ypx, prove that in spherical polar coordinates ̂ =
−i ∂/∂φ.
```