Physical Chemistry

1. Penetration. An electron with energy 1.60 eV runs into a potential step of height 1.90 eV. What is the

penetration depth of the electron?

2. Tunneling. (A) Plot the transmission probability T of an electron across a potential barrier of width a

and height Vo as a function of the electron energy E. To generate the plot, use Vo = 2.00 eV and a

700 pm and an energy range of 0 S E S 3Vo. (B) Plot the transmission probability Tclass for the case

of a classical particle with the same mass as the electron. (C) Identity the regions ofT where it differs

from Tclass, and discuss the physical meaning.

3. Uncertainty. The velocity of a Ag atom in an atomic-beam apparatus is about 600 m s”, with a relative

uncertainty ofo.5%. According to the Heisenberg uncertainty principle, what is the uncertainty in the

position of the Ag atom along the direction of motion?

4. Uncertainty. A series of measurements of an observable Q on identically prepared quantum systems

gave the following integer-valued outcomes: qk= -6, -3, -2, -2, 0, 1, 4, 4, 4, 7, 11. (A) Compute the

expectation value (= mean), (Q)? (B) Compute the uncertainty (= standard deviation = root-mean-

square deviation), oQ. (C) What is the median? (D) What is the most likely value (= mode).

5. Uncertainty. (A) For a harmonic oscillator in its ground state, calculate the standard deviations ox

(/(x2) – (x)2 and Up = (/(pz) – (p)2, and their product axap. (B) Discuss the result in light of the

uncertainty principle.

6. Oscillator. The infrared absorption spectrum of gas-phase 12C”’O consists of an intense line centered

at about 2143 cm”. Calculate the vibration frequency, the period of vibration, the wavelength of the

absorbed photons, the reduced mass, and the force constant.

7. Oscillator. When carbon monoxide, l2C”’O, is bound to the muscle-tissue protein myoglobin, the

vibrational frequency of its stretching mode is 1945.2 cm”. Compute the frequency of the same

vibrational transition in I3C”’O, assuming the force constant is not affected by the isotope change.

8. Oscillator. The force constant of dinitrogen ”N2 is 2294 N m”. (A) Compute the relative populations

of the five lowest vibrational states at thermal equilibrium at 22.0 °C. (Set the relative population of the

ground state equal to one). (B) Discuss the result.

9. Oscillator. Consider a lH1271 molecule (force constant 313.8 N m”, equilibrium bond distance 161 pm)

in its vibrational ground state. What is the probability of finding the molecule with its bond length at

least 5.0% longer than its equilibrium value? Hint: Use software (Excel, Matlab, Maple, etc.) or an

online resource (Wolfram Alpha, etc.) to numerically evaluate the error function.

10. Oscillator. The energy eigenstates of the harmonic oscillator are described by wave functions that

contain the Hermite polynomials, Hn (2). Using Ho (2) = 1, H1(z) = 2z, and the recursion relationship

Hn+1(z) = 22 – Hn(z) – 2n – Hn_1(z), compute H2 (z) through H5 (2).

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