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can you please help me with problem 2 parts b and c. Thank you
can you please help me with problem 2 parts b and c.
Problem Set 2 (due Thu Sep 1, 2016).
1. Consider an electron with a mass of 9.11 x 10-31 kg and a 50-g tennis ball that are both moving with
a velocity of 20 m s-1.
(a) Calculate the momentum of the electron (p = mv).
(b) Calculate the momentum of the tennis ball.
(c) What is the uncertainty in the position of the electron (x) if the uncertainty in its momentum
(p) is equal to 1% of p for the electron?
(d) What is the uncertainty in the position of the tennis ball (x) if the uncertainty in momentum
(p) is equal to 1% of p for the tennis ball?
(e) Comment on how the uncertainty in position (x) compares to the overall size in each case.
2. Consider an electron in a one-dimensional particle-in-a-box with a box length of L=1 x 10-9 m and
mass equal to m 9.111031 kg.
(a) For the two lowest energy wavefunctions (1 and 2), evaluate the following integral:
0 2 1 dx
sin dx ?
(b) What is the probability of locating the electron in the region between L/4 and 3L/4 when the
electron is in its lowest energy state (1)?
(c) What is the probability of locating the electron in the region between L/4 and 3L/4 when the
electron is in the second-lowest energy state (2)?
(d) Calculate the energy difference between the n=2 and n=1 states (E = E2 ? E1).
(e) What is the frequency of a quantum transition from the n=1 to n=2 state?
(f) Calculate the wavelength of electromagnetic radiation (in nanometers) that will excite a
quantum transition from the n=1 to n=2 state.
3. When the spacing between translational energy levels is small compared to the thermal energy
(kBT), classical mechanics is a good approximation for quantum mechanics. Consider whether the
translational motion of each particle below can be treated classically. For each system below,
calculate the energy of the particle in a box for the n=1 and n=2 states and determine whether E (ie
E2 ? E1) is smaller or larger than the available thermal energy, kBT.
(a) A helium atom in a 1000 Å box at 298 K (L = 10-7 m).
(b) A protein with a molecular weight of 50 kDa in a 100 Å box at 298 K (L = 10-8 m).
(c) A helium atom in a 1 Å box at 1 K (L = 10-10 m).
) , where N is the
normalization constant and ?a? is a constant that depends on the mass (m) and force constant (k).
4. The ground state wavefunction of a harmonic oscillator is ( x) N exp(
(a) Normalize this wavefunction by setting 2 ( x)dx Ne 2
dx 1 and solve for N.
(b) A harmonic oscillator describes the one-dimensional periodic displacement (x) of two objects
(with mass = m) attached at each end of a spring. The net displacement is zero (x = 0) when the
spring is at equilibrium. The net displacement is positive (x > 0) when the spring is stretched
and the two objects move apart. The net displacement is negative (x < 0) when the spring is
compressed and the two objects move close together.
d ( x)
What is the most probable displacement (x)? (Hint: set
0 and solve for x)
(c) Calculate the fundamental vibrational frequency (in units of s-1) of carbon monoxide that
consists of a carbon and oxygen atom ( C O 1.14 1026 kg ) connected by a triple
bond with a force constant (k) of 1860 N m . (Hint: assume the C?O stretching vibration
behaves like a harmonic oscillator that has a fundamental vibrational frequency of
(d) In infrared spectroscopy it is common to convert vibrational frequency (units of s-1) into
vibrational wavenumber ( ~0 / c) that has units of cm-1. What is the vibrational wavenumber
of the C?O stretching vibration?
5. The electrons of metal porphyrins, such as the iron-heme of hemoglobin or the magnesiumporphyrin of chlorophyll, can be described energetically using a simple model of free electrons in a
h2 nx2 n y
two-dimensional particle in a box: Enx , n y
where nx 1, 2, 3 & n y 1, 2, 3
8m a 2 a 2
(a) The porphyrin square structure measures about 1 nm on each side (a = 1 nm). Calculate the
energy ( Enx , n y ) for each of the 15 lowest energy states in a two-dimensional box with a = 1 nm
(E11, E12, E21, E22, E23, E32, E33, E34, E43, E44, E54, E45, E55, E65, E56). Sketch an energy-level
diagram showing the proper energy spacing between each of the states.
(b) A heme porphyrin contains 26 electrons. Place the 26 electrons (2 at a time) in the 13 lowest
energy level states in your diagram from part a. Each state (called an orbital) holds two
electrons at a time, including the states that are degenerate in energy (E12 = E21 or E23 = E32).
(c) Calculate the wavelength of electromagnetic radiation (in units of nm) that will excite the lowest
energy absorption band (E55 E56) called a * transition. (Experimentally these bands
occur at about 600 nm).
This question was answered on: Feb 21, 2020
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