Homework 5

Due 23 Friday 11/15

1 Boundary conditions and musical instruments

Adapted from Q2M.1 from Chpt 2 of Unit Q, 3rd Edition

(a) A concert flute, shown above, is about 2 ft long. Its lowest pitch is middle C (about 262 Hz). On the basis of this evidence, should we consider a flute to be a pipe that is open at both ends, or at just one end? Support your argument with a quantitative comparison. (The end of the flute farthest from the mouth piece is clearly open. The other end of the flute seems to be closed, so if you claim that the flute is open at both ends, you should try to explain where the other open end is. Note that for wind instruments, an open end corresponds to a pressure node.)

(b) What are the lowest three resonant frequencies that can be played on a flute when all the finger holes are closed? Give you answer in units of Hz. Draw these frequencies on a frequency level diagram (vertical axis represents frequency, horizontal lines are placed at the resonant frequencies).

(c) The orchestra is warming up their instruments. The flute starts at 290 K and increases temperature to 300 K. How seriously does this affect the pitch of the flute? For reference, each step on a chromatic musical scale has a frequency 1.06 times higher than the one below it (1.06 = 21/12). The conductor of the orchestra will be upset if the flute shifts from its correct frequency by ±1%. The speed of sound in a gas is v = √(γP0/ρ0) where γ is a dimensionless constant, P0 is the ambient pressure and ρ0 is the gas’s density. As the gas warms up, its density drops (ambient pressure does not change).

2 Hot hydrogen atom

Find the wavelength of the photon emitted during a n = 5 → 4 transition in a hydrogen atom.

Note: The energy levels in a hydrogen atom are

where n = 1, 2, 3, ...

3 Wavelength from a charge on a spring

Suppose a charged particle is held in position by an electrostatic spring (i.e. the restoring force on the charge follows Hooke’s law, F = −kx). The mass of the charge, and the spring constant, are such that the system has a natural frequency ω = 1016 rad/s (ω is a fixed parameter in this question, not a variable). Find the wavelength of the photon emitted during a n = 1 → 0 transition.

By solving the Schrodinger equation for this situation, we know that the energy of the charged particle (i.e. the sum of the particle’s kinetic energy, plus any potential energy stored in the spring) is given by

where n = 0, 1, 2, ...

Note: Due to the shape/symmetry of the wavefunctions for particles trapped by a springlike force, optical transitions only occur when ∆n = ±1.

Sense making: Try approaching this question from a classical physics perspective. What wavelength of light would we expect from a charge that oscillates at ω = 1016 rad/s?

4 Glowing electron in a box

Suppose an electron is trapped in a box whose length is L = 1.2 nm. This is a coarse-grained model for an electron in a small molecule like cyanine (see Example Q11.1 in the textbook, and the figure above). If we solve the Schrodinger equation for this coarse-grained model, the possible energy levels for this electron are

where m is the mass of the electron and n = 1, 2, 3, ...

Draw a spectrum chart (like figure Q11.2) showing all the visiblelight emission lines from this system.

Note: Due to the shape/symmetries of electron wavefunctions in a box, optical transitions between energy levels only happen when ∆n = ninitial − nfinal, is an odd integer.