Homework Problems

1	In a television, suppose the electrons are accelerated from rest through a voltage difference of 104 V. What is their final wavelength?	√
2	Use the Heisenberg uncertainty principle to estimate the minimum velocity of a proton or neutron in a 208Pb nucleus, which has a diameter of about 13 fm (1 fm = 10-15 m). Assume that the speed is nonrelativistic, and then check at the end whether this assumption was warranted.	√
3	A free electron that contributes to the current in an ohmic material typically has a speed of 105 m/s (much greater than the drift velocity). (a) Estimate its de Broglie wavelength, in nm. (b) If a computer memory chip contains 108 electric circuits in a 1 cm2 area, estimate the linear size, in nm, of one such circuit. (c) Based on your answers from parts a and b, does an electrical engineer designing such a chip need to worry about wave effects such as diffraction? (d) Estimate the maximum number of electric circuits that can fit on a 1 cm2 computer chip before quantum-mechanical effects become important.	√
4	On page 95, I discussed the idea of hooking up a video camera to a visible-light microscope and recording the trajectory of an electron orbiting a nucleus. An electron in an atom typically has a speed of about 1% of the speed of light. (a) Calculate the momentum of the electron. (b) When we make images with photons, we can't resolve details that are smaller than the photons' wavelength. Suppose we wanted to map out the trajectory of the electron with an accuracy of 0.01 nm. What part of the electromagnetic spectrum would we have to use? (c) As found in homework problem 10 on page 39, the momentum of a photon is given by p = E/c. Estimate the momentum of a photon of having the necessary wavelength. (d) Comparing your answers from parts a and c, what would be the effect on the electron if the photon bounced off of it? What does this tell you about the possibility of mapping out an electron's orbit around a nucleus?	√
5	Find the energy of a particle in a one-dimensional box of length L, expressing your result in terms of L, the particle's mass m, the number of peaks and valleys n in the wavefunction, and fundamental constants.	√
6	The Heisenberg uncertainty principle, can only be made into a strict inequality if we agree on a rigorous mathematical definition of Δx and Δp. Suppose we define the deltas in terms of the full width at half maximum (FWHM), which we first encountered in Vibrations and Waves, and revisited on page 49 of this book. Now consider the lowest-energy state of the one-dimensional particle in a box. As argued on page 96, the momentum has equal probability of being h/L or -h/L, so the FWHM definition gives Δp = 2h/L. (a) Find x using the FWHM definition. Keep in mind that the probability distribution depends on the square of the wavefunction. (b) Find ΔxΔp.	√
7	If x has an average value of zero, then the standard deviation of the probability distribution D(x) is defined by where the integral ranges over all possible values of x. Interpretation: if x only has a high probability of having values close to the average (i.e., small positive and negative values), the thing being integrated will always be small, because x2 is always a small number; the standard deviation will therefore be small. Squaring x makes sure that either a number below the average (x < 0) or a number above the average (x > 0) will contribute a positive amount to the standard deviation. We take the square root of the whole thing so that it will have the same units as x, rather than having units of x2. Redo problem 6 using the standard deviation rather than the FWHM. Hints: (1) You need to determine the amplitude of the wave based on normalization. (2) You'll need the following definite integral:	∫ √
8	In section 4.6 we derived an expression for the probability that a particle would tunnel through a rectangular potential barrier. Generalize this to a barrier of any shape. [Hints: First try generalizing to two rectangular barriers in a row, and then use a series of rectangular barriers to approximate the actual curve of an arbitrary potential. Note that the width and height of the barrier in the original equation occur in such a way that all that matters is the area under the PE-versus-x curve. Show that this is still true for a series of rectangular barriers, and generalize using an integral.] If you had done this calculation in the 1930's you could have become a famous physicist.	∫ *