In a television, suppose the electrons are accelerated from rest
through a voltage difference of 104 V. What is their final wavelength?
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
A free electron that contributes to the current in an ohmic
material typically has a speed of 105 m/s (much greater than the
(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
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
(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?
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
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.
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:
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.