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35.7 Summary by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.

*wavefunction* — the numerical measure of an electron wave, or in general of the wave corresponding to any quantum mechanical particle

`ℏ`— Planck's constant divided by `2pi` (used only in optional section 35.6)

`Psi` — the wavefunction of an electron

Light is both a particle and a wave. Matter is both a particle and a wave. The equations that connect the particle and wave properties are the same in all cases:

Unlike the electric and magnetic fields that make up a photon-wave, the electron wave function is not directly measurable. Only the square of the wave function, which relates to probability, has direct physical significance.

A particle that is bound within a certain region of space is a standing wave in terms of quantum physics. The two equations above can then be applied to the standing wave to yield some important general observations about bound particles:

- The particle's energy is quantized (can only have certain values).
- The particle has a minimum energy.
- The smaller the space in which the particle is confined, the higher its kinetic energy must be.

These immediately resolve the difficulties that classical physics had encountered in explaining observations such as the discrete spectra of atoms, the fact that atoms don't collapse by radiating away their energy, and the formation of chemical bonds.

A standing wave confined to a small space must have a short wavelength, which corresponds to a large momentum in quantum physics. Since a standing wave consists of a superposition of two traveling waves moving in opposite directions, this large momentum should actually be interpreted as an equal mixture of two possible momenta: a large momentum to the left, or a large momentum to the right. Thus it is not possible for a quantum wave-particle to be confined to a small space without making its momentum very uncertain. In general, the Heisenberg uncertainty principle states that it is not possible to know the position and momentum of a particle simultaneously with perfect accuracy. The uncertainties in these two quantities must satisfy the approximate inequality

`DeltapDeltax≳h`.

When an electron is subjected to electric forces, its wavelength cannot be constant. The “wavelength” to be used in the equation `p=h/lambda` should be thought of as the wavelength of the sine wave that most closely approximates the curvature of the wavefunction at a specific point.

Infinite curvature is not physically possible, so realistic wavefunctions cannot have kinks in them, and cannot just cut off abruptly at the edge of a region where the particle's energy would be insufficient to penetrate according to classical physics. Instead, the wavefunction “tails off” in the classically forbidden region, and as a consequence it is possible for particles to “tunnel” through regions where according to classical physics they should not be able to penetrate. If this quantum tunneling effect did not exist, there would be no fusion reactions to power our sun, because the energies of the nuclei would be insufficient to overcome the electrical repulsion between them.

**The New World of Mr. Tompkins: George Gamow's Classic Mr. Tompkins in Paperback**, *George Gamow*.

Mr. Tompkins finds himself in a world where the speed of light is only `30` miles per hour, making relativistic effects obvious. Later parts of the book play similar games with Planck's constant.

**The First Three Minutes: A Modern View of the Origin of the Universe**, *Steven Weinberg*.

Surprisingly simple ideas allow us to understand the infancy of the universe surprisingly well.

**Three Roads to Quantum Gravity**, *Lee Smolin*.

The greatest embarrassment of physics today is that we are unable to fully reconcile general relativity (the theory of gravity) with quantum mechanics. This book does a good job of introducing the lay reader to a difficult, speculative subject, and showing that even though we don't have a full theory of quantum gravity, we do have a clear outline of what such a theory must look like.

**Key**

`sqrt` A computerized answer check is available online.

`int` A problem that requires calculus.

`***` A difficult problem.

**1**. In a television, suppose the electrons are accelerated from rest through a voltage difference of `10^4 V`. What is their final wavelength? `sqrt`

**2**. Use the Heisenberg uncertainty principle to estimate the minimum velocity of a proton or neutron in a ^{208}Pb 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. `sqrt`

**3**. A free electron that contributes to the current in an ohmic material typically has a speed of `10^5 m"/"s` (much greater than the drift velocity).

(a) Estimate its de Broglie wavelength, in nm. `sqrt`

(b) If a computer memory chip contains `10^8` electric circuits in a `1 cm^2` area, estimate the linear size, in nm, of one such circuit. `sqrt`

(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 cm^2` computer chip before quantum-mechanical effects become important.

**4**. On page 970, 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. `sqrt`

(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 12 on page 789, the momentum of a photon is given by `p=E/c`. Estimate the momentum of a photon having the necessary wavelength. `sqrt`

(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 mm, the number of peaks and valleys nn in the wavefunction, and fundamental constants. `sqrt`

**6**. The Heisenberg uncertainty principle, `Deltap Deltax≳h`, can only be made into a strict inequality if we agree on a rigorous mathematical definition of `Deltax` and `Deltap`. Suppose we define the deltas in terms of the full width at half maximum (FWHM), which we first encountered on p. 469 and revisited on page 925 of this book. Now consider the lowest-energy state of the one-dimensional particle in a box. As argued on page 971, the momentum has equal probability of being `h"/"L` or `-h"/"L`, so the FWHM definition gives `Deltap=(2h)/lambda`.

(a) Find `Deltax` using the FWHM definition. Keep in mind that the probability distribution depends on the square of the wavefunction.

(b) Find `DeltaxDeltap`. `sqrt`

**7**. If `x` has an average value of zero, then the standard deviation of the probability distribution `D(x)` is defined by

`sigma^2=sqrt(intD(x)x^2dx)`

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 `x^2` 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 `x^2`.

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:` int_(-pi"/"2)^(pi"/"2)u^2cos^2udu=(pi^3-6pi)"/"24`. `sqrt` `int`

**8**. In section 35.6 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. `int` `***`

**9**. The electron, proton, and neutron were discovered, respectively, in 1897, 1919, and 1932. The neutron was late to the party, and some physicists felt that it was unnecessary to consider it as fundamental. Maybe it could be explained as simply a proton with an electron trapped inside it. The charges would cancel out, giving the composite particle the correct neutral charge, and the masses at least approximately made sense (a neutron is heavier than a proton). (a) Given that the diameter of a proton is on the order of `10^(-15) m`, use the Heisenberg uncertainty principle to estimate the trapped electron's minimum momentum. `sqrt`

(b) Find the electron's minimum kinetic energy. `sqrt`

(c) Show via `E=mc^2` that the proposed explanation fails, because the contribution to the neutron's mass from the electron's kinetic energy would be many orders of magnitude too large.

35.7 Summary by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.