Homework assignments updated

Homework assignments updated Announcements Two are due on Fridays to allow for time after the exam. Chapter 40 (“Quantum Mechanics”) has two homework assignments. I am away at KEK (Japan) Oct 13-25. Professor Lam will replace me and continue the iclicker reading quizzes.

Today: Heisenberg’s Uncertainty Principle Some “left-overs” on Compton scattering/pair production. We will then read the original 1927 paper in German by Heisenberg. Not joking: Will have some extra (4) clicker questions towards the end of class to check your understanding of the Heisenberg uncertainty principle.

Left overs: Compton scattering in “real life”

Compton scattering example An photon with a wavelength of 0.100nm collides with an electron initially at rest. The x-ray’s final wavelength is 0.110 nm. What is the final kinetic energy of the electron ? Energy is conserved

Compton scattering example (result in Joules and keV) A photon with a wavelength of 0.100nm collides with an electron initially at rest. The x-ray’s final wavelength is 0.110 nm. What is the final kinetic energy of the electron ? Question:: How do you convert to eV ? Ans: Remember 1 e =1.6 x 10-19 C

Compton scattering conceptual question Question: Do you expect to notice a wavelength shift in Compton scattering from visible light ? Ans: No, the quantity h/(mec)= 2.42 x 10-12m = 0.00242nm. Compare to visible light wavelengths 380-750 nm. The effect is much more dramatic with X-rays.

Pair production by a single photon Question : Why are the two tracks curving in opposite directions ? Ans: Particle and antiparticle have opposite charges ?

Bohr’s principle of complementarity 1928: The wave descriptions and the particle descriptions of nature are “complementary”. We need both to describe nature, but we will never need to use both at the same time to describe a single part of an occurrence.

Scene from the foundations of quantum mechanics. 1928-1935: In this period, Niels Bohr and Albert Einstein had many discussions about quantum mechanics. Einstein: “God does not play dice with the universe” Bohr: Einstein stop it ! Stop telling God what to do !

Let’s think about formation of Optical Images For very low light intensities, one can see the statistical and random nature of image formation. Use a sensitive camera that can detect single photons. A. Rose, J. Opt. Sci. Am. 43, 715 (1953) Exposure time The point:Processes that seem to be continuous may, in fact, consist of many microscopic “bits”. (Just like water flow.) For large light intensities, image formation by an optical system can be described by classical optics. A. Rose, J. Opt. Sci. Am. 43, 715 (1953)

Diffraction and the particle nature of light • A diffraction pattern is the result of many photons hitting the screen. The pattern appears even if only one photon is present at a time in the experiment.

Diffraction and uncertainty • A diffraction pattern is the result of many photons hitting the screen. The pattern appears even if only one photon is present at a time in the experiment. Question: Diffraction is the result of the wave nature of light ? How is this possible ? Ans: Light is both a wave and a particle. Different measurements reveal different aspects. “Complementarity”.

Diffraction and the road to the uncertainty principle • When a photon passes through a narrow slit, its momentum becomes uncertain and the photon can deflect to either side A diffraction pattern is the result of many photons hitting the screen. The pattern appears even if only one photon is present at a time in the experiment.

Diffraction and uncertainty • Question: What is the location of the first minimum in single slit diffraction ? In the small angle approximation Although photons are point particles, we cannot predict their paths exactly. All we can do is predict probabilitiesThere are uncertainties in their positions and momenta

Uncertainty in momentum and position (rough argument) • What is the uncertainty in the momentum of each photon ? ~85% of the photons lie in the angular range between [+θ1 , -θ1] Using the small angle approximation The average value of py is zero. However, the value of py is uncertain

Uncertainty in momentum and position (rough argument) • How are uncertainties in the momentum and position of each photon related ? Here a is related to the uncertainty in the y-position of the photon while Δpy is the y-momentum uncertainty. Question: What happens to the width of the central maximum in diffraction when the slit width a decreases ? Ans: The peak becomes broader. What does this mean for uncertainties ?

Heisenberg Uncertainty Principle (exact result) • How are uncertainties in the momentum and position of each photon related ? Here Δx and Δpx are the standard deviation or (rms) uncertainties in x and px The quantity on the right is “h-bar” over 2. Warning: “h-bar” is not the same as h

Heisenberg Uncertainty Principle (another version) • How are uncertainties in the energy and time localization related ? Here ΔE and Δt are the standard deviation or (rms) uncertainties in E and t. The quantity on the right is “h-bar” over 2. In Chapter 39, we will see that the Heisenberg uncertainty principle also applies to matter particles

The Heisenberg Uncertainty Principle 1932 Nobel Prize in Physics • You cannot simultaneously know the position and momentum of a photon with arbitrarily great precision. The better you know the value of one quantity, the less well you know the value of the other. • In addition, the better you know the energy of a photon, the less well you know when you will observe it.

The Heisenberg Uncertainty Principle Can also be understood in terms of “wave packets”. Combine two waves with slightly different frequencies (slightly different momenta, slightly different energies)gives a wave packet with an energy spread but localized in time and space.

QM: Clicker question I A photon has a momentum uncertainty of 2.00 x 10–28 kg • m/s. If you decrease the momentum uncertainty to 1.00 x 10–28 kg • m/s, how does this change the position uncertainty of the photon? A. the position uncertainty becomes 1/4 as large B. the position uncertainty becomes 1/2 as large C. the position uncertainty is unchanged D. the position uncertainty becomes twice as large E. the position uncertainty becomes 4 times larger

QM Clicker question I A photon has a momentum uncertainty of 2.00 x 10–28 kg • m/s. If you decrease the momentum uncertainty to 1.00 x 10–28 kg • m/s, how does this change the position uncertainty of the photon? A. the position uncertainty becomes 1/4 as large B. the position uncertainty becomes 1/2 as large C. the position uncertainty is unchanged D. the position uncertainty becomes twice as large E. the position uncertainty becomes 4 times larger

QM: Clicker question II Through which of the following angles is a photon of wavelength λ most likely to be deflected when passing through a slit of width a ? A. θ=λ/a B. θ=3λ/2a C. θ=2λ/a D. θ=3λ/a E. Not enough information to decide.

QM: Clicker question II Through which of the following angles is a photon of wavelength λ most likely to be deflected when passing through a slit of width a ? A. θ=λ/a (zero in diffraction) B. θ=3λ/2a (non-zero intensity in the diffraction pattern !) C. θ=2λ/a (zero in diffraction) D. θ=3λ/a (zero in diffraction) E. Not enough information to decide.

QM: Clicker question 3 A photon has a position uncertainty of 2.00 mm. If you decrease the position uncertainty to 1.00 mm, how does this change the momentum uncertainty of the photon? A. the momentum uncertainty becomes 1/4 as large B. the momentum uncertainty becomes 1/2 as large C. the momentum uncertainty is unchanged D. the momentum uncertainty becomes twice as large E. the momentum uncertainty becomes 4 times larger

QM Clicker question 3 A photon has a position uncertainty of 2.00 mm. If you decrease the position uncertainty to 1.00 mm, how does this change the momentum uncertainty of the photon? A. the momentum uncertainty becomes 1/4 as large B. the momentum uncertainty becomes 1/2 as large C. the momentum uncertainty is unchanged D. the momentum uncertainty becomes twice as large E. the momentum uncertainty becomes 4 times larger

QM Clicker question 4 A beam of photons passes through a narrow slit. The photons land on a distant screen, forming a diffraction pattern. In order for a particular photon to land at the center of the diffraction pattern, it must pass A. through the center of the slit. B. through the upper half of the slit. C. through the lower half of the slit. D. impossible to decide

Homework assignments updated