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Quantum Physics I (8.04) Spring 2016
Assignment 2
Massachusetts Institute of Technology Physics Department Due Thu. February 18, 2016 February 11, 2016 5:00pm
Problem Set 2
- de Broglie wavelength [20 points]
(a) The de Broglie wavelength of a non-relativistic (nr) electron with kinetic energy Ekin can be written as as δ λnr = √^ ˚A. Ekin In this formula δ is a unit-free constant, and the value of the energy Ekin is entered in eV as a pure number. The answer comes out in Angstroms (˚A = 10 −^10 m). Give the value of the unit-free constant δ. (b) The de Broglie wavelength of a relativistic (r) electron with energy E can be calculated in terms of the γ factor of the electron: E = γmec^2. One finds
ℓ λr = √. γ^2 − 1
What is the value of ℓ in fm = 10−^15 m? Is this a well-known length? (c) Rewrite the expression for λnr in (a) in terms of ℓ and γ, using Ekin = (γ − 1)m (^) ec^2. Demonstrate that λr < λnr for any energy. (d) A few numerical calculations: i. What is the energy of an electron whose de Broglie wavelength is equal to its Compton wavelength? Is that electron relativistic: Is it moving faster than 0. 2 c? ii. The de Broglie wavelength of a particle gives you the rough idea of the dis- tance scale it can explore in a collision experiment. The International Linear Collider, which may be built in the near future, is expected to accelerate electrons to 1 TeV = 1000 GeV. What is the de Broglie wavelength of such electrons? Compare to the de Broglie wavelength of 7 TeV protons at the LHC at Geneva. iii. What is the maximum electron kinetic energy, and the associated β = v/c, for which the non-relativistic value of λ (in (a) or (c)) has an error less than or equal to 10%?
Physics 8.04, Quantum Physics 1, Spring 2016 2
- Bohr radius, electron Compton wavelength, and classical electron radius. [10 points] The classical electron radius r 0 is the radius obtained by setting the electrostatic energy associated to a charged ball of radius r 0 equal (up to constant factors) to the rest energy of the electron
e^2 r 0 = mec^2 → r 0 = e^2 . mec^2
Here e is the charge of the electron. The bar-Compton wavelength λ¯C of the electron is ℏ λ¯C =. mec Finally, the fine structure constant α, which measures the strength of the electromag- netic coupling is e^2 α = ℏc
(a) The Bohr radius a 0 is the length scale that can be constructed from e^2 , ℏ, and me and no extra numerical constants. Find the formula for the Bohr radius by consideration of units. Evaluate this length in terms of fm. (b) Show that the three lengths form a geometric sequence with ratio α:
a 0 : λ¯C : r 0 = 1 : α : α^2.
Use this to give the values of λ¯C and r 0 in fm.
- Two-by-two matrices and linear devices. [10 points] Consider the two-beam Mach-Zender interferometer and a beam represented by the two-component column vector u:
u =
u 1
, with |u 1 |^2 + u u 2
| 2 |^2 = 1.
Any linear optical element in the interferometer can be represented by a two-by-two matrix R such that with input u beam the output is a u′^ beam given by
u′^ = R u.
Show that conservation of probability for arbitrary u requires that R be a unitary matrix. A (finite size) matrix R is said to be unitary if R†R = 1 , where dagger denotes the operation of transposition and complex conjugation.
Physics 8.04, Quantum Physics 1, Spring 2016 4
- Plane waves for matter particles. [10 points] Assume we want to represent the wave for a matter particle moving in the x direction with momentum p = ℏk. A reasonable guess for such a wave is
Ψ(x, t) = cos(kx − ωt) + γ sin(kx − ωt) ,
where γ is a constant. A physical requirement is that an arbitrary displacement of x or an arbitrary shift of t should not alter the character of the wave. We will demand therefore that after the shift, whose effect is to change the phase by some constant ǫ, we have
cos(kx − ωt + ǫ) + γ sin(kx − ωt + ǫ) = a
[
cos(kx − ωt) + γ sin(kx − ωt)]
for some constant a that may depend on ǫ. Write the equations that follow from the above requirement. Find the two possible solutions for γ and the associated a. Which is the solution that corresponds to our conventional description of a matter wave?
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