PHYS2013 Quantum Physics Problem Set 1 (2015) Due Monday 2nd March, 5pm, please submit your solutions in a PDF document via the Wattle site. Please attach a completed cover sheet to your work.

All late submissions of homework will be penalised as explained on the Wattle site.

For an item to be marked correct you must supply working and an explanation as well as the correct answer. These must be complete and legible, otherwise zero marks may be given.

1. Stability of a classical atom.This problem requires recalling your understanding of classical physics.

8 marks. 1 mark each for (a) through (h). 1 mark if completely correct. Otherwise 0 or at the markers discretion.

By the end of the 19th century it was known that as a consequence of Maxwells equations an accelerating charge q loses energy E at the rate given by the Larmor formula:

dEdt

= 23

q2

40a2

c3,

where a is the magnitude of the acceleration. Consider a classical electron in a circular orbit of radius r about a proton, i.e. a hydrogen atom model. Assume non-relativistic physics.

(a) Find the kinetic energy of the electron as a function of radius. (b) Find the magnitude of the acceleration of the electron as a function of radius. (c) Find the total energy of the system as a function of radius. (d) Find the power radiated as a function of radius. (e) Find the rate of decrease of the radius, as a function of radius. This is a differential equation for r. Observe that the power radiated increases as the electron spirals into the proton. (f) For r = 5x10-10 m calculate values for all these quantities. (g) Solve the differential equation you found in (e) for r as a function of time t. Note that:

r2 drdt

= 13dr3

dt.

(h) Hence estimate the time it takes the electron to spiral into the proton from an initial orbit of radius r = 5x10-10 m. How does this compare to a typical experimentally measured excited 2p state lifetime of about 1.6 ns?

2. Fourier transforms.

3 marks. 1 mark each for part completely correct.

(a) Calculate the Fourier transform p( ) = 12 x( )e ipx/ dx

of the square wave function x( ) =1d for d / 2 x d / 2

0 elsewhere

(b) & (c) Plot or sketch each function.

3. (a) In thermal equilibrium the average kinetic energy of a particle is

p2

2m=

3

2kBT

where T is in Kelvin and kB is Boltzmanns constant. The lattice spacing in atypical solid is 0.3 nm. For what temperatures do we have to treat (a) electronsand (b) nuclei in solids quantum mechanically?

(b) Using the general gas law

PV = NkBT

to deduce the interatomic spacing, find out at what temperatures we need totreat atoms in an ideal gas at pressure P as quantum mechanical. (Answer:T < (1/kB)(h

2/3m)3/5P 2/5). Is the helium inside a childs balloon quantummechanical?

(c) At Lawrence Livermore National Laboratory in the USA, there is a facilityfor accelerating electrons to energies of about 100 MeV. Ignoring relativisticeffects, calculate the de Broglie wavelength of these electrons.

(d) The Large Electron-Positron collider (LEP) at CERN in Europe acceleratedelectrons to energies of around 100 GeV. Again ignoring relativistic effects,calculate the de Broglie wavelength of these electrons.

(e) Do you think the non-relativistic approximation is justified in either case?

(5 marks)

4. Show that a stationary electron cannot absorb all the energy of a photon (hint: consider conservation of energy and momentum). Is it sensible to worry about stationary electrons? (3 marks)

5. Often we want to calculate the average of a quantity which doesnt take on discretevalues, such as the position of a particle along the x-axis, or the maximum frequencyaudible to lecturers in the faculty of science. Then the probability of measuring avalue of x in the interval between a and b is

P (a < x < b) = b

a(x)dx (5)

Here (x) is the probability density. This has to be normalized so that the integral of(x) from to + is equal to 1 this is equivalent to requiring that the particlehas to be somewhere. The equations for the mean etc then become

x = +

x(x)dx (6)

f(x) = +

f(x)(x)dx (7)

2 = x2 x2 (8)

Consider a Gaussian distribution,

(x) = Ae(c(xa)2)

where a, A and c are constants. Such distributions are extremely common in allareas of science (and social science), and also play an important role in quantummechanics, so it is good to familiarise yourself with them.

Looking up any integrals you need (i.e. from sos maths, mathematica, or other similiar sources),

(a) Sketch a graph of the probability density. (1 marks)(b) Evaluate A in terms of the other constants. (2 marks)

(c) Find x, x2 and . (3 marks)