YSP Modern Physics – Summer 2012
Homework set Q3 -- due 3 July
3.1: test of correspondence principle:
consider an electron in a hypothetical macroscopic H– atom at a distance (radius of
orbit) of 1cm;
(a) according to classical electrodynamics,
 an electron moving in a circular orbit will radiate waves of frequency = its
frequency of revolution;
 calculate this frequency, using classical means (start with Coulomb force =
centripetal force, get speed of electron,..)
(b) Within the Bohr model,
 calculate the n-value for an electron at a radius of 1cm (use relationship
between Rn and Bohr radius ao).
 Calculate corresponding energy En
 calculate energy difference between state n and n-1, i.e. ΔE = En – En-1
(beware of rounding errors! Find algebraic expression for the difference, then
plug in numbers)
 calculate frequency of radiation emitted in transition from state n to state n-1
 compare with frequency from (a)
Needed formulae:
(1) Coulomb’s law:
 F = kq1q2/r2
(2) Centripetal force:
 F = mv2/r
(3) potential and kinetic energies of an electron for circular orbit:
2
2
2
 U = -kZe /r K = mev /2 = kZe /(2r)
2
 Total energy E = U + K = -kZe /(2r)
(4) radius for stationary orbit n
2 2
2
2
 Rn = n ħ /mekZe = n ao /Z
 a = ħ2/m ke2 = 0.53 x 10-10 m = “Bohr radius”
o
e
(5) Energy for stationary orbit n
2 2
2 4
2 2
2
2
 En = - k Z me e /2n ħ = - Z Eo /n
2
2 4
2
 Eo = k me e /2ħ = 13.6 eV
(6) values of constants
 k = 1/(4πε ) = 8.98· 109 N m2 /c2





o
me = 9.10938 ∙10-31 kg = 0.511 MeV/c2
-34
-21
h = 6.62607 · 10
J s = 4.1357 · 10
MeV s
-34
-22
ħ = h/2π = 1.0546 · 10
J s = 6.582 · 10
MeV s
-19
e = elementary charge = 1.602 · 10 C
Z = nuclear charge = 1 for hydrogen
3.2: uncertainty:
Consider two objects, both moving with velocity v = (100±0.01) m/s in the x-direction. This
tells you that the uncertainty of the velocity in the x-direction, vx = 0.01m/s. From this
you can calculate the uncertainty px , using the fact that px = mvx. Then, using the
uncertainty relation, calculate the uncertainty x in the x – coordinate,
assuming the object is
-31
2
(a) an electron (mass = 9.11x10 kg = 0.511 MeV/c )
(b) a ball of mass = 10 grams
(c) discuss (note that the discussion is the important part of this assignment)
 Does the electron behave like a “particle” in the Newtonian sense? Does the
ball?
 Why / why not?
 hint: compare position uncertainty with size of object; assume:
-18
 size of electron < 10 m,
 estimate the size (diameter) of the ball from its mass. To do this, use some
sensible value for the density of the ball. Hint: the density of water is 1g/cm3 .
Assume that the ball ‘s density is a bit more(e.g. twice, say) than that of
water.
 If the position uncertainty is << (much smaller) than the size of the object,
then it behaves similar to a classical particle. If the position uncertainty is of
the same order or bigger than the size of the object, then clearly the object
does not behave like something that we would call a particle in the classical
(Newtonian) sense.
Needed formulae:
(1) uncertainty relation: x•px ħ/2
(2) Volume of a sphere V=4πr3/3
(3) Mass vs density: density = mass/volume; volume = mass/density
(4) values of constants
a. me = 9.10938 ∙10-31 kg = 0.511 MeV/c2
-34
-21
b. h = 6.62607 · 10
J s = 4.1357 · 10
MeV s
-34
-22
c. ħ = h/2π = 1.0546 · 10
J s = 6.582 · 10
MeV s