Physics
325 Quiz 2 Mar. 3,
2003
Open
book and notes, calculator allowed. Show HOW you arrive at your answers. No
credit for random scratch work on the page. If you cannot get it to work out,
explain in words how you are going about it. M(proton)=1.6
x 10 –27 kg, M(electron)=9.1 x 10 –31 kg, h=6.6 x 10 –34
J-sec
1
eV=1.6 x –19 J.
1.
[25] Hydrogenlike iron consists of an electron orbiting an iron nucleus (Z=26).
For such a system, find (using the Bohr model for the ion)
a.
The energy of the ground state of the ion, in eV.
b.
The shortest wavelength radiation which the system could emit if it starts in
an excited state with n=3. What kind of
radiation is this (visible, infrared, …. ?)
c.
The radius of the orbit of the ground state.
d.
If the electron were replaced by a m meson (mass = 207 times the mass of an
electron) what would be the radius of the orbit in the ground state? How does
this compare to the radius of a nucleus? (Hint: the nucleus is still much
heavier than the orbiting particle.)
2.
[20] Two traveling waves of different wavelengths propagate in the positive x
direction on a string, given by y(x,t)=cos(10x-.10t) +
cos (11x-.12t) (x in m, t in sec, y in mm)
a)
Find the phase velocity of the resulting wave.
b)
Find the group velocity of the resulting wave.
c)
Find the distance between successive zeros of the “envelope” of the wave.
d)
Find the product of this distance and the “delta k” between the two waves.
3. [5] If a certain
measurement localizes an electron to within 0.2 nm, what is the minimum kinetic
energy the electron will have after the measurement is performed?
4.
[25] A proton is confined within an infinite square
well which has infinite walls
located at x= -L/2 and x=L/2, where L=10 fm.
a.
What is the ground state energy in MeV?
b.
Write the wave function for the ground state, normalized correctly (note that
the center of the well is at x=0).
c.
What energy photon, in MeV, is required to excite the proton from the ground
state to the third excited state in this well?
d.
If the infinitely high walls of the potential well were replaced by walls only
6 MeV high, would you expect the ground state energy to go up, go down or stay
the same? Why? To explain your answer, sketch the wave function for the
infinite well and again for the finite well and show how they
would differ.
e.
For the finite well, with 6 MeV high walls, find the ratio between the
probability for finding the proton within an interval of 0.01fm located right
at the wall of the well and that for finding it within a 0.01fm interval located
1 fm outside the well. Assume the proton is in its ground state and take the
ground state energy to be what you found for the infinite well. (hint: do not do any integrals and notice you only need a
ratio.)
5.
[15]A beam of electrons is incident from the left onto
a potential step which goes from 0 to 20 V at x=0. That is, the potential
energy of the electron is zero for x<0, 20 eV for x>0. If the electron
kinetic energy for x<0 is 10 eV,
a.
Sketch the wave function for x<0 and find the k (the incident momentum
divided by hbar) of this function, in inverse meters.
b.
Sketch the wave function for x>0 and find the distance over which the wave
function is reduced by 1/e.
c.
What fraction of the incident electrons reflect?