Download quantum, relativistic and classical physics

THE UNIVERSITY OF HULL
Department of Physical Sciences (Physics)
Level 2 Examination
January 2008
QUANTUM, RELATIVISTIC AND CLASSICAL PHYSICS
(Module No. 04211)
TUESDAY 22 JANUARY 2008
2 hours (09.30 – 11.30)
Answer THREE questions, ONE from Section A and TWO from
Section B.
Do not open or turn over this exam paper, or start to write anything until
told to do so by the Invigilator. Starting to write before permitted to do so
may be seen as an attempt to use Unfair Means.
Module 04211
CONTINUED
Page 1 of 4
SECTION A: QUANTUM PHYSICS II
1.
The one-dimensional time independent form of Schrödinger’s equation for a particle is
given by the expression:
d 2 ψ(x) 2m
 2 (E  V) ψ(x)  0
dx 2

where the symbols have their usual meaning. When E > V this has the solution
(x) = A sin(kx) + B cos(kx)
where k =
2m( E  V )

.
(i) What do you understand by (x) in the Schrödinger equation and how does this relate
to the probability of finding the particle?
[3 marks]
(ii) Consider an electron of energy E travelling in the positive x direction in a region where
x < 0 and the potential V = 0. At the position x = 0 the electron strikes a potential barrier of
height V and width L. For x > L the potential falls to V = 0 again.
Assuming that the height of the barrier V exceeds the energy E of the electron solve
Schrödinger’s equation for the region within the barrier. Hence show that although it is
classically forbidden for the electron to penetrate into this region it is actually possible since
the probability amplitude decreases exponentially within the barrier according to
x   Be  k1x . Hence explain why it is also possible for the electron to tunnel through the
barrier to the region x > L.
[7 marks]
(iii) From your solution in (ii) show that the probability density within the barrier is
proportional to e-2kx and that the penetration distance x over which the probability density
falls by e-1 is given by

x 
2 2mV  E 
Hence explain why barrier penetration is not observed for macroscopic particles.
[5 marks]
(iv) Assuming that the electron has an incident energy of 5eV and that the barrier is of
height V = 10eV and thickness L = 5  10- 10 m, calculate the probability that the electron
will be able to tunnel through the barrier.
[5 marks]
[h = 6.63  10-34Js; mass of the electron = 9.1  10 –31 kg;
electron charge e = 1.6  10 –19 C]
Module 04211
CONTINUED
Page 2 of 4
2.
(i) With reference to suitable sketches discuss briefly what is meant by the spatial
quantization of the electron orbital angular momentum L and the electron spin angular
momentum S which occurs when an atom is placed in a magnetic field aligned along the
z direction. In this connection explain the role of the quantum numbers m and ms and
define how L and S are related to the corresponding quantum numbers  and s.
[7 marks]
(ii) Briefly note the experimental observations which made it necessary to introduce the
concept of electron spin. Describe the Stern-Gerlach experiment explaining how this
experiment proved the existence of electron spin.
[6 marks]
(iii) Describe how the fine structure splitting observed in hydrogen spectral lines arises
and, given the equations below, show that for hydrogen the energy splitting in a field B is
given by E = 2BB. Hence calculate the splitting in the n = 2 level of a hydrogen atom if
the internal magnetic field in the n = 2 state is 0.4T.
[You may assume that the z component of the spin magnetic dipole moment is given by the
equation sz = -gsBms and also that the potential energy of orientation of a magnetic dipole
of moment  making an angle  to a magnetic field B is given by E = -Bcos.
The Bohr Magneton B = 9.27  10-24 JT-1; the spin g-factor gs = 2]
[7 marks]
SECTION B: MECHANICS AND RELATIVITY
3.
(i) The Lorentz transformation equations are
x    x  vt, y   y, z   z, t    t  vx / c 2 ,

where   1  v 2 / c 2

1 / 2
. What assumptions are made in deriving this transformation?
[4 marks]
(ii) A rod of proper length  o is at rest in a frame K . It lies in the x y  plane and makes
an angle sin-1(3/5) with the x axis. If K  moves with constant velocity v parallel to the
x axis of another frame K, and if, when the length of the rod is measured in K, its end
points are observed simultaneously,
(a) what is the value of v if, as measured in K, the rod makes an angle  / 4 with
the x axis?
[12 marks]
(b)
What is the length of the rod as measured in K under these circumstances?
[4 marks]
Module 04211
CONTINUED
Page 3 of 4
4.
(i) A particle of unit mass is moving in a plane under an attractive force F (u ) , where
u  1 / r , directed towards the origin O. If the radial and transverse components of
acceleration are
1d 2
r  r 2 and
( r  ),
r dt
where r and  are plane polar coordinates show that the differential equation of the
orbit is
d 2u
F (u )
u   2 2 ,
2
dt
h u
where h is the angular momentum of the particle about O.
[8 marks]
(ii) If F (u )  au 2  bu 3 and h 2  b, show that the equation of the orbit is
1/ 2


a
b 


u 2
 A cos 1  2     ,
h b


 h 

where A and  are constants.
[4 marks]
(iii) If the particle is projected from a point at a distance b / 4a from the centre of force
with a velocity 8a /( 3b)1/ 2 perpendicular to the radius vector, show that the equation of the
orbit is


bu  a 3  cos  ,
2
where  is measured from the initial direction of the radius vector to the particle.
[8 marks]
5.
(i) Define moment of inertia and show that the moment of inertia of a circular disc, of
mass m and radius a, about its axis is ma 2 / 2.
[8 marks]
(ii) Two metal discs, one of mass 0.8kg and radius 25mm and the other of mass 1.6kg
and radius 50mm, are welded together and mounted on a frictionless axle through their
common centre. What is the total moment of inertia of the two discs?
[6 marks]
(iii) If in part (ii) a light string is wrapped around the edge of the smaller disc and a
mass of 1.5kg is suspended from the free end of that string, find the speed of the mass
immediately before it strikes the floor if it is released from a height of 2m above the floor.
Assume the axis of the axle is horizontal.
[6 marks]
[g = 9.81ms-2]
Module 04211
END
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