3.2 Problems 141
3.22 The small binding energy of the deuteron (2.2 MeV) implies that the maximum
ofU(r) lies just inside the rangeRof the well. From this knowledge deduce
the value ofV 0 ifRis approximately 1.5 fm.
[Osmania University]
3.23 Given that the normalized wave function
ψ=(
1
r)(
α
2 π) 1 / 2
e−αr(1/α= 4 .3 fm) is a useful approximation to describe the ground state of the
deuteron, find the root mean square separation of the neutron and proton in
this nucleus.
[University of Durham 1972]3.24 A particle of massmetrapped in an infinite depth well of widthL=1nm.
Consider the transition from the excited staten=2 to the ground staten=1.
Calculate the wavelength of light emitted. In which region of electromagnetic
spectrum does it fall?
3.25 Consider a particle of massmtrapped in a potential well of finite depthV 0
V(x)=V 0 ,|x|>a
=0;|x|<a
Discuss the solutions and eigen values for the class I and II solutions graphi-
cally.3.26 Show that for deuteron, neutron and proton stay outside the range of nuclear
forces for 70% of the time. Take the binding energy of deuteron as 2.2 MeV.
3.27 Show that the results of the energy levels for infinite well follow from those
for the finite well.
3.28 Show that for deuteron excited states are not possible.
3.29 The small binding energy of the deuteron indicates that the maximum ofU(r)
lies only just inside the rangeRof the square well potential. Use this informa-
tion to estimate the value ofV 0 ifRis approximately 1.5 fm.
3.30 Consider a stream of particles with energyEtravelling in one dimension from
x =−∞to∞. The particles have an average spacing of distanceL.The
particle stream encounters a potential barrier atx=0. The potential can be
written as
V(x)=0ifx< 0
=Vif 0<x<a
=0ifx>a
Suppose the particle energy is smaller than the potential barrier, i.e.,<Vb.
(a) For each of the three regions, write down Schrodinger’s equation and cal-
culate the wave-functionψand its derivative dψ/dx.