3.2 Problems
141
3.22 The small binding energy of the deuteron (2.2 MeV) implies that the maximum
of U (r ) lies just inside the range R of the well. From this knowledge deduce
the value of V 0 if R is approximately 1.5 fm.
[Osmania University]
3.23 Given that the normalized wave function
ψ =
1
r
α
2π
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 mass m e trapped in an infinite depth well of width L = 1 nm.
Consider the transition from the excited state n = 2 to the ground state n = 1.
Calculate the wavelength of light emitted. In which region of electromagnetic
spectrum does it fall?
3.25 Consider a particle of mass m trapped in a potential well of finite depth V 0
V (x) = V 0 , |x| > a
= 0; |x| < a
Discuss the solutions and eigen values for the class I and II solutions graphically.
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 of U (r )
lies only just inside the range R of the square well potential. Use this information to estimate the value of V 0 if R is approximately 1.5 fm.
3.30 Consider a stream of particles with energy E travelling in one dimension from
x = −∞ to ∞. The particles have an average spacing of distance L. The
particle stream encounters a potential barrier at x = 0. The potential can be
written as
V (x) = 0 if x < 0
= V if 0 < x < a
= 0 if x > a
Suppose the particle energy is smaller than the potential barrier, i.e., < V b .
(a) For each of the three regions, write down Schrodinger’s equation and calculate the wave-function ψ and its derivative dψ/dx.
141
3.22 The small binding energy of the deuteron (2.2 MeV) implies that the maximum
of U (r ) lies just inside the range R of the well. From this knowledge deduce
the value of V 0 if R is approximately 1.5 fm.
[Osmania University]
3.23 Given that the normalized wave function
ψ =
1
r
α
2π
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 mass m e trapped in an infinite depth well of width L = 1 nm.
Consider the transition from the excited state n = 2 to the ground state n = 1.
Calculate the wavelength of light emitted. In which region of electromagnetic
spectrum does it fall?
3.25 Consider a particle of mass m trapped in a potential well of finite depth V 0
V (x) = V 0 , |x| > a
= 0; |x| < a
Discuss the solutions and eigen values for the class I and II solutions graphically.
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 of U (r )
lies only just inside the range R of the square well potential. Use this information to estimate the value of V 0 if R is approximately 1.5 fm.
3.30 Consider a stream of particles with energy E travelling in one dimension from
x = −∞ to ∞. The particles have an average spacing of distance L. The
particle stream encounters a potential barrier at x = 0. The potential can be
written as
V (x) = 0 if x < 0
= V if 0 < x < a
= 0 if x > a
Suppose the particle energy is smaller than the potential barrier, i.e., < V b .
(a) For each of the three regions, write down Schrodinger’s equation and calculate the wave-function ψ and its derivative dψ/dx.