1. Quantum Two

3. Bound States of a Central Potential

4. Bound States of a Central Potential General Considerations

5. We consider now the properties of a particle of mass m attracted to a force center derived from a fixed spherically symmetric potential V (r).
In most cases, such a problem arises from a system of two particles, with a Hamiltonian
in which the potential energy corresponds to an attractive central force that depends only on the separation distance between the two particles.
After a transformation to center of mass and relative coordinates
R=∑_{i}m_{i}r_{i}/M r=r₂-r₁,
the Hamiltonian in such circumstances takes a form . . .

6. We consider now the properties of a particle of mass m attracted to a force center derived from a fixed spherically symmetric potential V (r).
In most cases, such a problem arises from a system of two particles, with a Hamiltonian
which describes an attractive central force that depends only on the separation distance between the two particles.
After transforming to center of mass and relative coordinates
the Hamiltonian in such circumstances takes a form . . .

7. We consider now the properties of a particle of mass m attracted to a force center derived from a fixed spherically symmetric potential V (r).
In most cases, such a problem arises from a system of two particles, with a Hamiltonian
which describes an attractive central force that depends only on the separation distance between the two particles.
After transforming to center of mass and relative coordinates
the Hamiltonian in such circumstances takes a form . . .

8. We consider now the properties of a particle of mass m attracted to a force center derived from a fixed spherically symmetric potential V (r).
In most cases, such a problem arises from a system of two particles, with a Hamiltonian
which describes an attractive central force that depends only on the separation distance between the two particles.
After transforming to center of mass and relative coordinates
the Hamiltonian in such circumstances takes a form . . .

9. We consider now the properties of a particle of mass m attracted to a force center derived from a fixed spherically symmetric potential V (r).
In most cases, such a problem arises from a system of two particles, with a Hamiltonian
which describes an attractive central force that depends only on the separation distance between the two particles.
After transforming to center of mass and relative coordinates
the Hamiltonian in such circumstances takes a form . . .

10. identical to that of a system of two non-interacting particles.
The first (associated with the motion of the center of mass) has the same form as that of a free particle with a mass and a momentum
equal to the total mass and total momentum of the original two particle system.
The second, associated with the relative motion of the two particles is equivalent to that of a single particle of reduced mass
and momentum p, moving in a spherically symmetric attractive potential V(r) centered at the origin.

11. identical to that of a system of two non-interacting particles.
The first (associated with the motion of the center of mass) has the same form as that of a free particle with a mass and a momentum
equal to the total mass and total momentum of the original two particle system.
The second, associated with the relative motion of the two particles is equivalent to that of a single particle of reduced mass
and momentum p, moving in a spherically symmetric attractive potential V(r) centered at the origin.

12. identical to that of a system of two non-interacting particles.
The first (associated with the motion of the center of mass) has the same form as that of a free particle with a mass and a momentum
equal to the total mass and total momentum of the original two particle system.
The second, associated with the relative motion of the two particles is equivalent to that of a single particle of reduced mass
and momentum p, moving in a spherically symmetric attractive potential V(r) centered at the origin.

13. identical to that of a system of two non-interacting particles.
The first (associated with the motion of the center of mass) has the same form as that of a free particle with a mass and a momentum
equal to the total mass and total momentum of the original two particle system.
The second, associated with the relative motion of the two particles is equivalent to that of a single particle of reduced mass
and momentum p, moving in a spherically symmetric attractive potential V(r) centered at the origin.

14. identical to that of a system of two non-interacting particles.
The first (associated with the motion of the center of mass) has the same form as that of a free particle with a mass and a momentum
equal to the total mass and total momentum of the original two particle system.
The second, associated with the relative motion of the two particles is equivalent to that of a single particle of reduced mass
and momentum p, moving in a spherically symmetric attractive potential V(r) centered at the origin.

15. After factoring out the center of mass motion, the corresponding quantum system associated with the relative motion is governed by a Hamiltonian
Our goal is information about the bound state solutions |φ〉 to the energy eigenvalue equation , which in the position representation takes the standard form
We assume that sufficiently fast as , so that bound state solutions are identified as those square normalizable solutions
for which ε ≤ 0.

16. After factoring out the center of mass motion, the corresponding quantum system associated with the relative motion is governed by a Hamiltonian
Our goal is information about the bound state solutions |φ〉 to the energy eigenvalue equation
which in the position representation takes the standard form

17. After factoring out the center of mass motion, the corresponding quantum system associated with the relative motion is governed by a Hamiltonian
Our goal is information about the bound state solutions |φ〉 to the energy eigenvalue equation
which in the position representation takes the standard form

18. We assume that sufficiently fast as ,
so that bound state solutions are identified as those square normalizable solutions
for which ε ≤ 0.

19. We assume that sufficiently fast as ,
so that bound state solutions are identified as those square normalizable solutions
with negative energy ε ≤ 0.

20. The spherical symmetry of (and thus of H) suggests the use of spherical coordinates for which the identity
holds everywhere except at r = 0.
This last expression is equivalent to the relation
in which
is associated with the square of the (dimensionless) orbital angular momentum

21. The spherical symmetry of (and thus of H) suggests the use of spherical coordinates for which the identity
holds everywhere except at r = 0.
This last expression is equivalent to the relation
in which
is associated with the square of the (dimensionless) orbital angular momentum

22. The spherical symmetry of (and thus of H) suggests the use of spherical coordinates for which the identity
holds everywhere except at r = 0.
This last expression is equivalent to the relation
in which
is associated with the square of the (dimensionless) orbital angular momentum

23. The spherical symmetry of (and thus of H) suggests the use of spherical coordinates for which the identity
holds everywhere except at r = 0.
This last expression is equivalent to the relation
in which
represents the square of the (dimensionless) orbital angular momentum operator

24. Thus, as in classical mechanics, the kinetic energy
separates into: a radial part, and a rotational part, in which
represents the (properly Hermitian-symmetrized) radial component
of the momentum.

25. Thus, as in classical mechanics, the kinetic energy
separates into: a radial part, and a rotational part, in which
represents the (properly Hermitian-symmetrized) radial component
of the momentum.

26. Thus, as in classical mechanics, the kinetic energy
separates into: a radial part, and a rotational part, in which
represents the (properly Hermitian-symmetrized) radial component
of the momentum.

27. Thus, as in classical mechanics, the kinetic energy
separates into: a radial part, and a rotational part, in which
represents the (properly Hermitian-symmetrized) radial component
of the momentum.

28. Thus, as in classical mechanics, the kinetic energy
separates into: a radial part, and a rotational part, in which
represents the (properly Hermitian-symmetrized) radial component
of the momentum.

29. Thus, as in classical mechanics, the kinetic energy
separates into: a radial part, and a rotational part, in which
represents the (properly Hermitian-symmetrized) radial component
of the momentum.

30. Indeed, with the formulae above we can write the energy eigenvalue equation of interest in the position representation in the form
We now make the observation that the Hamiltonian is a scalar valued function of the scalar observables and
The Hamiltonian is therefore a scalar with respect to rotations, itself.
Thus, as is easily confirmed directly, it commutes with and (or any other component of ).
Thus, since [H,ℓ²]=[ℓ²,ℓ_{z}]=[H,ℓ_{z}]=0
we know there exists a basis of eigenstates common to the three operators , , and .

31. Indeed, with the formulae above we can write the energy eigenvalue equation of interest in the position representation in the form
We now make the observation that the Hamiltonian is a scalar valued function of the scalar observables and
The Hamiltonian is therefore a scalar with respect to rotations, itself.
Thus, as is easily confirmed directly, it commutes with and (or any other component of ).
Thus, since [H,ℓ²]=[ℓ²,ℓ_{z}]=[H,ℓ_{z}]=0
we know there exists a basis of eigenstates common to the three operators , , and .

32. Indeed, with the formulae above we can write the energy eigenvalue equation of interest in the position representation in the form
We now make the observation that the Hamiltonian is a scalar valued function of the scalar observables and
The Hamiltonian is itself, therefore, a scalar with respect to rotations.
Thus, as is easily confirmed directly, it commutes with and (or any other component of ).
Thus, since [H,ℓ²]=[ℓ²,ℓ_{z}]=[H,ℓ_{z}]=0
we know there exists a basis of eigenstates common to the three operators , , and .

33. Indeed, with the formulae above we can write the energy eigenvalue equation of interest in the position representation in the form
We now make the observation that the Hamiltonian is a scalar valued function of the scalar observables and
The Hamiltonian is itself, therefore, a scalar with respect to rotations.
Thus, as is easily confirmed directly, it commutes with and (or any other component of ).
Thus, since [H,ℓ²]=[ℓ²,ℓ_{z}]=[H,ℓ_{z}]=0
we know there exists a basis of eigenstates common to the three operators , , and .

34. Indeed, with the formulae above we can write the energy eigenvalue equation of interest in the position representation in the form
We now make the observation that the Hamiltonian is a scalar valued function of the scalar observables and
The Hamiltonian is itself, therefore, a scalar with respect to rotations.
Thus, as is easily confirmed directly, it commutes with and (or any other component of ).
Thus, since [H,ℓ²]=[ℓ²,ℓ_{z}]=[H,ℓ_{z}]=0
we know there exists a basis of eigenstates common to the three operators , , and .

35. Indeed, with the formulae above we can write the energy eigenvalue equation of interest in the position representation in the form
We now make the observation that the Hamiltonian is a scalar valued function of the scalar observables and
The Hamiltonian is itself, therefore, a scalar with respect to rotations.
Thus, as is easily confirmed directly, it commutes with and (or any other component of ).
Thus, since [H,ℓ²]=[ℓ²,ℓ_{z}]=[H,ℓ_{z}]=0
we know there exists a basis of eigenstates common to the three operators , , and .

36. These common eigenstates, which we will denote by , satisfy the eigenvalue equations:
and we denote by
the wave functions in the position representation associated with these states.
Note that the eigenvalues of H are labeled by the principle quantum number n and by the total angular momentum quantum number ℓ, but not by the azimuthal quantum number m, which for a given value of ℓ takes on the values m=-ℓ,…,+ℓ

37. These common eigenstates, which we will denote by , satisfy the eigenvalue equations:
and we denote by
the wave functions in the position representation associated with these states.
Note that the eigenvalues of H are labeled by the principle quantum number n and by the total angular momentum quantum number ℓ, but not by the azimuthal quantum number m, which for a given value of ℓ takes on the values m=-ℓ,…,+ℓ

38. These common eigenstates, which we will denote by , satisfy the eigenvalue equations:
and we denote by
the wave functions in the position representation associated with these states.
Note that the eigenvalues of H are labeled by the principle quantum number n and by the total angular momentum quantum number ℓ, but not by the azimuthal quantum number m, which for a given value of ℓ takes on the values m=-ℓ,…,+ℓ

39. These common eigenstates, which we will denote by , satisfy the eigenvalue equations:
the wave functions in the position representation associated with these states.
Note that the eigenvalues of H are labeled by the principle quantum number n and by the total angular momentum quantum number ℓ, but not by the azimuthal quantum number m, which for a given value of ℓ takes on the values m=-ℓ,…,+ℓ

40. These common eigenstates, which we will denote by , satisfy the eigenvalue equations:
the wave functions in the position representation associated with these states.
Note that the eigenvalues of H are labeled by the principle quantum number n and by the total angular momentum quantum number ℓ, but not by the azimuthal quantum number m, which for a given value of ℓ takes on the values m=-ℓ,…,+ℓ

41. Each eigenvalue is therefore (at least) fold degenerate, reflecting the rotational degeneracy associated with the scalar operator
It is also clear from our earlier discussions that these common eigenfunctions of and can be written in a form
involving the spherical harmonics, with , and
Substitution of this assumed form into the energy eigenvalue equation results in the following ordinary differential equation
for the radial functions

42. Each eigenvalue is therefore (at least) fold degenerate, reflecting the rotational degeneracy associated with the scalar operator
It is also clear from our earlier discussions that these common eigenfunctions of and can be written in a form
where the denote the spherical harmonics, with , and
Substitution of this assumed form into the energy eigenvalue equation results in the following ordinary differential equation
for the radial functions

43. Each eigenvalue is therefore (at least) fold degenerate, reflecting the rotational degeneracy associated with the scalar operator
It is also clear from our earlier discussions that these common eigenfunctions of and can be written in a form
where the denote the spherical harmonics, with , and
Substitution of this assumed form into the energy eigenvalue equation results in the following ordinary differential equation
for the radial functions

44. Each eigenvalue is therefore (at least) fold degenerate, reflecting the rotational degeneracy associated with the scalar operator
It is also clear from our earlier discussions that these common eigenfunctions of and can be written in a form
where the denote the spherical harmonics, with , and
Substitution of this assumed form into the energy eigenvalue equation results in the following ordinary differential equation
for the radial functions

45. Each eigenvalue is therefore (at least) fold degenerate, reflecting the rotational degeneracy associated with the scalar operator
It is also clear from our earlier discussions that these common eigenfunctions of and can be written in a form
where the denote the spherical harmonics, with , and
Substitution of this assumed form into the energy eigenvalue equation results in the following ordinary differential equation
for the radial functions

46. This radial equation is independent of the eigenvalue m of , but does depend on , again confirming our labeling of the eigenvalues based upon their expected rotational degeneracy.
But since the differential equation does, generally, depend upon the quantum number ℓ, we can expect a different series of eigenvalues for each value of that quantum number.
So-called additional accidental degeneracies can arise, however.
They do arise for the 3D harmonic oscillator:
and for the Coulomb potential of a point charge:

47. This radial equation is independent of the eigenvalue m of , but does depend on , again confirming our labeling of the eigenvalues based upon their expected rotational degeneracy.
Indeed, since the differential equation does, generally, depend upon the quantum number ℓ we can, in general, expect a different series of eigenvalues for each value of that quantum number.
So-called additional accidental degeneracies can arise, however.
They do arise for the 3D harmonic oscillator:
and for the Coulomb potential of a point charge:

48. This radial equation is independent of the eigenvalue m of , but does depend on , again confirming our labeling of the eigenvalues based upon their expected rotational degeneracy.
Indeed, since the differential equation does, generally, depend upon the quantum number ℓ we can, in general, expect a different series of eigenvalues for each value of that quantum number.
So-called additional accidental degeneracies can arise, however.
They do arise for the 3D harmonic oscillator:
and for the Coulomb potential of a point charge:

49. This radial equation is independent of the eigenvalue m of , but does depend on , again confirming our labeling of the eigenvalues based upon their expected rotational degeneracy.
Indeed, since the differential equation does, generally, depend upon the quantum number ℓ we can, in general, expect a different series of eigenvalues for each value of that quantum number.
So-called additional accidental degeneracies can arise, however.
They do arise for the 3D harmonic oscillator:
and for the Coulomb potential of a point charge:

50. This radial equation is independent of the eigenvalue m of , but does depend on , again confirming our labeling of the eigenvalues based upon their expected rotational degeneracy.
Indeed, since the differential equation does, generally, depend upon the quantum number ℓ we can, in general, expect a different series of eigenvalues for each value of that quantum number.
So-called additional accidental degeneracies can arise, however.
They do arise for the 3D harmonic oscillator:
and for the Coulomb potential of a point charge:

51. As a further simplification it is useful to introduce a new “wave function”
which obeys the equation
Note that, like, itself, the new radial "wave function“ is defined only for r > 0.

52. As a further simplification it is useful to introduce a new “wave function”
which, after multiplying the equation above by the radial variable is found to obey
Note that, like, itself, the new radial "wave function“ is defined only for r > 0.

53. As a further simplification it is useful to introduce a new “wave function”
which, after multiplying the equation above by the radial variable is found to obey a 1D looking Schrodinger equation
Note that, like itself, the new radial "wave function“ is defined only for r > 0.

54. As a further simplification it is useful to introduce a new “wave function”
which, after multiplying the equation above by the radial variable is found to obey a 1D looking Schrodinger equation
Note that, like itself, the new radial "wave function“ is defined only for r > 0.

55. As a further simplification it is useful to introduce a new “wave function”
which, after multiplying the equation above by the radial variable is found to obey a 1D looking Schrodinger equation
Note that, like itself, the new radial "wave function“ is defined only for r > 0.

56. Moreover, the normalization condition on the total wave function
leads, along with the standard normalization condition for the spherical harmonics
to a normalization condition
for the functions identical to that of to a particle confined to the positive real axis.

57. Moreover, the normalization condition on the total wave function
leads, along with the standard normalization condition for the spherical harmonics
to a normalization condition
for the functions identical to that of to a particle confined to the positive real axis.

58. Moreover, the normalization condition on the total wave function
leads, along with the standard normalization condition for the spherical harmonics
to a normalization condition for the functions
that is identical to that of a particle in 1D, constrained to move only on the positive real axis.

59. Moreover, the normalization condition on the total wave function
leads, along with the standard normalization condition for the spherical harmonics
to a normalization condition for the functions
that is identical to that of a particle in 1D, constrained to move only on the positive real axis.

60. Moreover, the normalization condition on the total wave function
leads, along with the standard normalization condition for the spherical harmonics
to a normalization condition for the functions
that is identical to that of a particle in 1D, constrained to move only on the positive real axis.

61. As we will see, the boundary condition obeyed by these functions for most cases of interest corresponds to one in which and as as though there were an infinite barrier at the origin confining the particle to the region , where it obeys an "effective Schrödinger equation" of the form
equivalent to that of a particle moving on the positive real axis in an effective 1- dimensional potential
consisting of the original central potential to which has been added the so- called "centrifugal barrier"
which is actually just the rotational kinetic energy of the particle.

62. As we will see, the boundary condition obeyed by these functions for most cases of interest corresponds to one in which and as as though there were an infinite barrier at the origin confining the particle to the region , where it obeys an "effective Schrödinger equation" of the form
equivalent to that of a particle moving on the positive real axis in an effective 1- dimensional potential
consisting of the original central potential to which has been added the so- called "centrifugal barrier"
which is actually just the rotational kinetic energy of the particle.

63. As we will see, the boundary condition obeyed by these functions for most cases of interest corresponds to one in which and as as though there were an infinite barrier at the origin confining the particle to the region , where it obeys an "effective 1D Schrödinger equation" which we can write as
equivalent to that of a particle moving on the positive real axis in an effective 1- dimensional potential
consisting of the original central potential to which has been added the so- called "centrifugal barrier"
which is actually just the rotational kinetic energy of the particle.

64. As we will see, the boundary condition obeyed by these functions for most cases of interest corresponds to one in which and as as though there were an infinite barrier at the origin confining the particle to the region , where it obeys an "effective 1D Schrödinger equation" which we can write as
equivalent to that of a particle moving on the positive real axis in an effective 1- dimensional potential
consisting of the original central potential to which has been added the so- called "centrifugal barrier"
which is actually just the rotational kinetic energy of the particle.

65. As we will see, the boundary condition obeyed by these functions for most cases of interest corresponds to one in which and as as though there were an infinite barrier at the origin confining the particle to the region , where it obeys an "effective 1D Schrödinger equation" which we can write as
equivalent to that of a particle moving on the positive real axis in an effective 1- dimensional potential
consisting of the original central potential to which has been added the so- called "centrifugal barrier"
which is actually just the rotational kinetic energy of the particle.

66. This effective 1D potential and the contributions to it for non-zero are schematically indicated below:

67. To understand the consequences of the effective potential on bound state solutions to the Schrödinger equation, we consider general properties of its solution for small values of the radial coordinate r.
This is a particularly useful first step in solving this equation using the usual “power series method”, since it allows us to figure out the lowest non-vanishing power appearing in a solution to the radial equation expanded in powers of the radial coordinate.
As a first step, we rewrite the effective eigenvalue equation in the form
where
and
and where we have assumed, appropriate to bound states, that

68. To understand the consequences of the effective potential on bound state solutions to the Schrödinger equation, we consider general properties of its solution for small values of the radial coordinate r.
This is a particularly useful first step in solving this equation using the usual “power series method”, since it allows us to figure out the lowest non-vanishing power appearing in a solution to the radial equation expanded in powers of the radial coordinate.
As a first step, we rewrite the effective eigenvalue equation in the form
where
and
and where we have assumed, appropriate to bound states, that

69. To understand the consequences of the effective potential on bound state solutions to the Schrödinger equation, we consider general properties of its solution for small values of the radial coordinate r.
This is a particularly useful first step in solving this equation using the usual “power series method”, since it allows us to figure out the lowest non-vanishing power appearing in a solution to the radial equation expanded in powers of the radial coordinate.
As a first step, we rewrite the effective eigenvalue equation in the form
where
and
and where we have assumed, appropriate to bound states, that

70. To understand the consequences of the effective potential on bound state solutions to the Schrödinger equation, we consider general properties of its solution for small values of the radial coordinate r.
This is a particularly useful first step in solving this equation using the usual “power series method”, since it allows us to figure out the lowest non-vanishing power appearing in a solution to the radial equation expanded in powers of the radial coordinate.
As a first step, we rewrite the effective eigenvalue equation in the form
where
and
and where we have assumed, appropriate to bound states, that

71. We further assume that:
V(r) is bounded except possibly near r = 0, and that
|V(r)| ≤ Mr⁻¹ as for some positive constant M.
In other words, the magnitude of V diverges at the origin no more quickly than the Coulomb potential.
With these assumptions, we now seek solutions that near the origin take the form of a power law in r, i.e., we assume that as
for some positive constants C and s.
Note that this corresponds to the actual radial wave function having the limiting behavior

72. We further assume that:
V(r) is bounded everywhere except possibly near r = 0, and that
|V(r)| ≤ Mr⁻¹ as for some positive constant M.
In other words, the magnitude of V diverges at the origin no more quickly than the Coulomb potential.
With these assumptions, we now seek solutions that near the origin take the form of a power law in r, i.e., we assume that as
for some positive constants C and s.
Note that this corresponds to the actual radial wave function having the limiting behavior

73. We further assume that:
V(r) is bounded everywhere except possibly near r = 0, and that
|V(r)| ≤ Mr⁻¹ as for some positive constant M.
In other words, the magnitude of V diverges at the origin no more quickly than the Coulomb potential.
With these assumptions, we now seek solutions that near the origin take the form of a power law in r, i.e., we assume that as
for some positive constants C and s.
Note that this corresponds to the actual radial wave function having the limiting behavior

74. We further assume that:
V(r) is bounded everywhere except possibly near r = 0, and that
|V(r)| ≤ Mr⁻¹ as for some positive constant M.
In other words, the magnitude of V diverges at the origin no more quickly than the Coulomb potential.
With these assumptions, we now seek solutions that near the origin take the form of a power law in r, i.e., we assume that as
for some positive constants C and s.
Note that this corresponds to the actual radial wave function having the limiting behavior

75. We further assume that:
V(r) is bounded everywhere except possibly near r = 0, and that
|V(r)| ≤ Mr⁻¹ as for some positive constant M.
In other words, the magnitude of V diverges at the origin no more quickly than the Coulomb potential.
With these assumptions, we now seek solutions that near the origin take the form of a power law in r, i.e., we assume that as
for some constants C and s.
Note that this corresponds to the actual radial wave function having the limiting behavior

76. We further assume that:
V(r) is bounded everywhere except possibly near r = 0, and that
|V(r)| ≤ Mr⁻¹ as for some positive constant M.
In other words, the magnitude of V diverges at the origin no more quickly than the Coulomb potential.
With these assumptions, we now seek solutions that near the origin take the form of a power law in r, i.e., we assume that as
for some constants C and s.
Note that this corresponds to the actual radial wave function having the limiting behavior

77. If we now substitute these limiting forms into the effective energy eigenvalue equation
we obtain the following algebraic relation
Clearly, as , the last two terms are at most of order
They can be neglected compared to the first two terms, which are of order
Thus, for this power law form to be consistent at small r we must have
This has two solutions : and

78. If we now substitute these limiting forms into the effective energy eigenvalue equation
we obtain the following algebraic relation
Clearly, as , the last two terms are at most of order and
They can both be neglected compared to the first two terms, which are of order
Thus, for this power law form to be consistent at small r we must have
This has two solutions : and

79. If we now substitute these limiting forms into the effective energy eigenvalue equation
we obtain the following algebraic relation
Clearly, as , the last two terms are at most of order and
They can both be neglected compared to the first two terms, which are of order
Thus, for this power law form to be consistent at small r we must have
This has two solutions : and

80. If we now substitute these limiting forms into the effective energy eigenvalue equation
we obtain the following algebraic relation
Clearly, as , the last two terms are at most of order and
They can both be neglected compared to the first two terms, which are of order
Thus, for this power law form to be consistent at small r we must have
This has two solutions : and

81. If we now substitute these limiting forms into the effective energy eigenvalue equation
we obtain the following algebraic relation
Clearly, as , the last two terms are at most of order and
They can both be neglected compared to the first two terms, which are of order
Thus, for this power law form to be consistent at small r we must have
This has two solutions : and

82. If we now substitute these limiting forms into the effective energy eigenvalue equation
we obtain the following algebraic relation
Clearly, as , the last two terms are at most of order and
They can both be neglected compared to the first two terms, which are of order
For this power law form to be consistent at small r we must have
This has two solutions : and

83. If we now substitute these limiting forms into the effective energy eigenvalue equation
we obtain the following algebraic relation
Clearly, as , the last two terms are at most of order and
They can both be neglected compared to the first two terms, which are of order
For this power law form to be consistent at small r we must have
This has two solutions : and

84. If we now substitute these limiting forms into the effective energy eigenvalue equation
we obtain the following algebraic relation
Clearly, as , the last two terms are at most of order and
They can both be neglected compared to the first two terms, which are of order
For this power law form to be consistent at small r we must have
This has two solutions : and

85. If we now substitute these limiting forms into the effective energy eigenvalue equation
we obtain the following algebraic relation
Clearly, as , the last two terms are at most of order and
They can both be neglected compared to the first two terms, which are of order
For this power law form to be consistent at small r we must have
This has two solutions : and

86. With one obtains a solution which, at small r, goes as
while with the radial functions have the limiting behavior
Of these, the solutions of this second type are physically unacceptable, since the divergence at the origin for integer makes for a divergent normalization integral.
For , on the normalization integral converges, but the kinetic energy operator acting on any function proportional to gives a delta function that is not cancelled by any other terms in the eigenvalue equation, i.e.,

87. With one obtains a solution which, at small r, goes as
while with the radial functions have the limiting behavior
Of these, the solutions of this second type are physically unacceptable, since the divergence at the origin for integer makes for a divergent normalization integral.
For , on the normalization integral converges, but the kinetic energy operator acting on any function proportional to gives a delta function that is not cancelled by any other terms in the eigenvalue equation, i.e.,

88. With one obtains a solution which, at small r, goes as
while with the radial functions have the limiting behavior
Of these, the solutions of this second type are physically unacceptable, since the divergence at the origin for integer makes for a divergent normalization integral.
For , on the normalization integral converges, but the kinetic energy operator acting on any function proportional to gives a delta function that is not cancelled by any other terms in the eigenvalue equation, i.e.,

89. With one obtains a solution for which, at small r, goes as
while with the radial functions have the limiting behavior
Of these, the solutions of this second type are physically unacceptable, since the divergence at the origin for integer makes for a divergent normalization integral.
For , the normalization integral converges, but the kinetic energy operator acting on any function proportional to gives a delta function that is not cancelled by any other terms in the eigenvalue equation, i.e.,

90. for r near zero,
Thus, this second "solution" for is not actually a valid solution to the eigenvalue equation.
Thus, in the end, under these conditions one obtains for each integer value of , exactly one regular radial function with the limiting behavior
as r→0
Note that this limiting form implies the following "boundary condition“
for the effective one-dimensional problem, making the wave function act as though there were an infinite potential barrier at

91. for r near zero,
Thus, this second "solution" for is not actually a valid solution to the eigenvalue equation.
Thus, in the end, under these conditions one obtains for each integer value of , exactly one regular radial function with the limiting behavior
as r→0
Note that this limiting form implies the following "boundary condition“
for the effective one-dimensional problem, making the wave function act as though there were an infinite potential barrier at

92. for r near zero,
Thus, this second "solution" for is not actually a valid solution to the eigenvalue equation.
Thus, in the end, under these conditions one obtains for each integer value of , exactly one regular radial function with the limiting behavior
as r→0
Note that this limiting form implies the following "boundary condition“
for the effective one-dimensional problem, making the wave function act as though there were an infinite potential barrier at

93. for r near zero,
Thus, this second "solution" for is not actually a valid solution to the eigenvalue equation.
Thus, in the end, under these conditions one obtains for each integer value of , exactly one regular radial function with the limiting behavior
as r→0
Note that this limiting form implies the following "boundary condition“
for the effective one-dimensional problem, which as we have said, makes the wave function act as though there were an infinite potential barrier at

94. We have gone about as far as we can go without actually specifying the potential.
We turn, therefore, to treat what is certainly the most important application of these ideas to atomic systems, namely, the so-called Coulomb problem associated with a charged particle moving in the potential energy well
generated by a point charge of the opposite sign.
This will allow us to deduce, in the non-relativistic limit, the energy spectrum and eigenfunctions of what are referred to as “hydrogenic atoms”, consisting of a charged nucleus of atomic number Z and a single atomic electron.

95. We have gone about as far as we can go without actually specifying the potential.
We turn, therefore, to what is certainly the most important application of these ideas to atomic systems, namely, the so-called Coulomb problem associated with a charged particle moving in the potential energy well
generated by a point charge of the opposite sign.
This will allow us to deduce, in the non-relativistic limit, the energy spectrum and eigenfunctions of what are referred to as “hydrogenic atoms”, consisting of a charged nucleus of atomic number Z and a single atomic electron.

96. We have gone about as far as we can go without actually specifying the potential.
We turn, therefore, to what is certainly the most important application of these ideas to atomic systems, namely, the so-called Coulomb problem associated with a charged particle moving in the potential energy well
generated by a point charge of the opposite sign.
This will allow us to deduce, in the non-relativistic limit, the energy spectrum and eigenfunctions of what are referred to as “hydrogenic atoms”, consisting of a charged nucleus of atomic number Z and a single atomic electron.