1. Quantum Two

3. Bound States of a Central Potential

4. Bound States of a Central Potential
The Coulomb Problem

5. We now consider the physically important case in which the potential of interest is that associated with the attractive Coulomb interaction
between a nucleus of charge +Zq and a single electron of charge –q.
This is often referred to as "the Coulomb problem".
It is historically important, being one of the first problems solved by Schrödinger himself, and its mathematical solution provided a convincing explanation for the absorption and emission spectrum observed in what are referred to as "hydrogenic atoms” having a single valence electron. r

6. We now consider the physically important case in which the potential of interest is that associated with the attractive Coulomb interaction
between a nucleus of charge +Zq and a single electron of charge –q.
This is often referred to as "the Coulomb problem".
It is historically important, being one of the first problems solved by Schrödinger himself, and its mathematical solution provided a convincing explanation for the absorption and emission spectrum observed in what are referred to as "hydrogenic atoms” having a single valence electron. r

7. We now consider the physically important case in which the potential of interest is that associated with the attractive Coulomb interaction
between a nucleus of charge +Zq and a single electron of charge –q.
This is often referred to as "the Coulomb problem".
It is historically important, being one of the first problems solved by Schrödinger himself, and its mathematical solution provided a convincing explanation for the absorption and emission spectrum observed in what are referred to as "hydrogenic atoms” having a single valence electron. r

8. As an initial notational simplification, we ignore the spin of the constituents, and rewrite the potential in the form
by introducing
Then, rather than use the dimensionless form previously obtained for the case in which the potential was unspecified, we use scaling arguments to generate, in advance, estimates of the energy scale ε and the length scale a of bound states of this atomic system.
These will allow us to express the eigenvalue equation of interest in an even more useful dimensionless form. r

9. As an initial notational simplification, we ignore the spin of the constituents, and rewrite the potential in the form
by introducing
Then, rather than use the dimensionless form previously obtained for the case in which the potential was unspecified, we use scaling arguments to generate, in advance, estimates of the energy scale ε and the length scale a of bound states of this atomic system.
These will allow us to express the eigenvalue equation of interest in an even more useful dimensionless form. r

10. As an initial notational simplification, we ignore the spin of the constituents, and rewrite the potential in the form
by introducing
Then, rather than use the dimensionless form previously obtained for the case in which the potential was unspecified, we use scaling arguments to generate, in advance, estimates of the energy scale ε and the length scale a of bound states of this atomic system.
These will allow us to express the eigenvalue equation of interest in an even more useful dimensionless form. r

11. These scales of interest arise from the fact that the mean energy
of a bound state depends both upon
its kinetic energy
and its potential energy
where a ∼ 〈r〉 represents a typical distance of the electron from the nucleus, which is related to its statistical spread 2Δr ∼ 2a in position.
Now, in any stationary state the mean position of the particle must itself be stationary, which requires that

12. These scales of interest arise from the fact that the mean energy
of a bound state depends both upon
its kinetic energy
and its potential energy
where a ∼ 〈r〉 represents a typical distance of the electron from the nucleus, which is related to its statistical spread 2Δr ∼ 2a in position.
Now, in any stationary state the mean position of the particle must itself be stationary, which requires that

13. These scales of interest arise from the fact that the mean energy
of a bound state depends both upon
its kinetic energy
and its potential energy
where a ∼ 〈r〉 represents a typical distance of the electron from the nucleus, which is related to its statistical spread 2Δr ∼ 2a in position.
Now, in any stationary state the mean position of the particle must itself be stationary, which requires that

14. These scales of interest arise from the fact that the mean energy
of a bound state depends both upon
its kinetic energy
and its potential energy
where a ∼ 〈r〉 represents a typical distance of the electron from the nucleus, which is related to its statistical spread 2Δr ∼ 2a in position.
Now, in any stationary state the mean position of the particle must itself be stationary, which requires that 2a

15. Using the uncertainty principle ΔrΔp ∼ ℏ, we also obtain the estimate
Thus, the kinetic energy will be on the order of
giving the total energy as
We now minimize the total energy with respect to the length scale a. Setting ∂〈ε〉/∂a = 0 , one finds or , where is the Bohr radius. 2a

16. Using the uncertainty principle ΔrΔp ∼ ℏ, we also obtain the estimate
Thus, the kinetic energy will be on the order of
giving the total energy as
We now minimize the total energy with respect to the length scale a. Setting ∂〈ε〉/∂a = 0 , one finds or , where is the Bohr radius. 2a

17. Using the uncertainty principle ΔrΔp ∼ ℏ, we also obtain the estimate
Thus, the kinetic energy will be on the order of
giving the total energy as
We now minimize the total energy with respect to the length scale a. Setting ∂〈ε〉/∂a = 0 , one finds or , where is the Bohr radius. 2a

18. Using the uncertainty principle ΔrΔp ∼ ℏ, we also obtain the estimate
Thus, the kinetic energy will be on the order of
giving the total energy as
We now minimize the total energy with respect to the length scale a. Setting ∂〈ε〉/∂a = 0 , one finds or , where is the Bohr radius. 2a

19. Using the uncertainty principle ΔrΔp ∼ ℏ, we also obtain the estimate
Thus, the kinetic energy will be on the order of
giving the total energy as
We now minimize the total energy with respect to the length scale a. Setting ∂〈ε〉/∂a = 0 , one finds or , where is the Bohr radius. 2a

20. Using the uncertainty principle ΔrΔp ∼ ℏ, we also obtain the estimate
Thus, the kinetic energy will be on the order of
giving the total energy as
We now minimize the total energy with respect to the length scale a. Setting ∂〈ε〉/∂a = 0 , one finds or , where is the Bohr radius. 2a

21. Using the uncertainty principle ΔrΔp ∼ ℏ, we also obtain the estimate
Thus, the kinetic energy will be on the order of
giving the total energy as
We now minimize the total energy with respect to the length scale a. Setting ∂〈ε〉/∂a = 0 , one finds or , where is the Bohr radius. 2a

22. Using the uncertainty principle ΔrΔp ∼ ℏ, we also obtain the estimate
Thus, the kinetic energy will be on the order of
giving the total energy as
We now minimize the total energy with respect to the length scale a. Setting ∂〈ε〉/∂a = 0 , one finds or , where is the Bohr radius. 2a

23. Using the uncertainty principle ΔrΔp ∼ ℏ, we also obtain the estimate
Thus, the kinetic energy will be on the order of
giving the total energy as
We now minimize the total energy with respect to the length scale a. Setting ∂〈ε〉/∂a = 0 , one finds or , where is the Bohr radius. 2a

24. Putting this back into our expression for the energy then gives an estimate
for the energy scale, where
is the binding energy of the hydrogen atom.
In addition, we can obtain an estimate for a typical velocity or speed for the problem through the relation

25. Putting this back into our expression for the energy then gives an estimate
for the energy scale, where
is the binding energy of the hydrogen atom.
In addition, we can obtain an estimate for a typical velocity or speed for the problem through the relation

26. Putting this back into our expression for the energy then gives an estimate
for the energy scale, where
is the binding energy of the hydrogen atom.
In addition, we can obtain an estimate for a typical velocity or speed for the problem through the relation

27. Putting this back into our expression for the energy then gives an estimate
for the energy scale, where
is the binding energy of the hydrogen atom.
In addition, we can obtain an estimate for a typical velocity or speed for the problem through the relation

28. Putting this back into our expression for the energy then gives an estimate
for the energy scale, where
is the binding energy of the hydrogen atom.
In addition, we can obtain an estimate for a typical velocity or speed for the problem through the relation

29. Putting this back into our expression for the energy then gives an estimate
for the energy scale, where
is the binding energy of the hydrogen atom.
In addition, we can obtain an estimate for a typical velocity or speed for the problem through the relation

30. Putting this back into our expression for the energy then gives an estimate
for the energy scale, where
is the binding energy of the hydrogen atom.
In addition, we can obtain an estimate for a typical velocity or speed for the problem through the relation

31. Putting this back into our expression for the energy then gives an estimate
for the energy scale, where
is the binding energy of the hydrogen atom.
In addition, we can obtain an estimate for a typical velocity or speed for the problem through the relation

32. Putting this back into our expression for the energy then gives an estimate
for the energy scale, where
is the binding energy of the hydrogen atom.
In addition, we can obtain an estimate for a typical velocity or speed for the problem through the relation

33. Putting this back into our expression for the energy then gives an estimate
for the energy scale, where
is the binding energy of the hydrogen atom.
In addition, we can obtain an estimate for a typical velocity or speed for the problem through the relation

34. In atomic physics it is common to introduce a set of units in which so that
We shall not follow that convention in the present treatment, but we will use the length and energy scales derived above to appropriately transform the problem.
For example, it follows from our general treatment that energy eigenfunctions for this problem will be of the form
in which the angular dependence is governed by the spherical harmonics and the radial functions can be determined from the functions , which satisfy the equation . . .

35. In atomic physics it is common to introduce a set of units in which so that
We shall not follow that convention in the present treatment, but we will use the length and energy scales derived above to appropriately transform the problem.
For example, it follows from our general treatment that energy eigenfunctions for this problem will be of the form
in which the angular dependence is governed by the spherical harmonics and the radial functions can be determined from the functions , which satisfy the equation . . .

36. In atomic physics it is common to introduce a set of units in which so that
We shall not follow that convention in the present treatment, but we will use the length and energy scales derived above to appropriately transform the problem.
For example, it follows from our general treatment that energy eigenfunctions for this problem will be of the form
in which the angular dependence is governed by the spherical harmonics and the radial functions can be determined from the functions , which satisfy the equation . . .

37. In atomic physics it is common to introduce a set of units in which so that
We shall not follow that convention in the present treatment, but we will use the length and energy scales derived above to appropriately transform the problem.
For example, it follows from our general treatment that energy eigenfunctions for this problem will be of the form
in which the angular dependence is governed by the spherical harmonics and the radial functions can be determined from the functions , which satisfy the equation . . .

38. In atomic physics it is common to introduce a set of units in which so that
We shall not follow that convention in the present treatment, but we will use the length and energy scales derived above to appropriately transform the problem.
For example, it follows from our general treatment that energy eigenfunctions for this problem will be of the form
in which the angular dependence is governed by the spherical harmonics and the radial functions can be determined from the functions , which satisfy the equation . . .

39. In atomic physics it is common to introduce a set of units in which so that
We shall not follow that convention in the present treatment, but we will use the length and energy scales derived above to appropriately transform the problem.
For example, it follows from our general treatment that energy eigenfunctions for this problem will be of the form
in which the angular dependence is governed by the spherical harmonics and the radial functions can be determined from the functions , which satisfy the equation . . .

40. In atomic physics it is common to introduce a set of units in which so that
We shall not follow that convention in the present treatment, but we will use the length and energy scales derived above to appropriately transform the problem.
For example, it follows from our general treatment that energy eigenfunctions for this problem will be of the form
in which the angular dependence is governed by the spherical harmonics and the radial functions can be determined from the functions , which satisfy the equation . . .

41. In atomic physics it is common to introduce a set of units in which so that
We shall not follow that convention in the present treatment, but we will use the length and energy scales derived above to appropriately transform the problem.
For example, it follows from our general treatment that energy eigenfunctions for this problem will be of the form
in which the angular dependence is governed by the spherical harmonics and the radial functions can be determined from the functions , which satisfy the equation . . .

42. Introducing a dimensionless position variable ρ = r/a and treating now as a function of ρ, rather than r = ρa, this last equation can be written in the form
Dividing this by reduces this to the dimensionless form appearing on the next page:

43. Introducing a dimensionless position variable ρ = r/a and treating now as a function of ρ, rather than r = ρa, this last equation can be written in the form
Dividing this by reduces this to the dimensionless form appearing on the next page:

44. Introducing a dimensionless position variable ρ = r/a and treating now as a function of ρ, rather than r = ρa, this last equation can be written in the form
Dividing this by reduces this to the dimensionless form appearing on the next page:

45. Introducing a dimensionless position variable ρ = r/a and treating now as a function of ρ, rather than r = ρa, this last equation can be written in the form
Dividing this by reduces this to the dimensionless form appearing on the next page:

46. where primes now denote derivatives with respect to ρ, and where
is a positive dimensionless parameter characterizing the energy
of the bound states we seek.
We now focus on the asymptotic behavior of the functions for large ρ, which is important in obtaining solutions that are square normalizeable.

47. where primes now denote derivatives with respect to ρ, and where
is a positive dimensionless parameter characterizing the energy
of the bound states we seek.
We now focus on the asymptotic behavior of the functions for large ρ, which is important in obtaining solutions that are square normalizeable.

48. where primes now denote derivatives with respect to ρ, and where
is a positive dimensionless parameter characterizing the energy
of the bound states we seek.
We now focus on the asymptotic behavior of the functions for large ρ, which is important in obtaining solutions that are square normalizeable.

49. where primes now denote derivatives with respect to ρ, and where
is a positive dimensionless parameter characterizing the energy
of the bound states we seek.
We now focus on the asymptotic behavior of the functions for large ρ, which is important in obtaining solutions that are square normalizeable.

50. To study the asymptotic behavior note that for large ρ we can neglect ρ⁻¹ and ρ⁻² relative to in the radial eigenvalue equation, which then has the asymptotic form
in which we have (as we will continue to do) suppressed the subscripts n and ℓ on .
This asymptotic equation has two linearly independent solutions
of which the one that diverges at large ρ is rejected as being not square- normalizeable.

51. To study the asymptotic behavior note that for large ρ we can neglect ρ⁻¹ and ρ⁻² relative to in the radial eigenvalue equation, which then has the asymptotic form
in which we have (as we will continue to do) suppressed the subscripts n and ℓ on .
This asymptotic equation has two linearly independent solutions
of which the one that diverges at large ρ is rejected as being not square- normalizeable.

52. To study the asymptotic behavior note that for large ρ we can neglect ρ⁻¹ and ρ⁻² relative to in the radial eigenvalue equation, which then has the asymptotic form
in which we have (as we will continue to do) suppressed the subscripts n and ℓ on .
This asymptotic equation has two linearly independent solutions
of which the one that diverges at large ρ is rejected as being not square- normalizeable.

53. To study the asymptotic behavior note that for large ρ we can neglect ρ⁻¹ and ρ⁻² relative to in the radial eigenvalue equation, which then has the asymptotic form
in which we have (as we will continue to do) suppressed the subscripts n and ℓ on .
This asymptotic equation has two linearly independent solutions
of which the one that diverges at large ρ is rejected as being not square- normalizeable.

54. We now introduce new radial functions u(ρ) by factoring out the asymptotic dependence, writing so that
To determine the function we evaluate
and substitute into our earlier equation, which after multiplying through by can be re-expressed as

55. We now introduce new radial functions u(ρ) by factoring out the asymptotic dependence, writing so that
To determine the function we evaluate
and substitute into our earlier equation, which after multiplying through by can be re-expressed as

56. We now introduce new radial functions u(ρ) by factoring out the asymptotic dependence, writing so that
To determine the function we evaluate
and substitute into our earlier equation, which after multiplying through by can be re-expressed as

57. We now introduce new radial functions u(ρ) by factoring out the asymptotic dependence, writing so that
To determine the function we evaluate
and substitute into our earlier equation, which after multiplying through by can be re-expressed as

58. We now introduce new radial functions u(ρ) by factoring out the asymptotic dependence, writing so that
To determine the function we evaluate
and substitute into our earlier equation, which after multiplying through by can be re-expressed as

59. We now introduce new radial functions u(ρ) by factoring out the asymptotic dependence, writing so that
To determine the function we evaluate
and substitute into our earlier equation, which after multiplying through by can be re-expressed as

60. We now introduce new radial functions u(ρ) by factoring out the asymptotic dependence, writing so that
To determine the function we evaluate
and substitute into our earlier equation, which after multiplying through by can be re-expressed as

61. The classical method of solving such an equation is to assume a power series solution
where in the second form we explicitly assume the first coefficient b₀ is not equal to zero, and used our previous limiting results that for small r the function goes to zero as , so upon expanding the exponential in the right hand side of the definition one finds that aas

62. The classical method of solving such an equation is to assume a power series solution
where in the second form we explicitly assume the first coefficient b₀ is not equal to zero, and used our previous limiting results that for small r the function goes to zero as , so upon expanding the exponential in the right hand side of the definition one finds that aas

63. The classical method of solving such an equation is to assume a power series solution
where in the second form we explicitly assume the first coefficient b₀ is not equal to zero, and used our previous limiting results that for small r the function goes to zero as , so upon expanding the exponential in the right hand side of the definition one finds that aas

64. The classical method of solving such an equation is to assume a power series solution
where in the second form we explicitly assume the first coefficient b₀ is not equal to zero, and used our previous limiting results that for small r the function goes to zero as , so upon expanding the exponential in the right hand side of the definition one finds that aas

65. The classical method of solving such an equation is to assume a power series solution
where in the second form we explicitly assume the first coefficient b₀ is not equal to zero, and used our previous limiting results that for small r the function goes to zero as , so upon expanding the exponential in the right hand side of the definition one finds that aas

66. Thus, we can write power series expansions for the function u(ρ) and its derivatives:
Substituting these expressions into the differential equation

67. Thus, we can write power series expansions for the function u(ρ) and its derivatives:
Substituting these expressions into the differential equation

68. Thus, we can write power series expansions for the function u(ρ) and its derivatives:
Substituting these expressions into the differential equation

69. Thus, we can write power series expansions for the function u(ρ) and its derivatives:
Substituting these expressions into the differential equation

70. and multiplying by one finds the equation
Shifting the summation variable from k to k′ = k + 1 in the sums involving and dropping any obviously vanishing terms, leads after a little algebra to the equation . . .

71. and multiplying by one finds the equation
Shifting the summation variable from k to k′ = k + 1 in the sums involving and dropping any obviously vanishing terms, leads after a little algebra to the equation . . .

72. and multiplying by one finds the equation
Shifting the summation variable from k to k ′ = k + 1 in the sums involving and dropping any obviously vanishing terms, leads after a little algebra to the equation . . .

73. in which the coefficient of each linearly independent power of ρ must vanish independently.
This leads to a recursion relation
that allows all coefficients to be expressed in terms of the single coefficient b₀. If the resulting power series solution does not terminate, then, applying the ratio test, one finds that
so such an infinite series converges absolutely for any value of ρ.

74. in which the coefficient of each linearly independent power of ρ must vanish independently.
This leads to a recursion relation
that allows all coefficients to be expressed in terms of the single coefficient b₀. If the resulting power series solution does not terminate, then, applying the ratio test, one finds that
so such an infinite series converges absolutely for any value of ρ.

75. in which the coefficient of each linearly independent power of ρ must vanish independently.
This leads to a recursion relation
that allows all coefficients to be expressed in terms of the single coefficient b₀. If the resulting power series solution does not terminate, then, applying the ratio test, one finds that
so such an infinite series converges absolutely for any value of ρ.

76. in which the coefficient of each linearly independent power of ρ must vanish independently.
This leads to a recursion relation
that allows all coefficients to be expressed in terms of the single coefficient b₀. If the resulting power series solution does not terminate, then, applying the ratio test, one finds that
so such an infinite series converges absolutely for any value of ρ.

77. in which the coefficient of each linearly independent power of ρ must vanish independently.
This leads to a recursion relation
that allows all coefficients to be expressed in terms of the single coefficient b₀. If the resulting power series solution does not terminate, then, applying the ratio test, one finds that
so such an infinite series converges absolutely for any value of ρ.

78. in which the coefficient of each linearly independent power of ρ must vanish independently.
This leads to a recursion relation
that allows all coefficients to be expressed in terms of the single coefficient b₀. If the resulting power series solution does not terminate, then, applying the ratio test, one finds that
Such an infinite series converges absolutely for any value of ρ.

79. Unfortunately, such a non-terminating solution converges to a function u(ρ) for which diverges as a function of ρ, as ρ → ∞.
This can be inferred from the power series expansion for the diverging exponential function
from which one notes that for this series the ratio
has the same limiting behavior as
For large ρ, the behavior of both of these two series will be dominated by terms with large values of k, for which they have the same limiting behavior.

80. Unfortunately, such a non-terminating solution converges to a function u(ρ) for which diverges as a function of ρ, as ρ → ∞.
This can be inferred from the power series expansion for the diverging exponential function
from which one notes that for this series the ratio
has the same limiting behavior as
For large ρ, the behavior of both of these two series will be dominated by terms with large values of k, for which they have the same limiting behavior.

81. Unfortunately, such a non-terminating solution converges to a function u(ρ) for which diverges as a function of ρ, as ρ → ∞.
This can be inferred from the power series expansion for the diverging exponential function
from which one notes that for this series the ratio
has the same limiting behavior as
For large ρ, the behavior of both of these two series will be dominated by terms with large values of k, for which they have the same limiting behavior.

82. Unfortunately, such a non-terminating solution converges to a function u(ρ) for which diverges as a function of ρ, as ρ → ∞.
This can be inferred from the power series expansion for the diverging exponential function
from which one notes that for this series the ratio
has the same limiting behavior as
For large ρ, the behavior of both of these two series will be dominated by terms with large values of k, for which they have the same limiting behavior.

83. Unfortunately, such a non-terminating solution converges to a function u(ρ) for which diverges as a function of ρ, as ρ → ∞.
This can be inferred from the power series expansion for the diverging exponential function
from which one notes that for this series the ratio
is the same as the limiting behavior of
For large ρ, the behavior of both of these two series will be dominated by terms with large values of k, for which they have the same limiting behavior.

84. Unfortunately, such a non-terminating solution converges to a function u(ρ) for which diverges as a function of ρ, as ρ → ∞.
This can be inferred from the power series expansion for the diverging exponential function
from which one notes that for this series the ratio
is the same as the limiting behavior of
For large ρ, the behavior of both of these two series will be dominated by terms at large values of k, for which they have the same limiting behavior.

85. Thus, if it does not terminate, then , and the power series generates one of the unacceptable diverging solutions that we have already attempted to reject.
To prevent this from happening, and to obtain acceptable solutions, the power series must terminate, with u(ρ) taking the form of a simple polynomial function of its argument.
Thus, acceptable solutions u(ρ) to the eigenvalue equation have power series that terminate: i.e., there exists some integer for which does not vanish, but for which
Clearly for an acceptable solution we must have

86. Thus, if it does not terminate, then , and the power series generates one of the unacceptable diverging solutions that we have already attempted to reject.
To prevent this from happening, and to obtain acceptable solutions, the power series must terminate, with u(ρ) taking the form of a simple polynomial function of its argument.
Thus, acceptable solutions u(ρ) to the eigenvalue equation have power series that terminate: i.e., there exists some integer for which does not vanish, but for which
Clearly for an acceptable solution we must have

87. Thus, if it does not terminate, then , and the power series generates one of the unacceptable diverging solutions that we have already attempted to reject.
To prevent this from happening, and to obtain acceptable solutions, the power series must terminate, with u(ρ) taking the form of a simple polynomial function of its argument.
Thus, acceptable solutions u(ρ) to the eigenvalue equation have power series that terminate: i.e., there exists some integer for which does not vanish, but for which
Clearly for an acceptable solution we must have

88. Thus, if it does not terminate, then , and the power series generates one of the unacceptable diverging solutions that we have already attempted to reject.
To prevent this from happening, and to obtain acceptable solutions, the power series must terminate, with u(ρ) taking the form of a simple polynomial function of its argument.
Thus, acceptable solutions u(ρ) to the eigenvalue equation have power series that terminate: i.e., there exists some integer for which does not vanish, but for which
Clearly for an acceptable solution we must have

89. Hence, for a given value of ℓ, allowed values of the dimensionless constant α are determined by the equation
Note that acceptable solutions can terminate at any value of beyond k = 0, since b₀ is non-zero by assumption.
Since ℓ and are both integers, this can also be written in the form
where is itself a positive integer, referred to as the principle quantum number.

90. Hence, for a given value of ℓ, allowed values of the dimensionless constant α are determined by the equation
Note that acceptable solutions can terminate at any value of beyond k = 0, since b₀ is non-zero by assumption.
Since ℓ and are both integers, this can also be written in the form
where is itself a positive integer, referred to as the principle quantum number.

91. Hence, for a given value of ℓ, allowed values of the dimensionless constant α are determined by the equation
Note that acceptable solutions can terminate at any value of beyond k = 0, since b₀ is non-zero by assumption.
Since ℓ and are both integers, this can also be written in the form
where is itself a positive integer, referred to as the principle quantum number.

92. Hence, for a given value of ℓ, allowed values of the dimensionless constant α are determined by the equation
Note that acceptable solutions can terminate at any value of beyond k = 0, since b₀ is non-zero by assumption.
Since ℓ and are both integers, this can also be written in the form
where is itself a positive integer, referred to as the principle quantum number.

93. Hence, for a given value of ℓ, allowed values of the dimensionless constant α are determined by the equation
Note that acceptable solutions can terminate at any value of beyond k = 0, since b₀ is non-zero by assumption.
Since ℓ and are both integers, this can also be written in the form
where is itself a positive integer, referred to as the principle quantum number.

94. Clearly, for a given fixed value of ℓ, there are an infinite number of acceptable values of the principle quantum number
corresponding to the infinity of values at which an acceptable series solution can terminate.
On the other hand, for any allowed value of the principle quantum number n, not all values of ℓ yield acceptable solutions. Since ≥ 1, it follows that for a given value of the principal quantum number n, values of the orbital angular momentum quantum number satisfy
in which, e.g., corresponds to ℓ = 0, and corresponds to ℓ = n - 1.

95. Clearly, for a given fixed value of ℓ, there are an infinite number of acceptable values of the principle quantum number
corresponding to the infinity of values at which an acceptable series solution can terminate.
On the other hand, for any allowed value of the principle quantum number n, not all values of ℓ yield acceptable solutions. Since ≥ 1, it follows that for a given value of the principal quantum number n, values of the orbital angular momentum quantum number satisfy
in which, e.g., corresponds to ℓ = 0, and corresponds to ℓ = n - 1.

96. Clearly, for a given fixed value of ℓ, there are an infinite number of acceptable values of the principle quantum number
corresponding to the infinity of values at which an acceptable series solution can terminate.
On the other hand, for any allowed value of the principle quantum number n, not all values of ℓ yield acceptable solutions.
Since ≥ 1, it follows that for a given value of the principal quantum number n, values of the orbital angular momentum quantum number satisfy
in which, e.g., corresponds to ℓ = 0, and corresponds to ℓ = n - 1.

97. Turning this back into an equation for the energies, we then find for each , the energy eigenvalues
that are independent of the orbital angular quantum number, except for the restriction (shown above) on n the values of ℓ that can occur for each value of n.
Recall that for each value of n and ℓ there is a fold degenerate multiplet {|n,ℓ,m〉 | m = - ℓ,⋯,ℓ} of states corresponding to the different values taken by the azimuthal quantum number m. Thus, taking all n multiplets of the nth energy level into account, the degeneracy of that level can be computed as the sum

98. Turning this back into an equation for the energies, we then find for each , the energy eigenvalues
that are independent of the orbital angular quantum number, except for the restriction (shown above) on n the values of ℓ that can occur for each value of n.
Recall that for each value of n and ℓ there is a fold degenerate multiplet {|n,ℓ,m〉 | m = - ℓ,⋯,ℓ} of states corresponding to the different values taken by the azimuthal quantum number m. Thus, taking all n multiplets of the nth energy level into account, the degeneracy of that level can be computed as the sum

99. Turning this back into an equation for the energies, we then find for each , the energy eigenvalues
that are independent of the orbital angular quantum number, except for the restriction (shown above) on the n values of ℓ that can occur for each value of n.
Recall that for each value of n and ℓ there is a fold degenerate multiplet {|n,ℓ,m〉 | m = - ℓ,⋯,ℓ} of states corresponding to the different values taken by the azimuthal quantum number m. Thus, taking all n multiplets of the nth energy level into account, the degeneracy of that level can be computed as the sum

100. Turning this back into an equation for the energies, we then find for each , the energy eigenvalues
that are independent of the orbital angular quantum number, except for the restriction (shown above) on the n values of ℓ that can occur for each value of n.
Recall that for each value of n and ℓ there is a fold degenerate multiplet {|n,ℓ,m〉 | m = - ℓ,⋯,ℓ} of states corresponding to the different values taken by the azimuthal quantum number m. Thus, taking all n multiplets of the nth energy level into account, the degeneracy of that level can be computed as the sum

101. Turning this back into an equation for the energies, we then find for each , the energy eigenvalues
that are independent of the orbital angular quantum number, except for the restriction (shown above) on the n values of ℓ that can occur for each value of n.
Recall that for each value of n and ℓ there is a fold degenerate multiplet {|n,ℓ,m〉 | m = - ℓ,⋯,ℓ} of states corresponding to the different values taken by the azimuthal quantum number m.
Thus, taking all n multiplets of the nth energy level into account, the degeneracy of that level can be computed as the sum

102. Turning this back into an equation for the energies, we then find for each , the energy eigenvalues
that are independent of the orbital angular quantum number, except for the restriction (shown above) on the n values of ℓ that can occur for each value of n.
Recall that for each value of n and ℓ there is a fold degenerate multiplet {|n,ℓ,m〉 | m = - ℓ,⋯,ℓ} of states corresponding to the different values taken by the azimuthal quantum number m.
Thus, taking all n multiplets of the nth energy level into account, the degeneracy of that level can be computed as the sum

103. Turning this back into an equation for the energies, we then find for each , the energy eigenvalues
that are independent of the orbital angular quantum number, except for the restriction (shown above) on the n values of ℓ that can occur for each value of n.
Recall that for each value of n and ℓ there is a fold degenerate multiplet {|n,ℓ,m〉 | m = - ℓ,⋯,ℓ} of states corresponding to the different values taken by the azimuthal quantum number m.
Thus, taking all n multiplets of the nth energy level into account, the degeneracy of that level can be computed as the sum

104. The fact that there exist multiple solutions with the same energy (except for n = 0) corresponding to different values of ℓ constitutes what is referred to as an accidental degeneracy, i.e., one that is not required by the rotational invariance of the problem.
This additional degeneracy, which does not generally occur with other central potentials, reflects other symmetries inherent in the hydrogen atom problem that are not evident in the present treatment.
The energy levels for the hydrogen atom are shown on the next page as horizontal bars, for different values of the orbital angular momentum quantum number (as indicated horizontally), and different values of the principal quantum number (as indicated vertically in the legend).

105. The fact that there exist multiple solutions with the same energy (except for n = 0) corresponding to different values of ℓ constitutes what is referred to as an accidental degeneracy, i.e., one that is not required by the rotational invariance of the problem.
This additional degeneracy, which does not generally occur with other central potentials, reflects other symmetries inherent in the hydrogen atom problem that are not evident in the present treatment.
The energy levels for the hydrogen atom are shown on the next page as horizontal bars, for different values of the orbital angular momentum quantum number (as indicated horizontally), and different values of the principal quantum number (as indicated vertically in the legend).

106. The fact that there exist multiple solutions with the same energy (except for n = 0) corresponding to different values of ℓ constitutes what is referred to as an accidental degeneracy, i.e., one that is not required by the rotational invariance of the problem.
This additional degeneracy, which does not generally occur with other central potentials, reflects other symmetries inherent in the hydrogen atom problem that are not evident in the present treatment.
The energy levels for the hydrogen atom are shown on the next page as horizontal bars, for different values of the orbital angular momentum quantum number (as indicated horizontally), and different values of the principal quantum number (as indicated vertically in the legend).

107. The energy levels for the hydrogen atom as horizontal bars, for different values of the orbital angular momentum quantum number (as indicated horizontally), and different values of the principal quantum number (as indicated by different color horizontal bars, as indicated in the legend).

108. Having determined the energies, and the corresponding values of , the coefficients of the power series expansion for the radial functions can then be determined for each value of n and ℓ through the recursion relation already derived, i.e.,
For given values of n and ℓ ≤ n - 1 this relation is easily iterated.
Thus, e.g., for any n, when ℓ = n - 1, so that n = ℓ + 1,the series terminates at
For this special case the only non-zero term in the series is the first, giving the result

109. Having determined the energies, and the corresponding values of , the coefficients of the power series expansion for the radial functions can then be determined for each value of n and ℓ through the recursion relation already derived, i.e.,
For given values of n and ℓ ≤ n - 1 this relation is easily iterated.
Thus, e.g., for any n, when ℓ = n - 1, so that n = ℓ + 1,the series terminates at
For this special case the only non-zero term in the series is the first, giving the result

110. Having determined the energies, and the corresponding values of , the coefficients of the power series expansion for the radial functions can then be determined for each value of n and ℓ through the recursion relation already derived, i.e.,
For given values of n and ℓ ≤ n - 1 this relation is easily iterated.
Thus, e.g., for any n, when ℓ = n - 1, so that n = ℓ + 1,the series terminates at
For this special case the only non-zero term in the series is the first, giving the result

111. Having determined the energies, and the corresponding values of , the coefficients of the power series expansion for the radial functions can then be determined for each value of n and ℓ through the recursion relation already derived, i.e.,
For given values of n and ℓ ≤ n - 1 this relation is easily iterated.
Thus, e.g., for any n, when ℓ = n - 1, so that n = ℓ + 1, the series must terminate at
For this special case the only non-zero term in the series is the first, giving the result

112. Having determined the energies, and the corresponding values of , the coefficients of the power series expansion for the radial functions can then be determined for each value of n and ℓ through the recursion relation already derived, i.e.,
For given values of n and ℓ ≤ n - 1 this relation is easily iterated.
Thus, e.g., for any n, when ℓ = n - 1, so that n = ℓ + 1, the series must terminate at
For this special case the only non-zero term in the series is the first, giving the result

113. Similarly for n > 1, when ℓ = n - 2, or n = ℓ + 2, the series terminates at .
In this situation, the only two nonzero coefficients are and
Thus we have for this special case

114. Similarly for n > 1, when ℓ = n - 2, or n = ℓ + 2, the series terminates at .
In this situation, the only two nonzero coefficients are and
Thus we have for this special case

115. Similarly for n > 1, when ℓ = n - 2, or n = ℓ + 2, the series terminates at .
In this situation, the only two nonzero coefficients are and
Thus we have for this special case

116. Similarly for n > 1, when ℓ = n - 2, or n = ℓ + 2, the series terminates at .
In this situation, the only two nonzero coefficients are and
Thus we have for this special case

117. Similarly for n > 1, when ℓ = n - 2, or n = ℓ + 2, the series terminates at .
In this situation, the only two nonzero coefficients are and
Thus we have for this special case

118. As another special case, we note that for sufficiently large n when ℓ = 0, the series terminates at k₀ = n, so that the nonzero coefficients are at b₀ ,
from which one inductively deduces that for this case the kth coefficient is given in terms of by the expression
Thus, one finds that for ℓ = 0,

119. As another special case, we note that for sufficiently large n when ℓ = 0, the series terminates at k₀ = n, so that the nonzero coefficients are at b₀ ,
from which one inductively deduces that for this case the kth coefficient is given in terms of by the expression
Thus, one finds that for ℓ = 0,

120. As another special case, we note that for sufficiently large n when ℓ = 0, the series terminates at k₀ = n, so that the nonzero coefficients are at b₀ ,
from which one inductively deduces that for this case the kth coefficient is given in terms of by the expression
Thus, one finds that for ℓ = 0,

121. As another special case, we note that for sufficiently large n when ℓ = 0, the series terminates at k₀ = n, so that the nonzero coefficients are at b₀ ,
from which one inductively deduces that for this case the kth coefficient is given in terms of by the expression
Thus, one finds that for ℓ = 0,

122. As another special case, we note that for sufficiently large n when ℓ = 0, the series terminates at k₀ = n, so that the nonzero coefficients are at b₀ ,
from which one inductively deduces that for this case the kth coefficient is given by the expression
Thus, one finds that for ℓ = 0,

123. As another special case, we note that for sufficiently large n when ℓ = 0, the series terminates at k₀ = n, so that the nonzero coefficients are at b₀ ,
from which one inductively deduces that for this case the kth coefficient is given by the expression
Thus, one finds for ℓ = 0, an explicit sum:

124. It turns out that the polynomials generated by this recursion relation can be expressed in terms of the generalized Laguerre polynomials , which for integer q and ν can also be generated through the Rodrigues relation (one of several similar formulae)
In terms of these functions, the appropriately normalized radial wave functions are given by the expression

125. It turns out that the polynomials generated by this recursion relation can be expressed in terms of the generalized Laguerre polynomials , which for integer q and ν can also be generated through the Rodrigues relation (one of several similar formulae)
In terms of these functions, the appropriately normalized radial wave functions are given by the expression

126. It turns out that the polynomials generated by this recursion relation can be expressed in terms of the generalized Laguerre polynomials , which for integer q and ν can also be generated through the Rodrigues relation (one of several similar formulae)
In terms of these functions, the appropriately normalized radial wave functions are given by the expression

127. As we have already indicated, the total hydrogenic wavefunctions then take the form
In spectroscopic notation the bound states associated with orbital angular momentum quantum number ℓ = 0, 1, 2, 3, 4, 5,⋯ are indicated with a letter s, p, d, f, g, h,⋯.
Thus refers to a 2p state with principal quantum number n = 2, orbital angular quantum number ℓ = 1, and azimuthal quantum number m.
It is one of the three states {|n, ℓ, m|〉 = |2,1,-1〉, |2,1,0〉, |2,1,1〉} associated with this 3-fold degenerate multiplet.

128. As we have already indicated, the total hydrogenic wavefunctions then take the form
In spectroscopic notation the bound states associated with orbital angular momentum quantum numbers ℓ = 0, 1, 2, 3, 4, 5,⋯ are often indicated by letters s, p, d, f, g, h,⋯.
Thus refers to a 2p state with principal quantum number n = 2, orbital angular quantum number ℓ = 1, and azimuthal quantum number m.
It is one of the three states {|n, ℓ, m|〉 = |2,1,-1〉, |2,1,0〉, |2,1,1〉} associated with this 3-fold degenerate multiplet.

129. As we have already indicated, the total hydrogenic wavefunctions then take the form
In spectroscopic notation the bound states associated with orbital angular momentum quantum numbers ℓ = 0, 1, 2, 3, 4, 5,⋯ are often indicated by letters s, p, d, f, g, h,⋯.
Thus refers to a 2p state with principal quantum number n = 2, orbital angular quantum number ℓ = 1, and azimuthal quantum number m.
It is one of the three states {|n, ℓ, m|〉 = |2,1,-1〉, |2,1,0〉, |2,1,1〉} associated with this 3-fold degenerate multiplet.

130. As we have already indicated, the total hydrogenic wavefunctions then take the form
In spectroscopic notation the bound states associated with orbital angular momentum quantum numbers ℓ = 0, 1, 2, 3, 4, 5,⋯ are often indicated by letters s, p, d, f, g, h,⋯.
Thus refers to a 2p state with principal quantum number n = 2, orbital angular quantum number ℓ = 1, and azimuthal quantum number m.
It is one of the three states {|n, ℓ, m|〉 = |2,1,-1〉, |2,1,0〉, |2,1,1〉} associated with this 3-fold degenerate multiplet.

131. Let’s look at the hydrogenic wave functions for n = 1 and n = 2.

132. First, we have the single n = 1 state.

133. First, we have the single n = 1 state.

134. Next we have four n = 2 states.

135. The single n = 2 state with ℓ = 0.

136. The single n = 2 state with ℓ = 0.

137. Then the n = 2 states with ℓ = 1.

138. Then the n = 2 states with ℓ = 1.

139. Then the n = 2 states with ℓ = 1.

140. Then the n = 2 states with ℓ = 1.

141. This completes our study of the Coulomb problem.
Using the power series method that we employed for studying the eigenstates of the Coulomb potential, it is also possible to study the bound states of other spherically symmetric systems, such as the isotropic 3D harmonic oscillator.
Such a study, which exploits the spherical symmetry of the harmonic oscillator potential, is also useful for understanding the angular momentum of harmonic oscillator eigenstates.
Indeed, it allows for the construction of a standard representation of angular momentum eigenstates, that are also eigenstates of the energy.
That exercise is left for the interested reader.

142. This completes our study of the Coulomb problem.
Using the power series method that we employed for studying the eigenstates of the Coulomb potential, it is also possible to study the bound states of other spherically symmetric systems, such as the isotropic 3D harmonic oscillator.
Such a study, which exploits the spherical symmetry of the harmonic oscillator potential, is also useful for understanding the angular momentum of harmonic oscillator eigenstates.
Indeed, it allows for the construction of a standard representation of angular momentum eigenstates, that are also eigenstates of the energy.
That exercise is left for the interested reader.

143. This completes our study of the Coulomb problem.
Using the power series method that we employed for studying the eigenstates of the Coulomb potential, it is also possible to study the bound states of other spherically symmetric systems, such as the isotropic 3D harmonic oscillator.
Such a study, which exploits the spherical symmetry of the harmonic oscillator potential, is also useful for understanding the angular momentum of harmonic oscillator eigenstates.
Indeed, it allows for the construction of a standard representation of angular momentum eigenstates, that are also eigenstates of the energy.
That exercise is left for the interested reader.

144. This completes our study of the Coulomb problem.
Using the power series method that we employed for studying the eigenstates of the Coulomb potential, it is also possible to study the bound states of other spherically symmetric systems, such as the isotropic 3D harmonic oscillator.
Such a study, which exploits the spherical symmetry of the harmonic oscillator potential, is also useful for understanding the angular momentum of harmonic oscillator eigenstates.
Indeed, it allows for the construction of a standard representation of angular momentum eigenstates, that are also eigenstates of the energy.
That exercise is left for the interested reader.

145. This completes our study of the Coulomb problem.
Using the power series method that we employed for studying the eigenstates of the Coulomb potential, it is also possible to study the bound states of other spherically symmetric systems, such as the isotropic 3D harmonic oscillator.
Such a study, which exploits the spherical symmetry of the harmonic oscillator potential, is also useful for understanding the angular momentum of harmonic oscillator eigenstates.
Indeed, it allows for the construction of a standard representation of angular momentum eigenstates, that are also eigenstates of the energy.
That exercise is left for the interested reader.