In this note I want to explore some aspects of the solution of Schrödinger’s three-dimensional wave equation in spherical polar coordinates which pertain to the three main quantum numbers characterising the electron in a hydrogen atom: the electron’s principal, orbital and magnetic quantum numbers. In particular, I have noticed that many physics and mathematics sources tend to gloss over some (or all) of the mathematical details of how permissibility conditions on the three quantum numbers emerge naturally from validity constraints on the relevant underlying differential equations. I want to bring out these details clearly in the present note.
In general, four quantum numbers are needed to fully describe atomic electrons in many-electron atoms. These four numbers and their permissible values are:
Principal quantum number
Orbital quantum number
Magnetic quantum number
Spin magnetic quantum number
The principal quantum number determines the electron’s energy, the orbital quantum number its orbital angular-momentum magnitude, the magnetic quantum number its orbital angular-momentum direction, and the spin magnetic quantum number its spin direction.
I have noticed that it is often not explained clearly why, for example, the orbital quantum number cannot exceed the principal quantum number minus one, or why the magnitude of the magnetic quantum number cannot exceed that of the orbital quantum number. I will explore issues like these in depth in the context of the time-independent Schrödinger equation for the hydrogen atom, which only involves the first three quantum numbers. I will not discuss the spin magnetic quantum number in the present note.
Schrödinger’s wave equation for the hydrogen atom
In Cartesian coordinates, Schrödinger’s three-dimensional equation for the electron in the hydrogen atom is
where denotes the electron mass. The potential energy is the electric potential energy of a charge given that it is at distance from another charge , namely
It is necessary to change variables in Schrödinger’s equation since the potential energy is a function of radial distance rather than the Cartesian coordinate variables , and . Given the spherical symmetry of the atom, it is sensible to proceed by changing the variables in Schrödinger’s equation to those of spherical polar coordinates (rather than changing the variable in to Cartesian coordinates using ). Only the variables in the Laplacian part of Schrödinger’s equation need to be changed, so we can use the approach in my previous note on changing variables in Laplace’s equation to get
I will now temporarily simplify things by using the representation of the square of the angular momentum operator in spherical polar coordinates which I obtained in another previous note, namely
Using this to replace the two middle terms in Schrödinger’s equation and rearranging we get
This equation can now be solved by the usual separation of variables approach. We assume that the function can be expressed as a product
and then substitute this back into the wave equation to get
Dividing through by we get
Since the left-hand side of this equation depends only on while the right-hand side depends only on and , both sides must be equal to some constant which we can call . Setting the left and right-hand sides equal to in turn and rearranging slightly we finally get the radial equation
and the angular equation
We can now apply separation of variables again to the angular equation. Rewriting the operator in full the angular equation becomes
which simplifies to
We assume that the function can be written as the product
Substituting this into the angular equation gives
Multiplying through by and rearranging we get
Since the left-hand side of this equation depends only on while the right-hand side depends only on , both sides must be equal to some constant which we can provisionally call . Setting the left and right-hand sides equal to in turn and rearranging we get
We now have three ordinary differential equations that need to be solved, one for , one for and one for . We will solve each of them in turn.
The equation for
The equation for is a straightforward second-order differential equation with auxiliary equation
implying if and if . Therefore it has a general solution of the form
if and if , where and are arbitrary constants. Now, the azimuth angle can take any value in but the function must take a single value at each point in space (since this is a required property of the quantum wave function which is a constituent of). It follows that the function must be periodic since it must take the same value at and for any given . This imposes two constraints on the form of the general solution: (1) it cannot consist only of exponential functions with real arguments since these are not periodic (thus ruling out the first general solution above and thereby implying that the separation constant must be nonnegative); (2) must be an integer. Given these constraints, it is customary in quantum mechanics to denote by the letter (it is called the magnetic quantum number) and to specify the separation constant in the angular equations as , which guarantees its nonnegativity. We then state the general solution of the equation for as
In practice, a particular electron wave function will involve only a single value of so only the first of the two terms in the general solution will be necessary. We can therefore state the general solution of the equation for for a given magnetic quantum number as
The equation for
Given that we now know the separation constant for the angular equations is either zero or a positive square number , we can write the equation for as
Expanding the first term we get
I am now going to multiply and divide the first two terms by to get
Now we can make the change of variable which implies and therefore
Using these in the amended form of the equation together with the fact that = , the equation becomes
We will solve this equation first for the case (the solutions will be Legendre polynomials) and use these results to construct solutions for the case (the solutions here will be the associated Legendre functions). Setting we get
which has the form of a well known differential equation known as Legendre’s equation. It can be solved by assuming a series solution of the form
and then differentiating it term by term twice to get
We now substitute these into Legendre’s equation and set the coefficient of each power of equal to zero (because must satisfy Legendre’s equation identically). We find that the coefficient of the term satisfies
This formula makes it possible to find any even coefficient as a multiple of and any odd coefficient as a multiple of . The general solution of our Legendre equation is then a sum of two series involving two arbitrary constants and :
Both of the series in this sum converge for but in general they do not converge for . This is a problem for us because in our change of variables we set and we want solutions that converge for all possible values of including those that result in . It turns out that the only way to get such solutions is to choose integer values of that make either the or the series in the above sum terminate (the other series will generally be divergent so we remove it by setting the corresponding arbitrary constant equal to zero). This requires to take values in the quadratic sequence , , , , , , , The -th term of this sequence is , so the separation constant must be of this form, i.e., for some . When takes an even value the series will terminate and we can set to make the other series vanish. Conversely, when takes an odd value the series will terminate and we can set to make the other series vanish.
From the eigenvalue equation for given earlier () it is clear that the magnitude of the orbital angular momentum is . It is interesting to see how the form of this arises mathematically from considering series solutions to Legendre’s equation above. The parameter is called the orbital angular momentum quantum number.
Note that negative integral values of are allowed but they simply give solutions already obtained for positive values. For example, gives and this makes the series terminate, yielding the polynomial solution
This is exactly the same solution as the one that would be obtained if . It is therefore customary to restrict to nonnegative values. Each possible value of gives a polynomial solution to Legendre’s equation. For we get , for we get , for we get , and so on. If the value of or in each polynomial equation is selected so that when the resulting polynomials are called Legendre polynomials, denoted by . Given that for each we have the first few Legendre polynomials are
These are the physically acceptable solutions to Legendre’s equation for above.
We now consider the solutions for
of the equation
We now know that so we can write this in and we can also add the subscript to as the solutions to this equation will involve a link the between the values of the orbital angular momentum and magnetic quantum numbers. The equation we need to solve becomes
The link between and arises from the fact that we are constrained in trying to solve this equation: it encompasses the case for which the physically acceptable solutions are the Legendre polynomials . Therefore the physically allowable solutions for the above equation must include the Legendre polynomials as a special case. We can find these by using the series approach again and it turns out that the physically acceptable solutions are the so-called associated Legendre functions which take the form
Now, each Legendre polynomial is a polynomial of degree . Therefore the -th order derivative in will equal zero if , so for physically acceptable solutions we must impose the constraint in the differential equation for . This is where the link between the quantum numbers and comes from in the quantum theory of the hydrogen atom: given a value of the acceptable values of are integers in the range .
Finally, note two things: (1) reduces to the Legendre polynomial when , which is what we needed. (2) A negative value for does not change in the original differential equation so a solution for positive is also a solution for the corresponding negative . Thus many references define the associated Legendre function for as .
To conclude, given values for the quantum numbers and , the general solution of the equation for can be written as
The radial equation for R
To clarify where the principal quantum number comes from, the final equation we need to deal with is the radial equation
Writing and replacing with the formula for the potential energy we get
We are only interested in solutions for which the electron is bound within the atom, so we take (the negative energy of the electron is the amount of energy that must be supplied to it to free it from the atom). In order to solve the above equation it is customary to make the change of variable
and define the dimensionless constant
If we then specify we have
Using these results we can rewrite the differential equation as
To make further progress we consider the behaviour of this differential equation as . It reduces to
which is a straightforward second-order differential equation with auxiliary equation
The positive solution to the auxiliary equation implies a term in the general solution of the form which is unacceptable since it explodes as . Therefore we only accept the negative solution to the auxiliary equation and the general solution for as must be of the form
This suggests we can try an exact solution of the full differential equation of the form
Differentiating this twice we get
Substituting these into the differential equation
We can now try to solve this latest version of the differential equation by the method of Frobenius, which involves assuming a generalised power series solution of the form
Differentiating twice we get
Then the terms appearing in the differential equation have the generalised power series forms
Summing these terms (remembering that the sum must be identically equal to zero) we find the coefficient of to be
Now, when the first term of the power series for is which explodes as . This is unacceptable so we discard this solution and set .
For the coefficient of we get
Setting and rearranging gives us the recurrence equation
From this recurrence equation we observe that
as . We deduce from this that the series for becomes like as and therefore becomes like . However, this diverges as which is unacceptable, so we conclude that the series for must terminate at some value of which we will call . In this case we have which the recurrence equation tells us can only happen if
This is how the principal quantum number first appears. Now, we have
These are the famous bound-state energy eigenvalues for . This is the same formula for the energy levels of the hydrogen atom that Niels Bohr obtained by intuitive means in his 1913 ‘solar system’ model of atomic structure.
As stated above, the integer is called the principal quantum number. Recall that and cannot be smaller than zero. It follows that
This explains why for a given value of the allowable values of are .
Returning to the solution of
the above discussion suggests that we should look for a solution of the form
where is a polynomial (rather than an infinite series). Differentiating this twice gives
Substituting these into the differential equation and setting we get
where and . This last form is a well known differential equation whose physically acceptable solutions in the present context are associated Laguerre polynomials given by the formula
For given quantum numbers and , the solution of the radial equation for is then
Final form of the electron wave function
Putting everything together, for given principal quantum number , orbital quantum number and magnetic quantum number , the wave function of the electron in the hydrogen atom is