Hydrogen Atom: Quantum Numbers from Schrödinger's Equation

Introduction

In our last lesson, we uncovered a profound principle: energy quantization is not an arbitrary rule but an inevitable consequence of describing a bound particle, like an electron in an atom, with a wave function. We saw that only for specific energy eigenvalues are the solutions to the Schrödinger equation physically tenable—meaning they don't diverge to infinity and can represent a probability distribution.

Today, we apply this principle to the hydrogen atom. Our goal is to see how the act of solving the Schrödinger equation for this real-world system naturally gives birth to a set of integer labels that define the electron's state. These are the quantum numbers $n, l,$ and $m_{l}$ . You have likely encountered these before as a set of rules to be memorized. Our purpose here is to derive them from first principles, understanding each one as a direct consequence of a fundamental physical or mathematical constraint. We will see that they are not arbitrary rules, but the inherent structure of the solutions.

The Problem in Three Dimensions: The Power of Symmetry

The Schrödinger equation for the hydrogen atom involves a potential energy that depends only on the distance $r$ between the electron and the proton: $V (r) = - \frac{e ^{2}}{4 π ϵ _{0} r}$ . This potential is spherically symmetric. Attempting to solve the 3D Schrödinger equation in Cartesian coordinates $(x, y, z)$ would be a mathematical ordeal because the potential term $V (x^{2} + y^{2} + z^{2})$ mixes all three variables.

The guiding principle here, which you will find throughout physics and mathematics, is to choose a coordinate system that respects the symmetry of the problem. For the hydrogen atom, this means using spherical polar coordinates: $r$ (distance from the origin), $θ$ (polar angle), and $ϕ$ (azimuthal angle).

Solving the Schrödinger Equation for the Hydrogen Atom — This diagram illustrates the core strategy. Due to the spherically symmetric potential $U(r)$ of the hydrogen atom, we use spherical coordinates $(r, \theta, \phi)$. The key is the **separation of variables** method, where we assume the total wave function $\psi(r, \theta, \phi)$ can be factored into a product of three simpler functions, one for each coordinate. This process breaks one formidable equation into three manageable ones, and the solution to each gives rise to one of the three quantum numbers.

The central mathematical technique is called separation of variables. We propose a solution of the form:
$ψ (r, θ, ϕ) = R (r) \cdot Θ (θ) \cdot Φ (ϕ)$
Here, $R (r)$ is the "radial part" depending only on distance, while $Θ (θ)$ and $Φ (ϕ)$ together form the "angular part," often combined into a single function called a spherical harmonic, $Y (θ, ϕ)$ .

By substituting this form into the Schrödinger equation and performing some algebraic manipulation, we can separate the single, complex partial differential equation into three simpler ordinary differential equations. Let's examine how the physical constraints on the solution to each of these three equations give rise to a quantum number.

The Hydrogen Atom

To see the formal overview of this separation, you can refer to the following document. You don't need to follow every mathematical step, but seeing the structure will be very helpful.

Skim the first six pages, focusing on the section titled '7.2: Solution of the Schrödinger Equation for Hydrogen'. Observe how the equation is systematically broken down. Notice how one equation (Eq. 7.3) is separated into three (Eqs. 7.8, 7.10, and 7.11), each involving only one coordinate variable and a 'separation constant'. This process is the source of the quantum numbers.

The Birth of the Quantum Numbers

Each of the three separated equations must be solved with specific boundary conditions to ensure the resulting wave function is physically realistic. It is these conditions that force quantization upon us.

1. The $ϕ$ Equation and the Magnetic Quantum Number, $m_{l}$

The simplest of the three equations is the one for $Φ (ϕ)$ . It has the form:
$\frac{d ^{2} Φ}{d ϕ ^{2}} = - C \cdot Φ$
Where $C$ is a constant. The solutions are of the form $Φ (ϕ) = e^{i m_{l} ϕ}$ , where $m_{l}^{2} = C$ .

The crucial physical constraint here is that the wave function must be single-valued. As $ϕ$ is an angle, a rotation by $2 π$ radians brings you back to the same physical point. Therefore, the wave function must have the same value at $ϕ$ and $ϕ + 2 π$ .
$Φ (ϕ) = Φ (ϕ + 2 π)$
$e^{i m_{l} ϕ} = e^{i m_{l} (ϕ + 2 π)} = e^{i m_{l} ϕ} \cdot e^{i m_{l} 2 π}$
This requires that $e^{i m_{l} 2 π} = 1$ . From Euler's formula, $e^{i x} = cos (x) + i sin (x)$ , this is only true if $m_{l}$ is an integer.

Insight: The magnetic quantum number, $m_{l} = 0, \pm 1, \pm 2, \dots$ , arises from the geometric requirement that the wave function must smoothly connect with itself as you circle the z-axis. A non-integer $m_{l}$ would create a discontinuity.

The following video brilliantly demonstrates why this quantization is necessary.

The Hydrogen Atom, Part 2 of 3: Solving the Schrodinger Equation

This clip from the video 'The Hydrogen Atom, Part 2 of 3' by Richard Behiel visually shows what happens when the quantization condition for 'M' (our m_l) is not met.

Please watch these two short segments: (12:35 - 13:25): This part provides a powerful visual. It shows an animation where 'M' is varied continuously. Notice how the wave function fails to loop back onto itself and is discontinuous unless 'M' is an integer. This is the origin of the quantization of m_l. (30:29 - 32:52): This segment walks through the mathematical argument we just discussed, starting from the separation of variables and arriving at the solution for G(\phi) (our \Phi(\phi)). It explicitly shows how the continuity condition forces 'M' to be a real integer.

2. The $θ$ Equation and the Azimuthal Quantum Number, $l$

The equation for $Θ (θ)$ is more complex (it is the Associated Legendre Equation). We do not need to solve it, but simply understand the constraints on its solutions. It involves both the separation constant from the $ϕ$ equation, $m_{l}$ , and a new separation constant, which we cleverly write as $l (l + 1)$ .

For the wave function to be physically well-behaved—specifically, for it to remain finite at the poles ( $θ = 0$ and $θ = π$ )—the solutions only exist if:

$l$ is a non-negative integer: $l = 0, 1, 2, 3, \dots$
The magnitude of $m_{l}$ is less than or equal to $l$ : $∣ m_{l} ∣ \leq l$ .

Insight: The azimuthal quantum number, $l$ , is quantized to prevent the wave function from "blowing up" at the north and south poles of the atom. It is intrinsically linked to $m_{l}$ , and together they define the shape of the wave function on the surface of a sphere. The combined angular solutions are the spherical harmonics, $Y_{l m_{l}} (θ, ϕ)$ .

3. The $r$ Equation and the Principal Quantum Number, $n$

Finally, we arrive at the radial equation for $R (r)$ . This equation contains the potential energy $V (r)$ , the total energy $E$ , and the angular momentum term characterized by $l (l + 1)$ .

The physical boundary condition here is the same one we encountered in our last lesson: for the electron to be bound to the atom, the wave function must be normalizable. This means $ψ$ must go to zero as $r \to \infty$ .

As before, solutions that satisfy this boundary condition only exist for a discrete set of energy values. When this radial equation is solved, these allowed energies are found to be:
$E_{n} = - \frac{μ e ^{4}}{32 π ^{2} ϵ _{0}^{2} ℏ ^{2}} \frac{1}{n ^{2}} \propto - \frac{1}{n ^{2}}$
The solutions also impose a final constraint on the quantum numbers:

$n$ must be a positive integer: $n = 1, 2, 3, \dots$
$l$ must be strictly less than $n$ : $l < n$ .

Insight: The principal quantum number, $n$ , arises from the requirement that the electron is actually bound to the nucleus. It directly quantizes the energy levels of the atom, recovering the same energy dependence as the Bohr model, but now from a rigorous foundation.

This entire process—from a single PDE to three ODEs and three quantum numbers—is a beautiful example of how mathematical structure and physical reality intertwine.

Schrödinger's equation relation to quantum numbers

The answers on this Physics Stack Exchange page provide excellent, concise summaries of the origin of quantum numbers, connecting them to the concepts of boundary conditions and symmetries that you appreciate.

Please read the top two answers (by Bill N and Peter). Bill N's answer reinforces the central importance of boundary conditions. It clearly states that without these physical constraints, there would be no quantization. Peter's answer introduces a deeper, more elegant perspective: quantum numbers are labels related to the symmetries of the system. For an atom, the spherical symmetry gives rise to the angular momentum quantum numbers (l, m_l), and time-translation symmetry gives rise to the energy quantum number (n).

Guided Discovery

Let's consolidate these ideas.

Problem 1: The Origin of Integers

In your own words, explain why the magnetic quantum number, $m_{l}$ , must be an integer. What would be the physical consequence of a non-integer $m_{l}$ ? Frame your answer in terms of the wave function being a description of a physical state.

Problem 2: The Rules of the Game

For each of the following sets of quantum numbers $(n, l, m_{l})$ , determine if it describes a valid state in a hydrogen atom. If it is invalid, explain which rule is violated.

a) $(2, 1, 2)$
b) $(3, 3, 0)$
c) $(4, 2, - 1)$
d) $(1, 0, 0)$

Problem 3: The Big Picture

The solution to the Schrödinger equation is often called an "eigenvalue problem". In our previous lesson, we saw that 'tuning' the energy $E$ was necessary to find a physically valid wave function. How does the more formal method of separating variables and applying boundary conditions relate to this idea of 'tuning'? Where do the eigenvalues ( $E$ ) and eigenfunctions ( $ψ$ ) appear in today's discussion?

Conclusion

Today, we have un-black-boxed the quantum numbers. We've traced their origins not to a list of memorized rules, but to the fundamental constraints imposed on the solutions of the Schrödinger equation for a physical system.

Key Takeaways:

Solving the Schrödinger equation for the hydrogen atom is made possible by separation of variables in spherical coordinates, a technique that exploits the problem's spherical symmetry.
This process breaks the problem into three separate equations, each corresponding to a coordinate ( $r, θ, ϕ$ ).
Applying physical boundary conditions to the solution of each equation is what forces the solutions to be described by integers—the quantum numbers.
- $m_{l}$ (magnetic) comes from the single-valuedness requirement in the $ϕ$ direction (periodicity).
- $l$ (azimuthal) comes from the requirement that the wave function remains finite at the poles in the $θ$ direction.
- $n$ (principal) comes from the normalizability requirement in the $r$ direction (the electron must be bound) and it directly quantizes the system's energy.
A stable state of the hydrogen atom, an eigenstate, is uniquely defined by a set of three quantum numbers: $ψ_{n, l, m_{l}}$ .

We now have the "address" for each possible state of the electron. But what do these states look like? In our next lesson, we will explore the physical meaning of $n, l,$ and $m_{l}$ by examining the shapes, sizes, and orientations of the atomic orbitals they define. We will also introduce a fourth quantum number, spin ( $m_{s}$ ), an intrinsic property of the electron that does not arise from the Schrödinger equation but is essential for building our model of the atom.

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