PARTICLE IN A CENTRAL FORCE FIELD AND THE HYDROGEN ATOM

Quantum-Mechanical Approach and Solution of the Schrödinger Equation

In our previous article, I discussed the fundamental topics of quantum physics. These included the meaning of the wave function ψ, probability density, operator notation and operators, Wave–Particle Duality and the de Broglie Hypothesis, the Time-Dependent Schrödinger Equation, and the Time-Independent Schrödinger Equation. In this article, I will discuss a particle in a central force field and the hydrogen atom. Prof. Dr. Faruk Karadağ's lecture notes were used as a reference for this article.

Particle in a Central Force Field and Reduction of the Problem

Let us examine how the "Two-Body Problem," which provides the foundation for understanding the hydrogen atom in quantum mechanics, is simplified.

1. Definition of a Central Force Field

Symmetry: The potential energy $V(r)$ depends on the particle's distance from the center of force; that is, it has the form $V=V(r)$. This means that the field has spherical symmetry.

Hamiltonian Operator: In a field with central symmetry, the total energy (Hamiltonian) of a single particle is expressed as follows:

$$\hat{H}=-\frac{\hbar^{2}}{2m}\nabla^{2}+V(r)$$

2. The Two-Body Problem: The Hydrogen Atom Model

System: The hydrogen atom is a classical two-body problem in which the nucleus and electron move under their mutual interaction.

Complexity: The system contains two different masses ($m_{1}$ and $m_{2}$) and two different position vectors ($r_{1}$ and $r_{2}$). This means there are six degrees of freedom to solve for.

Classical Energy: The total energy of the system is the sum of the kinetic energies of both particles and their interaction potential.

3. Transformation to Center-of-Mass and Relative Coordinates

To simplify the problem, the coordinate system is changed:

Center of Mass ($r_{0}$): Describes the overall (translational) motion of the system.

Relative Radius Vector (r): Describes the distance and direction between the two particles $(r=r_{2}-r_{1})$.

Mathematical Transformation: When the Laplacian operator $(\nabla^{2})$ is rewritten in terms of center-of-mass and relative coordinates, the Hamiltonian of the system separates into two independent parts.

4. The Concept of Reduced Mass (µ)

Definition: It is the mass that allows us to treat the interaction of two bodies as though it were the motion of a single fictitious particle.

Physical Interpretation: The two-body problem reduces to the motion of a freely moving center of mass with mass $(m_{1}+m_{2})$ and a fictitious particle of mass µ moving in a central force field.

Energy Separation: The total energy (E) is the sum of the overall kinetic energy of the system $(E_{0})$ and its internal energy $(E_{r})$.

REDUCTION OF THE TWO-BODY PROBLEM

Systems of two particles moving under mutual interaction can be transformed into an equivalent one-body problem in a central force field. By using the center of mass and relative radius vectors of the particles, the Hamiltonian operator of the system separates into two independent parts. This separation yields two distinct differential equations describing the translational motion of the center of mass and the internal motion of the system. The Hamiltonian of the system is expressed as follows:

$$\hat{H}=-\frac{\hbar^{2}}{2(m_{1}+m_{2})}\nabla_{0}^{2}-\frac{\hbar^{2}}{2\mu}\nabla_{r}^{2}+V(r)$$

Here, µ represents the reduced mass of the system.

Reduction of the Two-Body Problem and Mathematical Derivation

To solve a system such as the hydrogen atom in quantum mechanics, we must first reduce the complex two-particle system (electron and nucleus) to the motion of a single fictitious particle.

1. Central Force Field and the Initial Hamiltonian

Definition: If the potential energy of a particle depends only on its distance (r) from the center $(V=V(r))$, the field is centrally symmetric.

Two-Particle System: The total Hamiltonian operator for two mutually interacting particles with masses $m_{1}$ and $m_{2}$ is

$$\hat{H}=-\frac{\hbar^{2}}{2m_{1}}\nabla_{1}^{2}-\frac{\hbar^{2}}{2m_{2}}\nabla_{2}^{2}+V(r)$$

2. Coordinate Transformation (Beginning the Derivation)

To simplify the system, we transform from laboratory coordinates $(r_{1},r_{2})$ to Center-of-Mass $(r_{0})$ and Relative Radius (r) coordinates:

Center-of-Mass Coordinates $(x_{0},y_{0},z_{0})$:

$$x_{0}=\frac{m_{1}x_{1}+m_{2}x_{2}}{m_{1}+m_{2}}$$

Relative Coordinates (x, y, z): These form the difference vector between the particles:

$$x=x_{2}-x_{1}$$

3. Calculation of Partial Derivatives (Chain Rule)

The chain rule is applied to transform the $\nabla^{2}$ operators in the Hamiltonian to the new coordinates:

Taking the derivative with respect to $x_{1}$ gives

$$\frac{\partial}{\partial x_{1}}=\frac{\partial}{\partial x}\frac{\partial x}{\partial x_{1}}+\frac{\partial}{\partial x_{0}}\frac{\partial x_{0}}{\partial x_{1}}=-\frac{\partial}{\partial x}+\frac{m_{1}}{m_{1}+m_{2}}\frac{\partial}{\partial x_{0}}$$

Squaring this expression (taking the second derivative) produces similar terms for $x_{1}$ and $x_{2}$.

4. Combining the Laplacian Operators

When the derivatives are added, the cross terms cancel, and the Laplacian operator takes the form

$$\frac{1}{m_{1}}\nabla_{1}^{2}+\frac{1}{m_{2}}\nabla_{2}^{2}=\frac{1}{\mu}\nabla_{r}^{2}+\frac{1}{m_{1}+m_{2}}\nabla_{0}^{2}$$

Two critical concepts emerge here:

Total Mass: $M=m_{1}+m_{2}$ (for the overall motion of the system).

Reduced Mass (µ): The fictitious mass describing the internal motion of the particles.

$$\mu=\frac{m_{1}m_{2}}{m_{1}+m_{2}}$$

5. Result: Separation into Independent Motions

As a result of these operations, the Hamiltonian divides into two independent parts:

Center-of-Mass Motion: The motion of the total mass of the system, which behaves like a free particle.

Relative Motion: The motion of a fictitious particle of mass µ in a central force field.

THE SCHRÖDINGER EQUATION FOR THE HYDROGEN ATOM

The hydrogen atom is a two-body system consisting of an electron and a proton that interact with one another. The electrostatic potential energy of the system depends only on the radial distance:

$$V=-\frac{Ze^{2}}{4\pi\epsilon_{0}r}$$

Because of the central symmetry, it is preferable to solve the Schrödinger equation in spherical polar coordinates (r, θ, φ). The wave function is separated into a product of functions of three independent variables:

$$\psi(r,\theta,\varphi)=R(r)\Theta(\theta)\Phi(\varphi)$$

Let us examine the Schrödinger Equation for the Hydrogen Atom in detail through its physical principles and step-by-step mathematical derivation.

The Schrödinger Equation for the Hydrogen Atom and the Method of Separation of Variables

This section addresses the quantum-mechanical foundation of the hydrogen atom system, formed by the mutual interaction of an electron and a proton, and the strategy for solving its Schrödinger equation.

1. Definition of the System and the Fundamental Equation

System: The hydrogen atom is a two-body structure formed by an electron and a nucleus (proton). As noted in the previous section, using center-of-mass and relative coordinates, this structure has been transformed into the motion of a single body with reduced mass (µ).

General Schrödinger Equation: The time-independent Schrödinger equation describing the internal energy (E) and wave function (ψ) of the system is

$$\nabla^{2}\psi+\frac{2\mu}{\hbar^{2}}(E-V)\psi=0$$

Potential Energy (V): The electrostatic attraction (Coulomb) potential energy between the nucleus and the electron depends only on the radial distance (r) between them:

2. Transformation to Spherical Polar Coordinates

$$V=-\frac{Ze^{2}}{4\pi\epsilon_{0}r}$$

Physical Rationale: Because the potential energy $V(r)$ depends only on r and has no spatial (angular) dependence (central symmetry), it is mathematically much easier to solve the problem in Spherical Polar coordinates (r, θ, φ) rather than Cartesian coordinates (x, y, z).

Expanded Equation: Substituting the expansion of the $\nabla^{2}$ (Laplacian) operator in spherical coordinates into the Schrödinger equation gives the following differential equation:

$$\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{\partial\psi}{\partial r}\right)+\frac{1}{r^{2}\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial\psi}{\partial\theta}\right)+\frac{1}{r^{2}\sin^{2}\theta}\frac{\partial^{2}\psi}{\partial\varphi^{2}}+\frac{2\mu}{\hbar^{2}}(E-V)\psi=0$$

3. Mathematical Derivation: Putting the Equation into Solvable Form

To solve this equation, the method of separation of variables is used. The steps are as follows:

Step 1: Rearranging the Equation To eliminate the fractions and isolate the angular terms, both sides of the equation are multiplied by $r^{2}\sin^{2}\theta$:

$$\sin^{2}\theta\frac{\partial}{\partial r}\left(r^{2}\frac{\partial\psi}{\partial r}\right)+\sin\theta\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial\psi}{\partial\theta}\right)+\frac{\partial^{2}\psi}{\partial\varphi^{2}}+\frac{2\mu r^{2}\sin^{2}\theta}{\hbar^{2}}(E-V)\psi=0$$

Step 2: Separating the Wave Function The wave function $\Psi(r,\theta,\varphi)$ is assumed to be the product of three independent functions, each depending on only one coordinate:

$$\psi(r,\theta,\varphi)=R(r)\Theta(\theta)\Phi(\varphi)$$

Step 3: Calculating the Partial Derivatives The required partial derivatives are taken using the assumed function ψ. For example, when differentiating with respect to r, θ and φ are treated as constants:

$$\frac{\partial\psi}{\partial r}=\Theta\Phi\frac{dR}{dr} \qquad \frac{\partial\psi}{\partial\theta}=R\Phi\frac{d\Theta}{d\theta} \qquad \frac{\partial^{2}\psi}{\partial\varphi^{2}}=R\Theta\frac{d^{2}\Phi}{d\varphi^{2}}$$

Step 4: Substitution and Division The derivatives obtained are substituted into the equation in Step 1. The entire equation is then divided by the product RΘΦ:

$$\frac{\sin^{2}\theta}{R}\frac{\partial}{\partial r}\left(r^{2}\frac{dR}{dr}\right)+\frac{\sin\theta}{\Theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{d\Theta}{d\theta}\right)+\frac{1}{\Phi}\frac{d^{2}\Phi}{d\varphi^{2}}+\frac{2\mu r^{2}\sin^{2}\theta}{\hbar^{2}}(E-V)=0$$

Step 5: Separating the Variables The term $\frac{1}{\Phi}\frac{d^{2}\Phi}{d\varphi^{2}}$, which depends only on the angle φ, is moved to the right-hand side of the equation, while all remaining terms depending on r and θ are left on the left-hand side:

$$\frac{\sin^{2}\theta}{R}\frac{\partial}{\partial r}\left(r^{2}\frac{dR}{dr}\right)+\frac{\sin\theta}{\Theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{d\Theta}{d\theta}\right)+\frac{2\mu r^{2}\sin^{2}\theta}{\hbar^{2}}(E-V)=-\frac{1}{\Phi}\frac{d^{2}\Phi}{d\varphi^{2}}$$

The left-hand side of this final equality depends only on r and θ, whereas the right-hand side depends only on φ. The only way that functions of independent variables can be equal in all cases is for both sides to equal the same constant (the separation constant, $m_{l}^{2}$). This crucial step initiates the decomposition of the partial differential equation into solvable ordinary differential equations.

ANGULAR SOLUTIONS AND QUANTUM NUMBERS

Of the equations obtained by separating the wave function, the function $\Phi(\varphi)$ is required to be single-valued. This condition defines the magnetic quantum number $(m_{l})$, which takes the values $m_{l}=0,\pm1,\pm2,...$. The other equation for the angular part, $\Theta(\theta)$, has solutions expressed in terms of Associated Legendre functions. For physically meaningful solutions, the orbital (azimuthal) quantum number l must be an integer greater than or equal to the absolute value of $m_{l}$ $(l\ge|m_{l}|)$. The product of the functions $\Theta(\theta)$ and $\Phi(\varphi)$ is called a Spherical Harmonic $(Y_{l,m_{l}}(\theta,\varphi))$.

Let us examine Angular Solutions and Quantum Numbers in detail, together with the physical meaning of the mathematical derivation.

In the previous section, we separated the Schrödinger equation into its variables. At this stage, by solving the parts of the equation that depend only on the angles (θ and φ), we will obtain the quantum numbers that determine the shapes of atomic orbitals.

Angular Solutions and Quantum Numbers

1. The Azimuthal-Angle (φ) Equation and the Magnetic Quantum Number $(m_{l})$

Separation: When the Schrödinger equation is separated into its variables, the right-hand side becomes a function entirely dependent on the angle φ, and this expression is set equal to a separation constant such as $-m_{l}^{2}$.

Equation: The resulting simple differential equation is

$$\frac{d^{2}\Phi}{d\varphi^{2}}+m_{l}^{2}\Phi=0$$

Solution and Boundary Condition: The solution of this equation is an exponential function: $\Phi(\varphi)=Ae^{im_{l}\varphi}$

In quantum mechanics, it is physically necessary for the wave function to be single-valued; that is, when the electron returns to the same point in space, the function must have the same value $(\Phi(\varphi)=\Phi(\varphi+2\pi))$.

Result: To satisfy this condition, $m_{l}$ must be an integer. Thus, the Magnetic Quantum Number emerges: $m_{l}=0,\pm1,\pm2,...$. Normalization of the wave function gives the constant as $A=1/\sqrt{2\pi}$.

2. The Polar-Angle (θ) Equation and the Orbital Quantum Number (l)

Separation Constant: When solving the θ-dependent part of the equation, a new separation constant is chosen, as in classical physics, and it is generally written in the form $l(l+1)$.

$$\frac{1}{\sin\theta}\frac{d}{d\theta}\left(\sin\theta\frac{d\Theta}{d\theta}\right)+\left\{l(l+1)-\frac{m_{l}^{2}}{\sin^{2}\theta}\right\}\Theta=0$$

Solution and Boundary Condition: To solve this differential equation, the transformation $x=\cos\theta$ is made, converting it into the "Associated Legendre Equation." For the solutions not to diverge (that is, to remain finite and well-behaved), l must be an integer greater than or equal to $|m_{l}|$ $(l\ge|m_{l}|)$.

Result: This mathematical restriction limits the possible values of $m_{l}$ to $m_{l}=0,\pm1,...,\pm l$. The constant l is called the Orbital (Azimuthal) Quantum Number. The resulting solutions $\Theta_{l,m_{l}}(\theta)$ are called Associated Legendre Polynomials.

3. Spherical Harmonics $(Y_{l,m_{l}})$

The product of the angular wave functions (Θ and Φ) is called the Spherical Harmonic $Y_{l,m_{l}}(\theta,\varphi)$, which gives the angular distribution of the electron in space. These functions describe the three-dimensional "shapes" of the s, p, d, and f orbitals commonly encountered in chemistry.

Using the interactive 3D visualizer below, you can vary the orbital (l) and magnetic $(m_{l})$ quantum numbers and examine the probability-density shapes in space of the Spherical Harmonics (atomic orbitals) obtained from the solutions of the equations.

RADIAL EQUATION AND ENERGY QUANTIZATION

The acceptable asymptotic solutions $R(r)$ of the radial equation are obtained using Associated Laguerre polynomials. The requirement that the power-series solution terminate as a finite polynomial gives rise to the principal quantum number (n). This requirement shows that the energy of the electron in the hydrogen atom is quantized. The energy levels are given by

$$E=-(13.6~eV)\frac{Z^{2}}{n^{2}}$$

Below, I examine the Radial Equation and Energy Quantization in detail, together with the mathematical basis showing why the energy in the hydrogen atom is discrete (quantized).

This stage describes the radial (outward from the center) motion of the electron around the nucleus and mathematically demonstrates the origin of the orbital energy levels (n) used in chemistry.

The Radial Equation, Energy Quantization, and the Principal Quantum Number

1. Formulation of the Radial Schrödinger Equation

Fundamental Equation: Once the angular parts have been solved, the differential equation for the radial part of the wave function $(R(r))$ is obtained. For an electron in a bound state (energy E is negative), the equation is

$$\frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{dR}{dr}\right)+\left[\frac{2\mu(-E)}{\hbar^{2}}+\frac{2\mu Ze^{2}}{4\pi\epsilon_{0}\hbar^{2}r}-\frac{l(l+1)}{r^{2}}\right]R=0$$

Centrifugal Potential: The term $\frac{l(l+1)}{r^{2}}$ in the equation is interpreted as the "centrifugal potential energy" arising from the orbital motion of the electron. This term prevents the electron from approaching the nucleus too closely (except for s orbitals).

2. Mathematical Transformations and Boundary Conditions

To put the equation into a solvable form, the physical boundary conditions are examined:

Dimensionless Variable: To simplify the calculations, the distance r is converted into a dimensionless variable ρ through the transformation $\rho=\alpha r$.

Large-Distance Behavior $(r\rightarrow\infty)$: When the electron is very far from the nucleus, the wave function must decay. This boundary condition requires the asymptotic solution to have the form $R(\rho)=e^{-\rho/2}$ (the solution with a positive exponent is discarded as physically meaningless).

Behavior Near the Origin $(r\rightarrow0)$: To keep the function from diverging (tending to infinity) very near the nucleus, the solution must be proportional to $\rho^{l}$.

3. General Solution and Termination of the Series

Combining the two boundary conditions above, a general solution is proposed for the radial wave function:

Form of the Solution: $R(\rho)=e^{-\rho/2}\cdot\rho^{l}\cdot L(\rho)$. Here, $L(\rho)$ is a new function to be determined.

Power Series: The function $L(\rho)$ is expanded as a power series (an infinite sum), and the relation between its coefficients is determined.

Physical Requirement (Origin of Quantization): If this series continues indefinitely, the function $L(\rho)$ behaves like $e^{\rho}$, causing the wave function to diverge at large distances. This is physically unacceptable.

Termination of the Series: For the function to be valid, the series must terminate at a particular term and become a finite "polynomial."

4. Principal Quantum Number (n) and Energy Levels

Principal Quantum Number (n): For the series to terminate, the constant λ in the equation must be an integer. This integer is the Principal Quantum Number, defined as $n=n^{\prime}+l+1$ $(n=1,2,3,...)$

Quantized Energy: The requirement that λ equal the integer n proves, upon substitution into the energy formula, that the energy cannot take arbitrary values. For the hydrogen atom, the energy levels are therefore

$$E=-(13.6~eV)\frac{Z^{2}}{n^{2}}$$

In conclusion, the electron can occupy only certain (quantized) energy levels around the nucleus because the radial solution of the Schrödinger equation must remain physically meaningful (finite), a mathematical necessity.

5. Associated Laguerre Polynomials

The terminated $L(\rho)$ series becomes an "Associated Laguerre Polynomial." The complete form of the radial wave function is determined by these polynomials, which also reveal the number of nodes (radial distances at which the probability of finding the electron is zero).

Using the tool below, you can explore how the principal quantum number (n) and the orbital quantum number (l) affect the radial wave function and the probability distribution.

ELECTRON PROBABILITY DENSITY AND ITS SHAPE

The probability density of finding the electron at a particular point in space is proportional to the square of the wave function:

$$|\psi|^{2}=|R|^{2}|\Theta|^{2}|\Phi|^{2}$$

It is independent of the azimuthal angle. Since $|\Phi|^{2}=\frac{1}{2\pi}$ is constant, according to quantum mechanics the angular dependence of the electron charge density is determined by $|\Theta|^{2}$. For an s electron $(l=0)$, the charge distribution is completely spherical in space. For a p electron $(l=1)$, the charge distribution takes a dumbbell shape. This shows how the solutions of the Schrödinger equation give us a physical and geometric picture of where the electron may be found in space—that is, the shapes of the orbitals.

Electron Probability Density and Angular Dependence

1. The Complete Wave Function and the Definition of Probability Density

In quantum mechanics, the complete wave function describing the behavior of the electron, $\Psi_{n,l,m_{l}}(r,\theta,\varphi)$, is expressed as the product of radial and angular components:

$$\Psi_{n,l,m_{l}}(r,\theta,\varphi)=R_{n,l}(r)\Theta_{l,m_{l}}(\theta)\Phi_{m_{l}}(\varphi)$$

Physically, the wave function itself cannot be measured directly. According to Max Born's probability interpretation, however, the probability density of finding the electron at a particular point (r, θ, φ) is given by the squared modulus (absolute square) of the wave function:

$$|\psi|^{2}=|R|^{2}|\Theta|^{2}|\Phi|^{2}$$

2. Independence from the Azimuthal Angle (φ)

The part of the function depending on the azimuthal angle (φ) was found to be $\Phi(\varphi)=\frac{1}{\sqrt{2\pi}}e^{im_{l}\varphi}$. To find this function's contribution to the probability density, its product with its complex conjugate is calculated:

$$|\Phi|^{2}=\Phi^{*}\Phi=\left(\frac{1}{\sqrt{2\pi}}e^{-im_{l}\varphi}\right)\left(\frac{1}{\sqrt{2\pi}}e^{im_{l}\varphi}\right)=\frac{1}{2\pi}$$

This mathematical result has an important physical consequence: $|\Phi|^{2}$ is entirely independent of φ and has the constant value $(1/2\pi)$. This shows that the probability of finding the electron at a particular azimuthal angle is the same in every direction. In other words, the electron cloud has circular symmetry about the z-axis.

3. Shape of the Electron Cloud: The $|\Theta|^{2}$ Function

In quantum mechanics, the electron is regarded not as a point particle but as a "charge cloud" (charge density) spread through space. Since independence from φ has been established, the sole component determining the actual geometric shape (angular dependence) of this cloud in space is the function $|\Theta|^{2}$.

4. Shape Analysis by Orbital (Derivation)

s electron (Ground State, $l=0$, $m_{l}=0$):

The normalized angular function obtained from the Associated Legendre polynomials is $\Theta_{00}=\frac{1}{\sqrt{2}}$. Squaring it gives

$$|\Theta_{00}|^{2}=\frac{1}{2}$$

This value is also independent of θ. Because the probability of finding the electron is equal at every angle, the charge distribution of the s electron is completely spherical in space.

p electron ($l=1$, $m_{l}=0$ or $\pm1$): For example, for $m_{l}=0$, the function $\Theta_{10}=\frac{\sqrt{6}}{2}\cos\theta$ is obtained. When squared, the probability density becomes proportional to $\cos^{2}\theta$. This angular dependence causes the electron cloud to concentrate along a particular axis (the z-axis in this case) and the probability to fall to zero toward the center, creating a node. Consequently, the charge distribution of p electrons takes a dumbbell shape. For higher-energy states (d or f orbitals), the angular distribution functions have much more complex geometries (for example, multiple lobes).

MOST PROBABLE DISTANCE FROM THE NUCLEUS

The probability of finding the electron at a distance r from the nucleus (irrespective of angle) is given by the radial probability distribution. The radial probability function is defined by

$$P(r)dr=r^{2}|R_{nl}(r)|^{2}dr$$

For the ground state (the 1s orbital), the radial distance at which the probability of finding the electron is greatest is examined. Satisfying this condition shows that the electron's most probable distance equals the Bohr radius (a₀) $(r=0.53\text{Å})$. Based on the "Most Probable Distance of Electron From Nucleus" section of the source you provided, I explain Slide 7 and the calculus behind it step by step. This slide focuses not on the probability of finding the electron at any particular point in space, but on calculating the probability of finding it at a specific distance r from the nucleus (the probability within a spherical shell).

Most Probable Distance from the Nucleus and Radial Probability

1. Volume Element and Spherical Probability

Concept: In quantum mechanics, the absolute square of the wave function $(|\psi|^{2})$ gives the probability density per unit volume. The probability of finding the electron within an infinitesimal volume dτ at a particular point $(r,\theta,\varphi)$ is calculated from $|\psi|^{2}d\tau$.

Volume in Spherical Coordinates: In spherical coordinates, the volume element is defined as

$$d\tau=r^{2}\sin\theta~dr~d\theta~d\varphi$$

2. Derivation of the Radial Probability Function $P(r)$ for the Ground State (1s)

For the ground state (1s) of the hydrogen atom, the wave function is

$$\psi_{100}(r)=\frac{1}{\sqrt{\pi a_{0}^{3}}}e^{-r/a_{0}}$$

We are interested in the probability of finding the electron, regardless of its angular position (θ and φ), in a spherical shell at a distance r from the nucleus and of thickness dr. To find it, we integrate over all angles:

$$P(r)dr=\int_{0}^{2\pi}d\varphi\int_{0}^{\pi}\sin\theta~d\theta\left(\frac{1}{\pi a_{0}^{3}}e^{-2r/a_{0}}\right)r^{2}dr$$

The φ integral gives 2π, while the θ integral gives 2 $(2\pi\times2=4\pi)$. Substituting these values yields the radial probability-density function:

$$P(r)dr=\frac{4}{a_{0}^{3}}r^{2}e^{-2r/a_{0}}dr$$

3. Finding the Most Probable Distance (Differentiation)

Mathematical Condition: To find the maximum of a function, its first derivative must be set equal to zero. To determine the distance r at which the probability of finding the electron is greatest, the derivative of $P(r)$ with respect to r is taken:

$$\frac{d}{dr}P(r)=\frac{d}{dr}\left[\frac{4}{a_{0}^{3}}r^{2}e^{-2r/a_{0}}\right]=0$$

Applying the product rule gives

$$\frac{4}{a_{0}^{3}}\left[(2r\cdot e^{-2r/a_{0}})+\left(r^{2}\cdot\frac{-2}{a_{0}}e^{-2r/a_{0}}\right)\right]=0$$

Factoring out the common factors gives

$$\frac{4}{a_{0}^{3}}e^{-2r/a_{0}}\cdot2r\left[1-\frac{r}{a_{0}}\right]=0$$

For this equality to hold, the expression in square brackets must vanish (the other roots, $r=0$ and $r=\infty$, are points at which the probability is minimal).

4. Result and Physical Meaning

The result directly gives the most probable distance:

$$1-\frac{r}{a_{0}}=0 \implies r=a_{0}$$

Physical Interpretation: Quantum-mechanical calculations prove that the point at which the probability of finding the 1s electron is greatest lies exactly at the Bohr radius $(a_{0}=0.53~\text{Å})$. This result forms a remarkable mathematical bridge demonstrating the consistency of Schrödinger wave mechanics with the older Bohr model.

DEGENERACY OF HYDROGEN ENERGY LEVELS

A quantum state is uniquely characterized by a set of the quantum numbers n, l, and $m_{l}$. Owing to the symmetry of the Coulomb potential, different subshells with the same principal quantum number (n), such as 2s and 2p, have the same energy. Furthermore, as in every central-field potential, states with the same n and l but different values of $m_{l}$ have the same energy. For this reason, each energy level has a degeneracy of $2l+1$. An externally applied magnetic field lifts this degeneracy by splitting the energy levels; this is known as the Zeeman Effect.

Let us examine the reasoning behind the "Degeneracy of Hydrogen Energy Levels."

In quantum mechanics, "degeneracy" refers to the situation in which different quantum states (different wave functions) have exactly the same energy level.

Degeneracy of Hydrogen Energy Levels

1. Specification of Quantum States

In quantum mechanics, each electron state is characterized by a set of three quantum numbers $(n,l,m_{l})$.

Ground State: When the principal quantum number is $n=1$, the orbital quantum number must be $l=0$, and the magnetic quantum number must be $m_{l}=0$. This state is denoted by the wave function $\Psi_{100}$ and is the ground state of the hydrogen atom.

2. Coulomb Degeneracy (Degeneracy with Respect to n)

In the hydrogen atom, the total energy of the electron depends only on the principal quantum number (n). Consequently, different orbitals with the same value of n have the same energy.

As an explicit example, let us consider the $n=2$ level:

If $n=2$, the quantum number l may take the value 0 or 1. For $l=0$, there is a single state: $m_{l}=0$ ($\Psi_{200}$). For $l=1$, there are three distinct states: $m_{l}=1,0,-1$ ($\Psi_{211},\Psi_{210},\Psi_{21-1}$).

Thus, for $n=2$, there are four different wave functions (states) in total, but because the total energy depends only on n, all four states have exactly the same energy. This is called "fourfold degeneracy." This special type of degeneracy in hydrogen arises from the perfect symmetry of the pure Coulomb potential.

3. Screening Effect (Many-Electron Atoms)

In atoms other than hydrogen (many-electron atoms), the Coulomb potential is modified because electrons in inner orbitals "screen" the nucleus (the screening effect).

This modification breaks the symmetry and causes the energy to depend not only on n but also on l, which determines the shape of the orbital $(E_{2s}\ne E_{2p})$.

4. Central-Field Degeneracy and Space Quantization

In addition to the situation above, there is a second type of degeneracy that applies in all central force fields (fields in which the potential depends only on r). States that have the same n and l but different $m_{l}$ values share the same energy. Mathematically, because there are $2l+1$ possible $m_{l}$ values for a given l, each energy level has a $2l+1$-fold degeneracy.

5. Lifting Degeneracy: The Zeeman Effect

This equality of energies (degeneracy) is removed when a non-central field, such as an external magnetic field, is applied to the atom. The external magnetic field causes energy levels with different orientations (different $m_{l}$ values) to split into distinct energies. This splitting of energy levels into sublevels under an external magnetic field is known in the physics literature as the Zeeman Effect.

ANGULAR MOMENTUM AND SPACE QUANTIZATION

When the wave function ψ is acted upon by the squared angular-momentum operator $(\hat{L^{2}})$, it yields the eigenvalue

$$|L|^{2}=l(l+1)\hbar^{2}$$

This result states that the magnitude of the orbital angular momentum is quantized. The eigenvalue of the z-component angular-momentum operator $(L_{z})$ is $m_{l}\hbar$. The fact that the z-component can take only discrete values shows that the angular-momentum vector (and the associated magnetic moment) can have only certain orientations in space. This physical phenomenon is defined as "space quantization." Here, mathematical operators demonstrate that not only the magnitude of the electron's orbital angular momentum but also its orientation in space takes discrete (quantized) values.

Angular Momentum and Space Quantization

1. Squared Angular Momentum $(\hat{L^{2}})$ and the Orbital Quantum Number

Whereas angular momentum can take continuous values in classical mechanics, in quantum mechanics it is restricted by the eigenvalue equations of operators. The squared angular-momentum operator $(\hat{L^{2}})$ is defined in spherical coordinates as

$$\hat{L^{2}}=-\hbar^{2}\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial\varphi^{2}}\right]$$

When this operator is applied to the complete wave function $(\psi=R\Theta\Phi)$, the radial part (R) is unaffected:

$$\hat{L^{2}}\psi=-\hbar^{2}R\left[\frac{\Phi}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial\Theta}{\partial\theta}\right)+\frac{\Theta}{\sin^{2}\theta}\frac{\partial^{2}\Phi}{\partial\varphi^{2}}\right]$$

From the solution of the azimuthal equation, we know that $\frac{\partial^{2}\Phi}{\partial\varphi^{2}}=-m_{l}^{2}\Phi$. Substituting this value gives

$$\hat{L^{2}}\psi=-\hbar^{2}R\Phi\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial\Theta}{\partial\theta}\right)-\left(\frac{m_{l}^{2}}{\sin^{2}\theta}\right)\Theta\right]$$

The expression inside the square brackets is precisely the left-hand side of the Associated Legendre differential equation previously solved for the angle θ, and it equals $-l(l+1)\Theta$. Making this substitution simplifies the result dramatically:

$$\hat{L^{2}}\psi=l(l+1)\hbar^{2}\psi$$

Physical Result: A measurement of the square of angular momentum will always yield the value $l(l+1)\hbar^{2}$. It follows that the magnitude (length) of the angular-momentum vector is quantized:

$$|L|=\sqrt{l(l+1)}\hbar$$

2. The Z Component $(L_{z})$ and the Magnetic Quantum Number

The operator for the z component of angular momentum is much simpler and depends only on the angle φ:

$$\hat{L_{z}}=-i\hbar\frac{\partial}{\partial\varphi}$$

When this operator is applied to the wave function, only the function $\Phi(\varphi)=Ae^{im_{l}\varphi}$ is differentiated:

$$\hat{L_{z}}\psi=-i\hbar R\Theta\frac{\partial\Phi}{\partial\varphi}=-i\hbar R\Theta(im_{l}\Phi)$$

Since $i\times i=-1$, the minus signs cancel, giving

$$\hat{L_{z}}\psi=m_{l}\hbar\psi$$

Physical Result: The projection of the angular-momentum vector (L) onto the z-axis likewise cannot take arbitrary values; it can take only the values $m_{l}\hbar$.

3. Space Quantization and the Vector Model

Because both the length of the L vector $(|L|)$ and its projection onto the z-axis $(L_{z})$ are fixed, this vector cannot point in an arbitrary direction in space. The possible angles (θ) it can make with the z-axis are governed by definite rules and are given by

$$\cos\theta=\frac{m_{l}}{\sqrt{l(l+1)}}$$

Example 1 (p electron, $l=1$): The vector has length $|L|=\sqrt{2}\hbar$. Its z components may be $L_{z}=\hbar,0,-\hbar$. The vector can have only three distinct orientations in space.

Example 2 (d electron, $l=2$): The vector has length $|L|=\sqrt{6}\hbar$. Its z components may be $L_{z}=2\hbar,1\hbar,0,-1\hbar,-2\hbar$. The vector can have only five distinct orientations in space.

In the vector model of atomic theory, this is called "Space Quantization." Because the x and y components of the vector cannot be known precisely (the Uncertainty Principle), the L vector continuously wobbles in a circle at a fixed angle about the z-axis (it undergoes precessional motion).