3.1 Hydrogen Atom: Wavefunctions and Energy Levels

Quantum mechanics revolutionized our understanding of atoms. For hydrogen, the simplest atom, it describes the electron's behavior using a wavefunction. This function contains all the info about the electron's position and energy in the atom.

The Schrödinger equation is key to solving the hydrogen atom problem. It gives us atomic orbitals, which are characterized by quantum numbers. These orbitals show how the electron is distributed in space and its energy levels.

Quantum description of the hydrogen atom

Hydrogen atom and quantum mechanics

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• The hydrogen atom consists of a single electron bound to a proton, making it the simplest atomic system to study using quantum mechanics
• In quantum mechanics, the state of the electron in the hydrogen atom is described by a wavefunction, denoted as $\Psi(r, \theta, \phi)$, which is a complex-valued function of the electron's position coordinates
• The wavefunction contains all the information about the electron's behavior in the atom, and its square modulus, $|\Psi(r, \theta, \phi)|^2$, represents the probability density of finding the electron at a particular position

Schrödinger equation and atomic orbitals

• The wavefunction must satisfy the time-independent Schrödinger equation for the hydrogen atom, which takes into account the potential energy due to the Coulomb interaction between the electron and the proton
• The solutions to the Schrödinger equation for the hydrogen atom are called atomic orbitals, which are characterized by a set of quantum numbers: principal ($n$), angular momentum ($l$), and magnetic ($m$)
• Atomic orbitals describe the spatial distribution and energy of the electron in the hydrogen atom, with each orbital corresponding to a specific set of quantum numbers

Solving the Schrödinger equation for hydrogen

Time-independent Schrödinger equation

• The time-independent Schrödinger equation for the hydrogen atom in spherical coordinates is given by: $[-\hbar^2/(2m)\nabla^2 - e^2/(4\pi\varepsilon_0r)]\Psi(r, \theta, \phi) = E\Psi(r, \theta, \phi)$, where $\hbar$ is the reduced Planck's constant, $m$ is the electron mass, $e$ is the elementary charge, $\varepsilon_0$ is the permittivity of free space, and $E$ is the energy eigenvalue
• The Schrödinger equation can be solved by separating the variables, leading to a radial part and an angular part, each with its own differential equation

Energy levels and Bohr formula

• The solutions to the radial part of the Schrödinger equation yield the energy levels of the hydrogen atom, given by the Bohr formula: $E_n = -13.6 \text{ eV} / n^2$, where $n$ is the principal quantum number ($n = 1, 2, 3, ...$)
• The energy levels are quantized and depend only on the principal quantum number, with lower values of $n$ corresponding to lower energy states (ground state for $n=1$, excited states for $n>1$)

Spherical harmonics and complete wavefunction

• The solutions to the angular part of the Schrödinger equation are the spherical harmonics, $Y_l^m(\theta, \phi)$, which depend on the angular momentum quantum number ($l$) and the magnetic quantum number ($m$)
• The complete wavefunction for the hydrogen atom is the product of the radial and angular parts: $\Psi_{nlm}(r, \theta, \phi) = R_{nl}(r) \times Y_l^m(\theta, \phi)$
• The complete wavefunction describes the spatial distribution of the electron in the hydrogen atom, with the radial part determining the distance from the nucleus and the angular part determining the shape and orientation of the orbital

Hydrogen atom wavefunctions and quantum numbers

Radial wavefunctions

• The radial part of the wavefunction, $R_{nl}(r)$, depends on the principal quantum number ($n$) and the angular momentum quantum number ($l$). It determines the probability of finding the electron at a certain distance from the nucleus
• The radial wavefunctions have $n-l-1$ nodes (points where the wavefunction crosses zero) and decay exponentially as $r$ increases, ensuring the normalization of the wavefunction
• The number of radial nodes increases with increasing principal quantum number, leading to a more complex radial distribution for higher energy states

Angular wavefunctions and spherical harmonics

• The angular part of the wavefunction, $Y_l^m(\theta, \phi)$, is described by the spherical harmonics and depends on the angular momentum quantum number ($l$) and the magnetic quantum number ($m$)
• The angular wavefunctions determine the shape and orientation of the atomic orbitals. The quantum number $l$ determines the overall shape (s, p, d, f, etc.), while the quantum number $m$ determines the orientation of the orbital in space
• The angular wavefunctions are orthonormal, meaning that they are mutually orthogonal (integrate to zero when multiplied together) and normalized (integrate to one when squared)
• Examples of angular wavefunctions: $Y_0^0$ (s orbital), $Y_1^{-1}, Y_1^0, Y_1^{+1}$ (p orbitals), $Y_2^{-2}, Y_2^{-1}, Y_2^0, Y_2^{+1}, Y_2^{+2}$ (d orbitals)

Orbital angular momentum in hydrogen

Quantum mechanical property and quantization

• Orbital angular momentum is a quantum mechanical property of the electron in the hydrogen atom, arising from its motion around the nucleus. It is characterized by the angular momentum quantum number ($l$)
• The magnitude of the orbital angular momentum is given by $L = \sqrt{l(l+1)}\hbar$, where $l$ can take integer values from 0 to $n-1$ ($l = 0, 1, 2, ..., n-1$)
• The z-component of the orbital angular momentum is quantized and given by $L_z = m\hbar$, where $m$ is the magnetic quantum number, which can take integer values from $-l$ to $+l$ ($m = -l, -l+1, ..., 0, ..., l-1, l$)

Relation to atomic orbitals and spherical harmonics

• The quantization of orbital angular momentum leads to the concept of atomic orbitals with distinct shapes and orientations, such as s, p, d, and f orbitals
• The orbital angular momentum is related to the shape of the atomic orbital through the angular part of the wavefunction, $Y_l^m(\theta, \phi)$, which is described by the spherical harmonics
• The quantum numbers $l$ and $m$ determine the shape and orientation of the atomic orbitals, with increasing $l$ corresponding to more complex shapes and additional angular nodes

Electron probability distributions in hydrogen orbitals

Probability density and wavefunction

• The probability distribution for an electron in a hydrogen atom orbital is given by the square modulus of the wavefunction, $|\Psi_{nlm}(r, \theta, \phi)|^2$. It represents the probability of finding the electron at a particular position in space
• The probability distribution is determined by both the radial and angular parts of the wavefunction, with the radial part affecting the distance from the nucleus and the angular part affecting the shape and orientation of the orbital

Shapes of atomic orbitals

• The shapes of the atomic orbitals are determined by the angular part of the wavefunction, $Y_l^m(\theta, \phi)$, which depends on the angular momentum quantum number ($l$) and the magnetic quantum number ($m$)
• The s orbitals ($l = 0$) are spherically symmetric, with the probability distribution depending only on the radial distance from the nucleus. The 1s orbital has the highest probability density near the nucleus, while higher-energy s orbitals have additional radial nodes
• The p orbitals ($l = 1$) have a dumbbell shape, with two lobes oriented along the x, y, or z axis, depending on the value of the magnetic quantum number ($m = -1, 0, +1$). The p orbitals have a node at the nucleus
• The d orbitals ($l = 2$) have more complex shapes, such as cloverleaf or double-dumbbell, with different orientations depending on the value of the magnetic quantum number ($m = -2, -1, 0, +1, +2$). The d orbitals have two angular nodes
• Higher-energy orbitals (f, g, etc.) have increasingly complex shapes and additional angular nodes, reflecting the increased complexity of the angular wavefunctions for higher values of $l$