Quantum mechanical model Knowledge-Based Study

Study notes

1. The Bohr model successfully accounted for the discrete emission spectrum of atomic hydrogen by assuming that electrons travel in circular, fixed orbits around the nucleus. However, it failed to predict the spectra of multi-electron atoms or explain chemical bonding. Classical mechanics could not adequately describe the behavior of subatomic particles.

   Two revolutionary principles laid the foundation for modern quantum mechanics:

   • De Broglie Hypothesis: In 1924, Louis de Broglie proposed that particles such as electrons possess wave-like characteristics alongside particle properties. The wavelength of a particle is given by the relation:
     $\lambda = \frac{h}{p} = \frac{h}{mv}$
     where $h$ is Planck's constant, $m$ is mass, and $v$ is velocity.
   • Heisenberg Uncertainty Principle: Werner Heisenberg demonstrated that it is fundamentally impossible to simultaneously determine both the exact position ($x$) and exact momentum ($p$) of an electron. Mathematically:
     $\Delta x \cdot \Delta p \ge \frac{h}{4\pi}$
     Consequently, the classical idea of a well-defined trajectory or orbit must be replaced with statistical probabilities.

   In 1926, Erwin Schrödinger formulated a mathematical equation that treats the electron as a three-dimensional standing wave influenced by the electric field of the nucleus:

   $$\hat{H}\psi = E\psi$$

   Here, $\hat{H}$ is the Hamiltonian operator, $E$ is the total energy of the system, and $\psi$ is the wave function.

   Physical Meaning of the Wave Function

   While the wave function $\psi$ itself has no direct physical meaning, its square, $|\psi|^2$ (or probability density), represents the probability of finding an electron within a given volume of space around the nucleus (Max Born interpretation).

   Orbit vs. Orbital

   • Bohr Orbit: A distinct, two-dimensional circular path at a fixed distance from the nucleus where an electron was thought to revolve with known momentum.
   • Atomic Orbital: A three-dimensional region of space around the nucleus where there is a high probability (typically defined as 90% or 95%) of locating an electron.

   Solutions to the Schrödinger equation for an electron in an atom yield sets of mathematical parameters called quantum numbers. Three quantum numbers describe the orbital, while a fourth describes the intrinsic magnetic property of the electron:

   1. Principal Quantum Number ($n$): Defines the main energy level (shell) and the relative size of the orbital. Values: $n = 1, 2, 3, \dots$. Larger $n$ values correspond to greater average distance from the nucleus and higher energy.
   2. Angular Momentum Quantum Number ($l$): Defines the shape of the orbital (subshell). For a given $n$, $l$ can take integer values from $0$ to $n - 1$. Each numerical value corresponds to a specific subshell letter:
      • $l = 0$: s orbital
      • $l = 1$: p orbital
      • $l = 2$: d orbital
      • $l = 3$: f orbital
   3. Magnetic Quantum Number ($m_l$): Specifies the spatial orientation of the orbital in space relative to a magnetic field. Values range from $-l$ through $0$ to $+l$. The total number of orientations per subshell is $2l + 1$.
   4. Electron Spin Quantum Number ($m_s$): Independent of the spatial wave function, it describes the intrinsic angular momentum of the electron. It has two possible values: $+\frac{1}{2}$ or $-\frac{1}{2}$.

   Each type of atomic orbital exhibits a characteristic geometry determined by its angular wave function:

   • s Orbitals ($l = 0$): Spherically symmetrical around the nucleus. There is one s orbital per energy level ($m_l = 0$).
   • p Orbitals ($l = 1$): Dumbbell-shaped, consisting of two lobes separated by a nodal plane passing through the nucleus. There are three mutually perpendicular p orbitals per level: $p_x$, $p_y$, and $p_z$ ($m_l = -1, 0, +1$).
   • d Orbitals ($l = 2$): A set of five degenerate orbitals ($d_{xy}, d_{yz}, d_{xz}, d_{x^2-y^2}, d_{z^2}$) with mostly cloverleaf geometries, except $d_{z^2}$, which has two lobes along the z-axis encircled by a doughnut-shaped torus.
   • f Orbitals ($l = 3$): A set of seven complex, multi-lobed orbitals ($m_l = -3, -2, -1, 0, +1, +2, +3$).

   Nodes

   A node is a region where the probability density $|\psi|^2$ drops to zero:

   • Radial Nodes: Spherical shells where electron density is zero, given by $n - l - 1$.
   • Angular Nodes (Nodal Planes): Planar or conical surfaces given by $l$.
   • Total Nodes: The sum of radial and angular nodes, equal to $n - 1$.