Schrödinger Equation and Separation of Variables

Schrödinger Equation and Separation of Variables

• The Schrödinger equation in spherical coordinates is used to describe the behavior of the hydrogen atom: $-\frac{\hbar^2}{2m} \nabla^2 \psi(r, \theta, \phi) + V(r)\psi(r, \theta, \phi) = E\psi(r, \theta, \phi)$
• The Laplacian operator in spherical coordinates is: $\nabla^2 = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 \frac{\partial}{\partial r}) + \frac{1}{r^2 \sin\theta} \frac{\partial}{\partial \theta} (\sin\theta \frac{\partial}{\partial \theta}) + \frac{1}{r^2 \sin^2\theta} \frac{\partial^2}{\partial \phi^2}$
• The separation of variables allows the wavefunction to be expressed as a product of radial and angular functions: $\psi(r, \theta, \phi) = R(r)\Theta(\theta)\Phi(\phi)$

Radial and Angular Equations

• The radial equation is given by: $-\frac{\hbar^2}{2m} [\frac{1}{r^2} \frac{d}{dr} (r^2 \frac{dR(r)}{dr}) - \frac{l(l+1)}{r^2}R(r)] + V(r)R(r) = ER(r)$
• One of the angular equations is: $\frac{1}{\sin\theta} \frac{d}{d\theta} (\sin\theta \frac{d\Theta(\theta)}{d\theta}) + [l(l+1) - \frac{m^2}{\sin^2\theta}]\Theta(\theta) = 0$
• Another Angular Equations is: $\frac{d^2\Phi(\phi)}{d\phi^2} + m^2\Phi(\phi) = 0$

Spherical Harmonics

• Solutions to the angular equations are spherical harmonics: $Y_{l,m}(\theta, \phi) = \Theta(\theta)\Phi(\phi)$
• $l$ represents the azimuthal quantum number, taking values $0, 1, 2,...$
• $m$ is the magnetic quantum number, ranging from $-l$ to $l$ in integer steps: $m = -l, -l+1,..., 0,..., l-1, l$

Hydrogen Atom Specifics

• For the hydrogen atom, the potential is $V(r) = -\frac{e^2}{4\pi\epsilon_0r}$
• $n$ is the principal quantum number, with values $n = 1, 2, 3,...$
• $l$ ranges from $0$ to $n-1$: $l = 0, 1, 2,..., n-1$
• The energy levels are given by: $E_n = -\frac{13.6 \text{ eV}}{n^2}$
• The complete wavefunction is: $\psi_{n,l,m}(r, \theta, \phi) = R_{n,l}(r)Y_{l,m}(\theta, \phi)$

Quantum Number Significance

• $n$ dictates the energy level
• $l$ determines the shape of the orbital
• $m$ specifies the orbital's orientation in space

Atomic Orbitals

• s orbitals have $l = 0$ and are spherical
• p orbitals have $l = 1$ and are dumbbell-shaped, oriented along axes $p_x, p_y, p_z$
• d orbitals have $l = 2$ and exhibit more complex shapes, including $d_{xy}, d_{yz}, d_{xz}, d_{x^2-y^2}, d_{z^2}$
• f orbitals correspond to $l = 3$, possessing even more intricate shapes