Table of Contents
1. Schrödinger Equation
- The quantum equivalent of Euler-Lagrange equation.
1.1. Basis Independent Equation
- \[ \hat{H}|\psi\rangle =\hat{E}|\psi\rangle \]
1.2. Position Basis Equation
- \[
\hat{H}\psi=\hat{E}\psi
\]
- Where:
- \(\hat{H}\) is Hamiltonian operator
\[
\hat{H}=\hat{T}+\hat{V}
\]
\[
\hat{T}=\frac{\hat{p}^2}{2m}=-\frac{\hbar^2}{2m}\nabla^2
\]
\[
\hat{p} = -i\hbar\nabla
\]
- The potential operator \(\hat{V}\) is expressed in terms of the position operator \(\hat{x}\), as \(V(\hat{x})\).
- \(\hat{E}\) is energy operator. \[ \hat{E}= i\hbar\frac{\partial}{\partial t} \]
- \(\hat{H}\) is Hamiltonian operator
\[
\hat{H}=\hat{T}+\hat{V}
\]
\[
\hat{T}=\frac{\hat{p}^2}{2m}=-\frac{\hbar^2}{2m}\nabla^2
\]
\[
\hat{p} = -i\hbar\nabla
\]
- Where:
1.2.1. Time-Independent
- When \(\psi\) is of a form \(\psi(\mathbf{x})e^{-iEt/\hbar}\) , we
have \[
\hat{H}\psi = E\psi
\]
- Where \(\psi(\mathbf{x})e^{-iEt/\hbar}\) is called the energy eigenstate, since it can construct further wave functions.
1.3. For Hydrogen Atom
1.3.1. Why the electron doesn't fall to the nucleus?
- Quantum particles cannot be localized, because of quantum fuzziness.
- By uncertainty principle, if a particle happens to be localized, then it would have very high momentum for the particle to fly away.
- It takes energy to localize a particle.
1.3.2. Potential Energy Operator
- \[ \hat{V} = -\frac{e^2}{4\pi\varepsilon_0\hat{r}} \]
1.3.3. Solution
- \[
\psi=R_{n,l}(r)Y_l^m(\theta, \phi)e^{-iEt/\hbar}
\]
- Where,
- \(R_{n,l}(r)\) is radial function.
- \(Y_l^m(\theta, \phi)\) is sperical harmonics.
- Where,
1.3.3.1. Radial Function
- Starts positive at \(r=0\) , flips sign \(n-l-1\) times.
- flattens as \(n\) gets large.
1.3.3.2. Sperical Harmonics
- basis function for all sperical functions.
- smooth at every points.
- it is the orbitals in chemistry when the real part of the function is taken.
- it flips its phase \(l-|m|\) times, when going from north pole to the south.
- it rotates in phase \(m\) times, as circling along the azimuthal plane.