• used to solve problems for quantum mechanical particles that have mass
    • single electron moving slowly (much slower than the velocity of light), neglecting magnetic effects
  • exposes general concepts that come up repeatedly in quantum mechanics
    • underlying linearity of quantum mechanics
    • quantum mechanical amplitudes
    • eigenstates (quantum aspect of quantum mechanics)

from de broglie to schrödinger

electron as waves

  • de Broglie’s hypothesis is that the electron wavelength \( \lambda \) is given by
    • \( \lambda = \frac{h}{p} \)
    • where
      • \( p \) is the electron momentum
      • \( h \) is Planck’s constant
        • \( h = 6.626… \times 10^{-34} Js \)

helmholtz’s wave equation to schrödinger

  • considering only waves of one wavelength \( \lambda \) (i.e. monochromatic waves)
    • \( \frac{d^2 \psi}{d z^2} = -k^2\psi \)
      • \( k = \frac{2\pi}{\lambda} \)
    • has solutions of the form:
      • \( \sin(kz), \cos(kz), \exp(ikz) \)
      • \( \sin(-kz), \cos(-kz), \exp(-ikz) \)
  • in 3D, we can write this as
    • \( \nabla^2 \psi = \frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial x^2} = -k^2\psi \)
    • which has solutions like
      • \( \sin(\textbf{k}\cdot\textbf{r}), \cos(\textbf{k}\cdot\textbf{r}), \exp(i \textbf{k}\cdot\textbf{r}) \)
      • \( \sin(-\textbf{k}\cdot\textbf{r}), \cos(-\textbf{k}\cdot\textbf{r}), \exp(-i \textbf{k}\cdot\textbf{r}) \)
        • where \( \textbf{k} \) and \( \textbf{r} \) are vectors
  • with de broy’s hypothesis
    • \( \lambda = \frac{h}{p} \)
  • and the definition \( k = \frac{2\pi}{\lambda} \)
    • \( k = \frac{2 \pi p}{h} = \frac{p}{h} \)
      • where \( h = \frac{h}{2\pi} \)
      • so \( k^2 = \frac{p^2}{h^2} \)
  • hellholtz equation rewritten
    • \( \nabla^2 \psi = -\frac{p^2}{h^2} \psi \)
    • \( -h^2\nabla^2\psi = p^2\psi \)
  • considering an electron
    • \( -\frac{h^2}{2 m_0}\nabla^2\psi = \frac{p^2}{2m_0}\psi \)
      • dividing both sides by the mass of an electron
    • from classical mechanics
      • \( \frac{p^2}{2m_0} = \) kinetic energy of electron
    • in general:
      • \( E \text{ (total energy) } = \text{ kinetic energy } + \text{ potential energy } (V(\textbf{r})) \)
      • Kinetic Energy = Total Energy - Potential Energy
    • hellholtz Equation:
      • \( -\frac{h^2}{2m_0} \nabla^2 \psi = \frac{p^2}{2m_0} \psi \)
    • schrodinger’s equation
      • \( -\frac{h^2}{2m_0} \nabla^2 \psi = (E - V(\textbf{r})) \psi \)
      • equivalently: \( \left( -\frac{h^2}{2m_0} \nabla^2 + V(\textbf{r}) \right) \psi = E \psi \)
        • this is a time independent equation

schrodinger’s wave equation

  • for any mass \( m \):
    • \( \left( -\frac{h^2}{2m} \nabla^2 + V(\textbf{r}) \right) \psi = E \psi \)
    • this is the time independent shrodinger equation
  • there is no first-principles that precede schrodinger’s equation
    • it is only postulated
    • similar to newton’s gravity theory

probability densities

  • born’s postulates
    • the probability \( P(\textbf{r}) \) of finding an electron near any specific point \( \textbf{r} \) in space
    • probability is proportional to the modulus squared \( \lvert \psi(\textbf{r}) \rvert^2 \)
      • of the wave amplitude \( \lvert \psi(\textbf{r}) \rvert \)
    • \( \lvert \psi(\textbf{r}) \rvert^2 \): is the probability density
    • \( \psi(\textbf{r}) \): probability amplitude (quantum mechanical amplitude)