[QMSE] W02 - schrödinger’s Wave Equation

• 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)

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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
      • $h=6.626\dots \times {10}^{-34}Js$

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:
    • $\sin \left(kz\right),\cos \left(kz\right),\exp \left(ikz\right)$
    • $\sin (-kz),\cos (-kz),\exp (-ikz)$
• 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
    • $\sin (\text{k}\cdot \text{r}),\cos (\text{k}\cdot \text{r}),\exp (i\text{k}\cdot \text{r})$
    • $\sin (-\text{k}\cdot \text{r}),\cos (-\text{k}\cdot \text{r}),\exp (-i\text{k}\cdot \text{r})$
      • where $\text{k}$ and $\text{r}$ 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
  • $-\frac{h^2}{2m_0}{\nabla }^2\psi =\frac{p^2}{2m_0}\psi$
    • dividing both sides by the mass of an electron
  • from classical mechanics
    • $\frac{p^2}{2m_0}=$ kinetic energy of electron
  • in general:
    • $E\text{(total energy)}=\text{kinetic energy}+\text{potential energy}\left(V\right(\text{r}\left)\right)$
    • 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
    • $-\frac{h^2}{2m_0}{\nabla }^2\psi =(E-V(\text{r}\left)\right)\psi$
    • equivalently: $(-\frac{h^2}{2m_0}{\nabla }^2+V(\text{r}\left)\right)\psi =E\psi$
      • this is a time independent equation

schrodinger’s wave equation

• for any mass $m$:
  • $(-\frac{h^2}{2m}{\nabla }^2+V(\text{r}\left)\right)\psi =E\psi$
  • 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

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probability densities

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