• barriers exist in real systems
    • real barriers are not infinite
  • calls for boundary conditions to handle boundary conditions at finite walls
  • reasonable boundary conditions are applied on
    • the wave function at such finite height barriers
    • and the slope of the wave function

schrodinger’s equation

  • given schrodinger’s equation \[ -\frac{h^2}{2m} \frac{d^2\psi(x)}{dz^2} + V(z)\psi(z) = E\psi(z) \]

boundary conditions

  • presume that \( E, V \) and \( \psi \) are finite
    • \( E\): energy
    • \( V\): potential
    • \( \psi\): wave function
  • it follows that \( \frac{d^2\psi}{dz^2} \) is also finite
    • so slope of wavefunction \( \frac{d\psi}{dz}\) must be continuous
    • discontinuities and singularities in the slope
      • causes the second derivate to be infinite
  • slope \( \frac{d\psi}{dz} \) must also be finite
    • else, \( \frac{d^2\psi}{dz^2} \) could be infinite
      • being the limit of a difference involving infinite quantities
    • for \( \frac{d\psi}{dz} \) to be finite
      • \( \psi \) must be continuous
  • so the boundary conditions are
    • \( \psi \) must be continuous
    • \( \frac{d\psi}{dz} \) must be continuous
  • proceed to solve problems with finite heights of boundaries

mechanics - CM vs. QM

  • classical particle
    • a particle like a ball reaching a potential barrier like a wall
      • will bounce of the wall
    • even if the kinetic energy of the ball is more than the potential energy,
      • the ball will need to physically jump over the wall
      • to get over the wall
      • i.e. must have enough potential energy to go over the wall
    • if the barrier is a slope, then the ball’s kinetic energy is converted to potential
      • the ball will roll of over the slope
      • but not if the ball has less potential energy than needed to get over the slope
  • quantum mechanical particle
    • can get over a potential barrier even if energy is less than the height of the potential barrier
    • this is known as quantum tunneling

infinitely thick barrier

  • suppose we have a barrier of height \(V_0\)
    • with potential \(0\) to the left of the barrier

infintely-thick-barrier

  • a quantum mechanical wave is incident from the left
    • the energy \(E\) of this wave is positive \(E > 0\)
    • \(E < V_0\)
  • reflection from the barrier allowed in the region on the left
  • using general solution on the left with complex exponential waves \[ \psi_{\text{left}}(z) = C \exp{(ikz)} + D\exp{(-ikz)} \] - where \( k = \sqrt{\frac{2mE}{h^2}} \)
    • \( C\exp{(izk)} \) is the incident wave, going right
    • \( D\exp{(-izk)} \) is the reflected wave, going left

inside the barrier

  • the wave equation inside the barrier is \[ -\frac{h^2}{2m}\frac{d^2\psi(z)}{dz^2} + V_0\psi(z) = E\psi(z) \]
  • rearranging \[ \frac{d^2\psi(z)}{dz^2} = \frac{2m}{h^2}(V_0 - E ) \psi(z) \]
  • the general solution for this wave on the right is \[ \psi_{\text{right}}(z) = F\exp{(\kappa z)} + G \exp{(-\kappa z)} \]
    • where \( \kappa = \sqrt{\frac{2m(V_0 - E)}{h^2}} \)
    • this solution wave increases exponentially to the right for ever
    • to manage the math and keep it within our current understanding,
      • presume \(F = 0\)
  • the wave on the right inside the barrier is reduced to \[ \psi_{\text{right}}(z) = G \exp{(-\kappa z)} \]
    • where \( \kappa = \sqrt{\frac{2m(V_0 - E)}{h^2}} \)
  • so wave inside the barrier is non-zero
    • it falls off exponentially

applying boundary conditions

  • on the left: \[ \psi_{\text{left}}(z) = C \exp{(ikz)} + D\exp{(-ikz)} \]
  • on the right: \[ \psi_{\text{right}}(z) = G \exp{(-\kappa z)} \]
  • continuity of the wavefunction at \(z = 0\) gives
    • \( C + D = G \)
  • continuity of the wavefunction slope at \(z = 0\) gives
    • \( ikC - ikD = -\kappa G \)
  • rearranging: \[ \begin{align} C - D & = \frac{i\kappa}{k}G \\
    \end{align} \]

incident wave - barrier wave amplitude relationship

  • adding: \[ \begin{align} C + D & = G \\
    C - D & = \frac{i\kappa}{k} G \\
    2C & = \left( \frac{k+i\kappa}{k} \right)G \\
    G & = \frac{2k}{k+i\kappa}C \\
    & = \frac{2k(k-i\kappa)}{k^2 + \kappa^2}C \\
    \end{align} \]
  • this relates the amplitude of the wave inside the barrier \( G \)
    • to the amplitude of the incident wave \( C \)

incident wave - reflected wave amplitude relationship

  • subtracting: \[ \begin{align} 2D & = \left( 1 - \frac{i\kappa}{k}G \right) \\
    & = \left( \frac{k - i\kappa}{k}G \right) \\
    G & = \frac{2k}{k + i\kappa}C \\
    \text{ then } \\
    D & = \frac{k - i\kappa}{k + i\kappa}C \\
    \end{align} \]
  • this relates the amplitude \( D \) of the reflected wave
    • to the amplitude of the incident wave \( C \)

formal solution for quantum tunnelling

  • wave on the left: \[ \begin{align} \psi_{\text{left}}(z) & = C \exp{(ikz)} + D\exp{(-ikz)} \\
    \text{ where } \\
    D & = \frac{k - i\kappa}{k + i\kappa}C \\
    \end{align} \]

  • wave on the right: \[ \begin{align} \psi_{\text{right}}(z) & = G \exp{(-\kappa z)} \\
    \text{ where } \\
    G & = \frac{2k}{k + i\kappa}C \\\ \\
    \end{align} \]

  • where: \[ \kappa = \sqrt{\frac{2m(V_0 - E)}{h^2}} \\
    k = \sqrt{\frac{2mE}{h^2}} \\
    \]

  • assumptions:

    • \( \psi \) must be continuous
    • \( \frac{d\psi}{dz} \) must be continuous
    • solution wave increasing exponentially to the right forever is dropped
    • \( E, V \) and \( \psi \) are finite

quantum-tunnelling

  • electron energy: \( 1 eV \)
  • barrier energy: \( 2 eV \)
  • phase change of the wave in the barrier
    • indicated by change from red to blue
    • exponential decay of wave in barrier
  • phenomenon is called quantum tunneling or tunneling penetration
    • no mathematical idea of penetration
    • “a loose analogy”
  • reflection at the barrier is 100%, because \[ \left\vert \frac{D}{C} \right\vert^2 = \frac{k-i\kappa}{k+i\kappa} \frac{k+i\kappa}{k-i\kappa} = 1 \]
    • this means the amplitude of the wave going in and the amplitude of the wave going out
      • are of the same magnitude
    • there is a phase shift in the reflected wave
    • there is no classical analog to this matter wave phenomenon
  • a similar phenomena is “total internal reflection” for waves
    • although most of the light gets reflected within the denser medium
    • light penetrates a small distance into the rarer medium outside the rare-dense medium’s interface
  • penetration is practically negligible for reflection of what is seen off the interface
  • interface penetration effect important in the physics of optical fibers
    • the core of each fiber is only about \(10\mu m\) diameter
    • tenth of the diameter of human hair
    • exponential penetration is useful to understand behavior of light in fibers
  • this penetration behavior is not uncommon for waves at all
    • but unexpected for matter waves
    • for particles like electrons

penetration decay

  • for a barrier height of \( V_0 = 2 eV \) and electron energy \( E = 1eV \) \[ \begin{align} \kappa & = \sqrt{ \frac{2m(V_0-E)}{h^2} } \\
    & \simeq 5 \cdot 10^{9} m^{-1} \\
    \end{align} \]

tunnelling-penetration

  • the attenuation length of the wave amplitude into the barrier
    • the length to the fall to \( \frac{1}{e}\) of its initial value is
    • \( \frac{1}{\kappa} \cong 0.2 nm \equiv 2 Å \)

probability density in a barrier

  • \( P \propto \left\vert \psi_{right}(z) \right\vert^2 \)
    • \( \psi_{right}(z) \propto \exp{(-\kappa z)} \)
  • then, \( \left\vert \psi_{right}(z) \right\vert^2 \propto \exp{(-2\kappa z)} \)
    • falling by \(\frac{1}{e}\) in \( \frac{1}{2\kappa} \simeq 0.1 nm \equiv 1 Å \)

probability-density-in-a-barrier

particle energy increase

  • as the particle energy increases, the exponential decay gets longer
    • there is more phase change progressively, converging to zero phase change on reflection
  • if \(E > V_0\):
    • both, the reflected (left of barrier) and the through wave (right of barrier) will be oscillatory waves