[QMSE] W03 - Barriers and Boundary Conditions

• 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 $$-\frac{h^2}{2m}\frac{d^2\psi \left(x\right)}{d{z}^2}+V\left(z\right)\psi \left(z\right)=E\psi \left(z\right)$$

boundary conditions

• presume that $E,V$ and $\psi$ are finite
  • $E$: energy
  • $V$: potential
  • $\psi$: wave function
• it follows that $\frac{d^2\psi }{d{z}^2}$ 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, $\frac{d^2\psi }{d{z}^2}$ 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

• 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 $${\psi }_{\text{left}}\left(z\right)=C\exp \left(ikz\right)+D\exp (-ikz)$$ - where $k=\sqrt{\frac{2mE}{h^2}}$
  • $C\exp \left(izk\right)$ 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 $$-\frac{h^2}{2m}\frac{d^2\psi \left(z\right)}{d{z}^2}+V_0\psi \left(z\right)=E\psi \left(z\right)$$
• rearranging $$\frac{d^2\psi \left(z\right)}{d{z}^2}=\frac{2m}{h^2}(V_0-E)\psi \left(z\right)$$
• the general solution for this wave on the right is $${\psi }_{\text{right}}\left(z\right)=F\exp \left(\kappa z\right)+G\exp (-\kappa z)$$
  • 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 $${\psi }_{\text{right}}\left(z\right)=G\exp (-\kappa z)$$
  • 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: $${\psi }_{\text{left}}\left(z\right)=C\exp \left(ikz\right)+D\exp (-ikz)$$
• on the right: $${\psi }_{\text{right}}\left(z\right)=G\exp (-\kappa z)$$
• 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: $$\begin{array}{cc}C-D& =\frac{i\kappa }{k}G\end{array}$$

incident wave - barrier wave amplitude relationship

• adding: $$\begin{array}{cc}C+D& =G\\ C-D& =\frac{i\kappa }{k}G\\ 2C& =\left(\frac{k+i\kappa }{k}\right)G\\ G& =\frac{2k}{k+i\kappa }C\\ & =\frac{2k(k-i\kappa )}{k^2+{\kappa }^2}C\end{array}$$
• 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: $$\begin{array}{cc}2D& =(1-\frac{i\kappa }{k}G)\\ & =\left(\frac{k-i\kappa }{k}G\right)\\ G& =\frac{2k}{k+i\kappa }C\\ \text{then}& \\ D& =\frac{k-i\kappa }{k+i\kappa }C\end{array}$$
• this relates the amplitude $D$ of the reflected wave
  • to the amplitude of the incident wave $C$

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formal solution for quantum tunnelling

• wave on the left: $$\begin{array}{cc}{\psi }_{\text{left}}\left(z\right)& =C\exp \left(ikz\right)+D\exp (-ikz)\\ \text{where}& \\ D& =\frac{k-i\kappa }{k+i\kappa }C\end{array}$$

• wave on the right: $$\begin{array}{cc}{\psi }_{\text{right}}\left(z\right)& =G\exp (-\kappa z)\\ \text{where}& \\ G& =\frac{2k}{k+i\kappa }C\\ & \end{array}$$

• 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

• electron energy: $1eV$
• barrier energy: $2eV$
• 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”

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• reflection at the barrier is 100%, because $${\left|\frac{D}{C}\right|}^2=\frac{k-i\kappa }{k+i\kappa }\frac{k+i\kappa }{k-i\kappa }=1$$
  • 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

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penetration decay

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

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

probability density in a barrier

• $P\propto {\left|{\psi }_{right}\right(z\left)\right|}^2$
  • ${\psi }_{right}\left(z\right)\propto \exp (-\kappa z)$
• then, ${\left|{\psi }_{right}\right(z\left)\right|}^2\propto \exp (-2\kappa z)$
  • falling by $\frac{1}{e}$ in $\frac{1}{2\kappa }\simeq 0.1nm\equiv 1\AA$

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