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This corresponds to the expected classical behaviour or particles. @@ -157,4 +157,57 @@ of the penetrating wave sinks to $1/e$ of its initial value after a distance $x Particles with energy $E$ can penetrate into potential areas $\energy{0} > E$ with a certain probability, even if they shouldn't be able to according to classic particle physics. \end{tcolorbox} Once we accept that particles are described by waves, we can come to the conclusion that particles are allowed to exist in \textit{classically forbidden} locations. + +\subsubsection{(b) $E > \energy{0}$} +In this case, the kinetic energy $ekin = E$ of the incidental particle is greater than the potential jump $\energy{0}$, and in the classical model all particles would enter the area $x > 0$, +losing speed in the process, because their kinetic energy decreases to $\ekin = E - \energy{0}$. What does this look like in the wave model? The quantity $\alpha$ is now purely imaginary, so we introduce the real quantity +\begin{equation} + k' = \frac{\sqrt{2m(E - \energy{0})}}{\hbar} = i\alpha +\end{equation} +The solution~\eqref{eq:potentialstepsolution} then becomes +\begin{equation} + \psi_{\text{II}} = Ce^{-ik'x} + De^{ik'x} +\end{equation} +Since there are no particles moving in the $-x$ direction, we know that $C = 0$, and we are left with $\psi_{\text{II}} = De^{ik'x}$. +From the boundary conditions~\eqref{eq:boundaryconditions} we can deduce +\begin{align} + B = \frac{k - k'}{k + k'}A && \text{and} && D = \frac{2k}{k + k'}A +\end{align} +thus leaving us with the wave functions +\begin{equation} + \begin{split} + \psi_{\text{I}}(x) &= A \cdot \left(e^{ikx} + \frac{k - k'}{k + k'} \cdot e^{-ikx} \right) \\ + \psi_{\text{II}}(x) &= \frac{2k}{k + k'}Ae^{ik'x} + \end{split} +\end{equation} +The \textbf{\textit{reflection coefficient}} $R$, i.e.\ the fraction of reflected particles is then analogous to the result from optics +\begin{equation}\label{eq:reflectioncoeff} + R = \frac{\abs{B}^2}{\abs{A}^2} = \abs{\frac{k - k'}{k + k'}}^2 +\end{equation} +\textbf{Remark.} Since the wave number $k$ is proportional to the refractive index in optics ($k = n \cdot k_0$), the result~\eqref{eq:reflectioncoeff} can easily be transformed into the reflection coefficient +\[ + R = \abs{\frac{n_1 - n_2}{n_1 + n_2}}^2 +\] +of a lightwave encountering a boundary surface between two mediums with refractive indexes $n_1$ and $n_2$. + +To compute the amount of particles being transmitted per unit of time (i.e.\ the number of particles passing through the area $x = x_0 > 0$ divided by the amount of particles passing through an area $x < 0$ per unit of time) +one has to consider that the velocities in both areas are different. +The ratio $v' / v = k' / k = \lambda / \lambda'$ is given by the ratio of the wave numbers. That is why the \textbf{\textit{transmission coefficient}} +\begin{equation}\label{eq:transmissioncoeff} + T = \frac{v' \abs{D}^2}{v \abs{A}^2} = \frac{4k \cdot k'}{(k + k')^2} +\end{equation} +From comparing~\eqref{eq:reflectioncoeff} and~\eqref{eq:transmissioncoeff} we can see that +\[ + T + R = 1 +\] +which makes sense because the amount of particles overall has to be conserved. + +\textbf{\textit{Remarks.}} +\begin{itemize} + \item Total reflection also occurs for $E = \energy{0}$. In this case $\alpha = 0$ and $k' = 0$, and thus according to~\eqref{eq:reflectioncoefffirst} or~\eqref{eq:reflectioncoeff} we see that $R = 1$ + \item Instead of a positive potential barrier one can also consider a negative one with $\energy{0} < 0$, where reflection and transmission occurs as well. + One has to let the wave incide from the right in order to yield the same equations. The optical equivalent would be the transition from a medium with higher optical density into one with a lower optical density. +\end{itemize} + +\subsection{Quantum Tunnelling} \end{document} \ No newline at end of file diff --git a/chapters/fundamentals/schroedinger.tex b/chapters/fundamentals/schroedinger.tex index 14e3e39..ceb47cb 100644 --- a/chapters/fundamentals/schroedinger.tex +++ b/chapters/fundamentals/schroedinger.tex @@ -1,4 +1,4 @@ -% !TeX root = ../../script.tex +% !TeX root = ../script.tex \documentclass[../../script.tex]{subfiles} \begin{document} diff --git a/script.pdf b/script.pdf index ece4377061665b77ae62860581acc360f63a2162..6b644657f952521b3fb10e60443bf87b49d42d39 100644 GIT binary patch delta 115210 zcmbTcWk6J2*FVhA(v2X@I)r*tVwNC`?w3rMMe z5)$$rK(AY`=f3|RUOo)x?6dbdYyVd6wbtP`%IbCM3JWd_%M}W8I1gA#ipa{@(#71K z2nok*anNA0*!fU8#t;a^^Yq90pC7B!E60C+?EZOR!}HG@5SR`QIz}U;Fc<>X0t+eH zd6@k10TF0*AQuXRT~LOp=Po6~UvCAUuDs*nZgtxU!^C?jQ6CoU>ex47C^vgv88;U&Qc< zvtb^HMPdTDNHEjBq`-YS249xpY@Aji3S%!xg4qZs#WYEjQHj8SkDtORAweW`6%1g3 zVe;p}Oo)@>o!>o|DmXH7o zU3Y?lkthu0h?idU6o=m*QMGcm@vsFW;KG=vG(70{3F0mvAtYUU0Yp*n%yc=H9*kao zjGf_LP?sGVS04rHT16pUGrklj1?7yr78sg&THXQXR?hkU%`{$Dq_?O#h556cn7F~c zobD9S2+iCvURi3l>_zyfe?r-N&ZBYNRLWFi64~l8z11J`WY^n?O1DP$E|d_NBarjA z1=iZp+JwZP*=>tGpEF}Dmu?ZHi;kOkC(hfXk;Zw6lkkgV_}2;DR3|Qa6I{ZHFE&Z= zqwBD}+qIKd2Kn?7`CY~5p^1dUYJ$4s9j}zNvC|G>$V%@|Hx@W=Hxfhp(6t8I-4iiI zO&r+ILY;rEKl=95iFjsO)yJIV$nra@J;A;C5HvUn_fuQ-XQz~}1donQX{`I(6Xvs5 zC#-g+SBO99i#pu9V86#^e`U{3BKwoChOO9r=GBLOR!{Gs6KaRDzlgE4SaDUCI6u@) zQOCW%KahXKjn^9TEQ2kFa=r@FTjn70qJ*bv8(es_TaRt%F$?NDVSQv86{~bFP=xlO zya+AAr!X;b@}~Q7Q8WM2n%dQiBlPTBgMAPh>+nH&dEX4G@4R!}Y2j1zDSi^ZPX@k# zti61nU^Hc@F-1yL|0uv2ga1+hQIS&x5dWnB|A@TknWlh6Po+)^EDAsWcP?0>=Q1UF zE>0#GBl(*c4OtazQ5YcA*D*nIgqSa~3K$FE+p)Hyz`ZIaO6pXSWiWK|nwYt8RZO+K z)|n2dFvA2PC?ptxLJC5KVPKfBs305#0}De$1Vs_R`>u0<;F!K^Os7IFDvF`I%8LO7 zC^G|7QRfs2DGI~Ez>0Ija3J|>D@Gz;qz z8Kd(G@w#s2q=?^f)BNc+`}0Rg?GwM4oOtAOBT)fEAF+tV7OBf$aeX=N25?^$noW4s z-+7GdJxUuqv~ESa5wwTOrc4`z zvaUyuW=dP_FH2Ydpsa!DNx*e3y{mi?Rh%OW zU+_l2Zy7~ar);)A9%{#E#_d#V1ODl-$q zMdIFIJDr4%9NsU3teQ-pOv6^~YS&YJjaUetfqrh~u2cMUkPLx;k!!YDg0X8e`W@E+58rfdUA^ zo@7J*<1BtV925#8o5D&1ft-rfpY8%Bda5n5|JD}NxqhI|^#gUTAAne&3MJ~?UHs`M zP;fv|R4_5u2r$P=3fL&buUj=hH&ifO*UxoB8Pj=P3-i`jg&6@8L_om^C`=FqXd(@4);Hv2nspXj^CCX&>dh$r|v>SnG%G;0ImYVtU`iOxhlu_ z|5hVx2%?%rWtId73W1+OLJK+D;-8Q}A+S?SIL6-@rT7)}F_^| zafY9A;2$*qC!_vg!D-Z@fe|}m#mt72CVer(!LMF*&?Uu&BhKU!;Nrhj6$%)A0Cb`M z13K)S*-+RyZ~p=rcB+}m7$!drY}lz*Dr4&Xv@n4Y=i&%_i^-o4bMfInw+8_YYM5CI5iehQiNf3V3oRowhXCwQ zcv{i~E$+Dlc137l>>|%a)8e5P=1s~uX2ADP?a<~!%2QHsBE?SsNk{Tg{aY-7;ebd! 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