From 921427371fb1f0f232683f79774932c73aae0c82 Mon Sep 17 00:00:00 2001 From: Lauchmelder Date: Fri, 14 Oct 2022 01:28:13 +0200 Subject: [PATCH] vrzgh --- chapters/fundamentals/examples.tex | 63 ++++++++++++++++++ chapters/fundamentals/fundamentals.tex | 27 ++++++++ chapters/fundamentals/schroedinger.tex | 87 +++++++++++++++++++++++++ script.pdf | Bin 75134 -> 258016 bytes script.tex | 17 +++++ 5 files changed, 194 insertions(+) create mode 100644 chapters/fundamentals/examples.tex create mode 100644 chapters/fundamentals/fundamentals.tex create mode 100644 chapters/fundamentals/schroedinger.tex diff --git a/chapters/fundamentals/examples.tex b/chapters/fundamentals/examples.tex new file mode 100644 index 0000000..e2ae95b --- /dev/null +++ b/chapters/fundamentals/examples.tex @@ -0,0 +1,63 @@ +% !TeX root = ../../script.tex +\documentclass[../../script.tex]{subfiles} + +\begin{document} +\section[Examples of the Stationary Schrödinger Equation]{Examples of the Stationary Schrödinger Equation} + +We now want to solve the Schrödinger equation +\[ + \frac{-\hbar^2}{2m} \dv[2]{\psi}{x} + \epot\psi = E\psi +\] +for a few simple, one-dimensional problems. These examples will illustrate the description of classical particles as waves and the following physical consequences. + +\subsection{The Free Particle} + +A particle is said to be free, if it is moving in a constant potention $\phi_0$, because then $\vec{F} = -\grad{\epot}$ means that no forces are acting on the particle. +Through a suitable choice of the zero point energy we can set $\phi_0 = 0$, i.e. $\epot = 0$, and thus get the Schrödinger equation for a free particle +\begin{equation}\label{eq:onedschroedinger} + \frac{-\hbar^2}{2m} \dv[2]{\psi}{x} = E\psi +\end{equation} +The total energy $E = \ekin + \epot$ is because of $\epot$ now +\[ + E = \frac{p^2}{2m} = \frac{\hbar^2 k^2}{2m} +\] +Thus~\eqref{eq:onedschroedinger} gets reduced to +\[ + \dv[2]{\psi}{x} = -k^2 \psi +\] +which has the general solution +\begin{equation}\label{eq:onedsolution} + \psi(x) = Ae^{ikx} + Be^{-ikx} +\end{equation} +The time-dependent wave function +\begin{equation} + \psi(x, t) = \psi(x) \cdot e^{-i\omega t} = Ae^{i(kx - \omega t)} + Be^{-i(kx + \omega t)} +\end{equation} +represents the superposition of a planar wave travelling in the $+x$ and $-x$ direction. + +The coefficients $A$ and $B$ are the amplitudes of those waves, which are determined by the boundary conditions. +For example, the wave function of electrons which are emitted from a cathode in $+x$ direction towards a detector, will have $B = 0$, since there are no particles moving in $-x$. +From this experimental setup we know that the electrons are found along the length $L$ of the path between cathode and detector. This means their wave function can only be different from zero in this region of space. +Using the normalization condition we get +\begin{align*} + &\int_0^L \abs{\psi(x)}^2 \dd{x} = 1 \\ + &\implies A^2 \cdot L = 1 \implies A = \frac{1}{\sqrt{L}} +\end{align*} +To determine the location of a particle at time $t$ more accurately, we will have to construct \textit{wave packets} in place of planar waves~\eqref{eq:onedsolution} +\begin{equation} + \psi(x, t) = \int_{k_0 - \Delta k / 2}^{k_0 + \Delta k / 2} A(k) e^{i(kx - \omega t)} \dd{k} +\end{equation} +The location uncertainty of this packet at $t = 0$ is +\[ + \Delta x \ge \frac{\hbar}{2 \Delta p_x} = \frac{1}{2\Delta k} +\] +and depends on the pulse width $\Delta p_x = \hbar \Delta k$. The larger $k$ is, the more certainly $\Delta x(t = 0)$ can be determined, but the faster the wave packet spreads. + +Experimentally, this can be illustrated as follows: If we apply a short voltage pulse to the cathode at time $t = 0$, then electrons can start travelling towards the detector at this instance. +The emitted electrons have a velocity distribution $\Delta v$, such that electrons with differing velocities $v$ will not necessarily be in the same location $x$ at a later point in time $t$. +Instead they are spread over the interval $\Delta x(t) = t \cdot \Delta v$. The velocity distribution is described by $\Delta v \propto \Delta k$ of the wave packet, such that the location uncertainty $\Delta x$ +\[ + \dv{(\Delta x(t))}{t} = \Delta v(t = 0) = \frac{\hbar}{m} \Delta k(t = 0) +\] +changes proportionally to the initial impulse uncertainty. +\end{document} \ No newline at end of file diff --git a/chapters/fundamentals/fundamentals.tex b/chapters/fundamentals/fundamentals.tex new file mode 100644 index 0000000..b01d2cb --- /dev/null +++ b/chapters/fundamentals/fundamentals.tex @@ -0,0 +1,27 @@ +% !TeX root = ../script.tex +\documentclass[../../script.tex]{subfiles} + +\begin{document} + \chapter{Fundamentals of Quantum Physics} + \minitoc + \vspace*{\fill}\par + \pagebreak + + Due to the uncertainty principle, location and impulse of an atomic particle cannot be both stated with arbitrairy precision. + The classic trajectory, represented in the model of mass points by a well-defined curve in space $\vec{r}(t)$, is replaced by the probability + \begin{equation} + W(x, y, z, t) \dd{v} = \abs{\psi(x, y, z, t)}^2 \dd{V} + \end{equation} + to find the particle in the volume element $\dd{V} = \dd{x}\dd{y}\dd{z}$ at the time $t$. This probability depends on the absolute square of the matter wave function $\psi(x, y, z, t)$. + + In this chapter we wnat to show how this wave function can be calculated for simple examples. These examples will also demonstrate the physical fundamentals of quantum mechanics and its + differences to classical particle mechanics, elaborate on the concept of \textit{quantum numbers} and show under which conditions quantum mechanical results can be transitioned into classical physics. + This is supposed to clarify that classic (i.e.\ pre-quantum) mechanics are contained in quantum mechanics as a limiting case for very small de Broglie wavelengths $\lambda_{dB} \rightarrow 0$. + + These examples should also demonstrate that almost all insights of quantum mechanics are already known in classic wave optics. This means: the actual novel concepts in quantum mechanics + is the description of classic particles with matter waves. The deterministic description of the temporal development of location and impulse of a particle is thus replaced with a statistical + treatment, by which we can only discuss probabilities of the results of a measurement. A fundamental uncertainty occurs when we observe location and impulse at the same time. + + \subfile{schroedinger.tex} + \subfile{examples.tex} +\end{document} \ No newline at end of file diff --git a/chapters/fundamentals/schroedinger.tex b/chapters/fundamentals/schroedinger.tex new file mode 100644 index 0000000..14e3e39 --- /dev/null +++ b/chapters/fundamentals/schroedinger.tex @@ -0,0 +1,87 @@ +% !TeX root = ../../script.tex +\documentclass[../../script.tex]{subfiles} + +\begin{document} +\section[The Schrödinger Equation]{The Schrödinger Equation} + +In this section we will outline the fundamental equation of quantum mechanics, which was established by \textit{Erwin Schrödinger} (1887--1961) in 1926. +The solutions of this equation are the desired wave functions $\psi(x, y, z, t)$. However, these solutions can only be derived in analytical form for a few very simple physical problems. +Very fast computers are usually able to numerically compute solutions for complex problems. + +First, we will consider the mathematically simplest case.A free particle of mass $m$ which moves at a constant velocity $\vec{v}$ in direction $x$. +With $\vec{p} = \hbar\vec{k}$ and $E = \hbar \omega = \ekin$ (because $\epot = 0$), we know the wave function must be of the form +\begin{equation}\label{eq:wavefuncform} + \psi(x, t) = Ae^{i(kx-\omega t)} = Ae^{(i/\hbar)(px-\ekin t)} +\end{equation} +Here we use the fact that $\ekin = p^2 / 2m$, which is the kinetic energy of the particle. Since the mathematical representation is absolutely identical to that of an electromagnetic wave, +it makes sense to start with the wave equation +\begin{equation}\label{eq:1dwaveeq} + \pdv[2]{\psi}{x} = \frac{1}{u^2} \pdv[2]{\psi}{t} +\end{equation} +for waves propagating with phase velocity $u$ in direction $x$. For \textit{stationary} problems where $\vec{p}$ and $E$ are time-independent the wave function can be split into a +strictly location-dependent factor $\psi(x) = Ae^{ikx}$, and a strictly time-dependent phase factor $e^{-i\omega t}$. Thus we can write +\begin{equation}\label{eq:splitwave} + \psi(x, t) = \psi(x) \cdot e^{-i\omega t} = A e^{ikx} \cdot e^{-i\omega t} +\end{equation} +If we use the ansatz~\eqref{eq:splitwave} in the wave equation~\eqref{eq:1dwaveeq}, and the fact that $k^2 = p^2 / \hbar^2 = 2m\ekin/\hbar^2$, we get the equations +\begin{equation}\label{eq:values} + \begin{split} + &\pdv[2]{\psi}{x} = -k^2\psi = -\frac{2m}{k} \cdot \ekin \cdot \psi \\ + &\pdv[2]{\psi}{t} = -\omega^2 \psi + \end{split} +\end{equation} +Comparison with~\eqref{eq:1dwaveeq} gives us +\[ + u^2 = \frac{\omega^2}{k^2} \implies u = \frac{\omega}{k} +\] +Note that the particle velocity $v_\text{P} = v$ +\[ + v = \frac{p}{m} = \frac{\hbar k}{m} = \pdv{\omega}{k} +\] +is different from the phase velocity $u = v_{\text{ph}} = \omega / k$. + +In the general case the particle can move in a force field. If it is conservative then we can assign each point a potential energy, with the condition that the total energy $E = \ekin + \epot$ remains constant. +Using $\ekin = E - \epot$ and~\eqref{eq:values} we then receive the one-dimensional stationary Schrödinger equation +\begin{tcolorbox}[ams equation] + \frac{-\hbar^2}{2m} \pdv[2]{\psi}{x} + \epot\psi = E\psi +\end{tcolorbox} +For the general case where the particle is moving freely in three-dimensional space we can use the three-dimensional wave equation +\[ + \laplacian\psi = \frac{1}{u^2} \pdv[2]{\psi}{t} +\] +and the ansatz $\psi(x, y, z, t) = \psi(x, y, z) \cdot e^{-i\omega t}$, we can establish the three-dimensional stationary Schrödinger equation +\begin{tcolorbox}[ams equation]\label{eq:stationaryschroedinger} + \frac{-\hbar^2}{2m} \laplacian \psi = \epot \psi = E \psi +\end{tcolorbox} +If we differentiate~\eqref{eq:wavefuncform} partially for time we receive +\[ + \pdv{\psi}{t} = -\frac{i}{h} \ekin \cdot \psi +\] +and with~\eqref{eq:values} we can find the time-dependent equation for a free particle with $\epot = 0$ (i.e. $\ekin = \const$) +\begin{equation} + -\frac{\hbar^2}{2m} \pdv[2]{\psi(x, t)}{x} = i\hbar \pdv{\psi(x, t)}{t} +\end{equation} +The three-dimensional representation is then +\begin{tcolorbox}[ams equation] + -\frac{\hbar^2}{2m} \laplacian \psi(\vec{r}, t) = i\hbar \pdv{\psi(\vec{r}, t)}{t} +\end{tcolorbox} +There are some remarks to be made: +\begin{itemize} + \item In this ``derivation'' we have used the de Broglie-relationship $\vec{p} = \hbar \vec{k}$, which is only supported by experiments and has no mathematical justification. + \item The law of conservation of energy of quantum mechanics is $E\psi = \ekin\psi + \epot\psi$. Like in classical mechanics, there is no derivation for this law, and is accepted as truth from experience. + \item While electromagnetic waves have a linear dispersion relation $\omega(k) = kc$, the matter wave $\psi(\vec{r}, t)$ of a free particle has a \textit{quadratic} dispersion relation $\omega(k) = (\hbar/2m) \cdot k^2$. This results from $E = \hbar\omega = p^2/2m$. + \item The Schrödinger equations are a \textit{linear} homogeneous differential equation. Because of this, different solutions of the equation can be superpositioned. This means, if $\psi_1$ and $\psi_2$ are solutions to the equation, then $\psi_3 = a\cdot\psi_1 + b\cdot\psi_2$ is also a solution. + \item Since the time-dependent Schrödinger equation is a complex equation, the wave functions $\psi$ may also be complex. The absolute square $\abs{\psi}^2$ however, which represents the probability of the presence of a particle, is always real. +\end{itemize} +For non-stationary problems (i.e. $E = E(t)$ and $p = p(t)$), the dispersion relation $\omega(t)$ also becomes time-dependent. This means that $\partial^2\psi/\partial t^2$ can no longer be written as $-\omega^2 \psi$, and cannot be derived from the wave equation for matter waves of particles. + +Schrödinger postulated (!), that even for time-dependent potential energy $\epot(\vec{r}, t)$ the equation +\begin{tcolorbox}[ams equation]\label{eq:schroedinger} + \frac{-\hbar^2}{2m} \laplacian \psi(\vec{r}, t) + \epot(\vec{r}, t) \psi(\vec{r}, t) = i\hbar\pdv{\psi(\vec{r}, t)}{t} +\end{tcolorbox} +holds. The general time-dependent Schrödinger equation has since been verified in numerous experiments, and is generally considered correct, even if no mathematical justification exists. +This equation is the fundamental equation of quantum mechanics. + +For stationary problems we can separate $\psi(\vec{r}, t)$ into $\psi(\vec{r}, t) = \psi(\vec{r}) \cdot e^{-i(E/\hbar) \cdot t}$. Inserting this into~\eqref{eq:schroedinger} yields the +stationary Schrödinger equation~\eqref{eq:stationaryschroedinger} for $\psi(\vec{r})$. +\end{document} \ No newline at end of file diff --git a/script.pdf b/script.pdf index 9a8eef661b32099d17d860b97d58108019115387..c9063d332947a6d3adca8794cc8ad25f680205c8 100644 GIT binary patch delta 202634 zcmeFZ2RxVG`#+Adv-gOXmCbAMLNc>gHrab*uL#|g6{3jDvQqY5S&=A2Nhq5T8Ig*t z|Ls-Y>0R&jd4Im&-+w*I{kqpV_jS&B-q(Fy&+9s8xAW-4ILS4-_ysJ8mW+(6FT@B6 zfkI$V^q&cVpuDSxvyPve4FpCYsAl2i;Q-zN1MdK@?{C37v}`Y?psVPyjmJZ+X{o7qVtUbS; zKXe6@4ffn zI{sgQq3C}E!~IhK3owN2>&inq5$xQ9f%;jo{ZGO0V1)ljBOXZfPm%Lykp5dRJgDGT z@5jG{A@Xm+a9<;S(@ZdE{3I5)_P_lO!}dSTKYxh-N-v^|{SP7%`1XzdKZQu(H*4Yl zjA8`78Xf<_c=&gSem<;>M@!x~-Kl@0-{=ruo)LMVTUK9cqArvZ%1UnHl z7r%a_56(e;+u7Haza9Lv^G}rLLE%GX4jGdG+9v&#zB^P-_HP)If2KPBq$=P*^BsJ2 z_igciP}f26zf_!X0U;<9j{41{1iNG3l9c{)Q_{=B`>!m?n*~gmO#c;#!uK2Vv%dV% zak+V&9FroxknGrBc`yG78qpTx!DxNQ-!H)k{C52ivi|Z@eo*u{-?e8zu=O9!@TDI1etW0OA-JH-miZL7#9Ez zpE>}-UqKyYm%lT&`(N*$D+b5%ZxNlqugKtkVi5dGqLb6+PnphdS{U))@#_))nO~1C z_P^lQ|Hw1_0r&ntfYJXIzy3#z{(y8)<^7@k{5PQf1NaXrM(ff8Rr(hgRoGXZKdaBb z%f=sI6#iea6cE3*U;m1GLKpjA!07)yxThb69Q{@P;miRD{)>7PI23;RGav0Iyna)} zzcL;E2}V!-D~!fVR{eL#D8#Qrlz)lN{bK*0LFb?h#=Y;HebcVUeYLocATDDBN*Me@8G~e`{Qky!f8(;hvEJWU>~Ea+H#Ykl z^aZcqE;#M)dG2qFp-Au#3U%Gz*#34#A3W0BKSEcFuIzgi=yT}04@#feztw(!obVClFsQp{*=xZMPhsFqD z;r(B92cRF1mf3Fq9J(<2F0jkKHd$`}y!-yG z;rwY_&=2t4f57+AaN9pbcRyO9KN|z^MzlEJ8xI_aj&A+;;rzYSzdaJ&VDymuP!c^V z|Fqx08vcWR6M}#3H*d>BUH9X-3mvG%zccP=tfQNS?mzVAplxUrqPrS>!}ngHMmP9t zum3ipLc-TcX=%}61(33K@Y=V)4{$1sD3cV$#JnHxhgnFQPS{wwf8odQzYr?kR0~2O z{LN7Lp1vX+e_e}T3H8m$R`783c7veyom`L_mUeM*_1Y&)h4$S{Er<}>Lxc!_`$6CJ zHM!y23oi@<&8c2?6#Q%X?;$-=CfI0NE2|&O5}lr~*E2sJ8_T_%s#%!S$6AJ8?$yFaAe%^66sm?)nR6mIU(e4OBKBEc z+GO?a`sA61G>|YYk@?qHadPs6amDpf;vPIJb-Nr(< zY1Ke^=xNjvTu?T`e523Wd-A;Gd~p$GOZK(BliBI@i>)u8(rKs$HKj3Stx;w!kP(|z zJpCXBl!(PFAuiZ^<0hJG)x0nF7~f*@nv(VmK@5CR7|2;~#)OR@(JaFwKbO~r%T-i< z=ADMk)mr`Y$)CqW9`o_Ov#jpvJsRpe!?C9vOk>`5G*qr*`;*qa_nThhkKEUw#~Y3W zC+_(E3}4h&p2IQTT8o|%JwPBlzk?Wsig&}}Ge1D_Z@5LF;SNFV!$k{B>-qZg6$OV0 zL-c(IrVcth^%eVIs*Ja#*M1_HpsIt*x%f#6YGD6O`I-S{1t$22Km~-6aEK5JC4fYN z@g5OT0a4Jb7Xp7_2o#8SX>^c68zQLSs^bb~P(gn8sr>#Hqow>yAuxX>g(wQ77K9)| z!UzEoQ4t6nDJp;x5(cFb5)cv*`K1)!_1o7^_8~bAzeRu08ApEI8wYZLaTf$q6oL?j z35dWzIfUT?P=x4T$no_Pbx4l=ar%q?_yaj$aPWbm;DZq&0-|sj*dgGth$ut|DI$Or z`2#_IrSrc_BeJjU4x|B94jPeK5J3aOvk(zj{F2sjED(jqv{mUv(7F`VGC~N52wD?@ z`(L%pzIqY)24!?6);INn1i#4pS%bjS8)_QUP|~oKO)Aey5}{kG3RCa0sPP3dX7$`7 zUP~1?6XfItQ2^=gY1Eo!1ca@xCKQ#4rpMH4CzV`7tDUk5$RU6qgy|MVqYs3UP zW0VulXF{hFu)Etf%P%Z0=*6fmAMG4ogWDJ;(1^D45oIR@o#0TlF14&N)KO~$uH(q* z7u9fYO$t0pHA(RCQ9D~}sw_L@U7zAg!YrlJ;;zd4K1S}f^mEzHn8EnBUv4-V+iK9Q zX%f|bo{ZKLAni_d@+q)r@xPgofR8QynUkS3Or+Q%Wt8#3BHMVbF|dpZ5H};5IChju ze3OgxijE7P$T0$L_T+nKVn{s&I{}-g5`=8IAMW|SYEq(c=hI8wy2a;u^|p_0jiuf5 zZd1(ep;{W{n6v>&KE`==Yl_(y3%55OUC7dHd3}nDUh1k@{;a!+D@9zBiHHyWZK@Z= z3OZFKcPHN%J)%)03X4yl8b22Qo|8@@)lDP7ONFsE4F_lT0i-IbO17T1E50q)VEBS> zeWFf8{*vAY??AqG*4??RVbZ3m+tBlyWMd`rBh)#HTfp(viy?Jf_jI3S?nP-%ctF

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colframe=red,colback=red!10!white,boxsep=0pt} + } + \usepackage{subfiles} +\DeclareMathOperator{\const}{const} + +\newcommand{\energy}[1]{E_{\text{#1}}} +\newcommand{\ekin}{\energy{kin}} +\newcommand{\epot}{\energy{pot}} + +\renewcommand{\vec}{\vb*} +\renewcommand{\laplacian}{\Delta} + \begin{document} \begin{titlepage} \begin{center} @@ -61,6 +76,8 @@ urlcolor=black } +\subfile{chapters/fundamentals/fundamentals.tex} + \pagestyle{headings} \end{document}