$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$ ##Appendix 4.C. Harmonic Oscillator ----------------------------------- **Definition 1.** *Quantum harmonic oscillator* is an operator on $\mathbb{R}$ \begin{equation} L:=-\frac{1}{2}\partial\_x^2 +\frac{1}{2} x^2 \label{equ-4.C.1} \end{equation} It is defined in the space $L^2(\mathbb{R})$ of square integrable functions on $\mathbb{R}$. **Remark 1.** Operator \begin{equation} L\_{\alpha\beta}:=-\frac{\alpha^2}{2}\partial\_x^2 +\frac{\beta^2}{2} x^2 \label{equ-4.C.2} \end{equation} can be reduced to (\ref{equ-4.C.1}) by change of variables $x:= x\gamma$ with $\gamma=\sqrt{\beta/\alpha}$ and division by $\sqrt{\alpha\beta}$. Observe that \begin{equation} L=\frac{1}{2}Z^\*Z+\frac{1}{2}=\frac{1}{2}ZZ^\*-\frac{1}{2} \label{equ-4.C.3} \end{equation} with \begin{equation} Z:= \partial_x +x ,\qquad Z^\* =-\partial _x +x \label{equ-4.C.4} \end{equation} and $Z^\*$ is adjoint to $Z$: $(Zu,v)=(u,Z^\* v)$. Note that (\ref{equ-4.C.3}) implies that the lowest eigenvalue of $L$ is $\frac{1}{2}$ with eigenfunction which is "annihilated" by $Z$, i.e. $u\_0(x):= e^{-\frac{x^2}{2}}$. To find other eigenvalues and eigenfunctions observe that $[Z^\*,Z]=-2$ and therefore \begin{equation} LZ^\* = Z^\* (L+1),\qquad LZ = Z (L-1). \label{equ-4.C.5} \end{equation} The first equality implies that if $u$ is an eigenfunction with an eigenvalue $\lambda$, then $Z^\*u$ is an eigenfunction with an eigenvalue $\lambda+1$; therefore we have a sequence of eigenvalues $\lambda\_n =(n+\frac{1}{2})$, $n=0,1,2,\ldots$ and eigenfunctions $u\_n$ defined \begin{equation} u\_n = Z^\* u_{n-1},\qquad n=1,2,\ldots. \label{equ-4.C.6} \end{equation} **Theorem 1.** 1. There are no other than above eigenvalues and eigenfunctions; 2. $u\_n(x)= H\_n (x) e^{-\frac{x^2}{2}}$ where $H\_n(x)$ is a polynomial of degree $n$ (and it is even/odd for even/odd $n$); 3. All $u\_n(x)$ are orthogonal; $\\|u_n\\|=\sqrt{\pi n!}$. 4. System $\\{u\_n\\}$ is complete. *Proof.* 1. The second of equalities (\ref{equ-4.C.5}) implies that if $u$ is an eigenfunction with an eigenvalue $\lambda$, then $Zu$ is an eigenfunction with an eigenvalue $\lambda-1$; however since eigenvalues start from $\frac{1}{2}$ there are no eigenvalues in $(\frac{1}{2}, \frac{3}{2})$; so the next eigenvalue is $\frac{3}{2}$ and if $u$ is a corresponding eigenfunction then $Z u=c u_0$. But then $Z^\*Z u = cZ^\* u\_0$; but $Z^\*Z u=(L-\frac{1}{2})u= u$ and $u=c Z^\* u\_0= c u\_1$. Continuing these arguments we conclude that there are no eigenvalues in $(\frac{3}{2}, \frac{5}{2})$; so the next eigenvalue is $\frac{5}{2}$ and $u=c\_2$ and so on. 2. By induction; 3. Due to $L^\*=L$ functions are orthogonal; on the other hand \begin{multline\*} \\|u\_n\\|^2=\\|Z^\* u\\|^2 = (Z^\* u\_{n-1}, Z^\* u\_{n-1})= (ZZ^\* u,u)= ((L+\frac{1}{2}) u\_{n-1},u\_{n-1})=\\\\ (\lambda+\frac{1}{2})\\|u\_{n-1}\\|^2=n\\|u\\|^2 \end{multline\*} and by induction it is equal to $n! \\|u\_0\\|^2= n! \pi$. Here we used the fact that $\|u\_0\|^2=\int\_{-\infty}^\infty e^{-x^2}\,dx=\pi$. **Definition 2.** Functions $u\_n$ are *Hermite functions*, $H\_n(x)$ are *Hermite polynomials*. One can prove \begin{equation} H\_n(x) = n! \sum\_{m=0}^{\lfloor \tfrac{n}{2} \rfloor} \frac{(-1)^m}{m!(n - 2m)!} (2x)^{n - 2m}. \label{equ-4.C.7} \end{equation} Then \begin{align\*} &H\_0(x)=1,\\\\ &H\_1(x)=2x,\\\\ &H\_2(x)=4x^2-2,\\\\ &H\_3(x)=8x^3-12x,\\\\ &H\_4(x)=16x^4-48x^2+12,\\\\ &H\_5(x)=32x^5-160x^3+120x,\\\\ &H\_6(x)=64x^6-480x^4+720x^2-120,\\\\ &H\_7(x)=128x^7-1344x^5+3360x^3-1680x,\\\\ &H\_8(x)=256x^8-3584x^6+13440x^4-13440x^2+1680,\\\\ &H\_9(x)=512x^9-9216x^7+48384x^5-80640x^3+30240x,\\\\ &H\_{10}(x)=1024x^{10}-23040x^8+161280x^6-403200x^4+302400x^2-30240 \end{align\*} **Remark 2.** In the toy-model of QFT (Quantum Field Theory) $u\_n$ is considered as $n$-particle state, in particular $u\_0$ is a vacuum state; operators $a=\frac{1}{\sqrt{2}}Z$ and $a^+=\frac{1}{\sqrt{2}}Z^*$ are operators of annihilation and creation respectively, $N=a^+a= L-\frac{1}{2}$ is an operator of number of the particles (actually, it is true only for bosons). ####References [http://en.wikipedia.org/wiki/Hermite_polynomials](http://en.wikipedia.org/wiki/Hermite_polynomials) [http://mathworld.wolfram.com/HermitePolynomial.html](http://mathworld.wolfram.com/HermitePolynomial.html) See plots for Hermite polynomial and Hermite functions. Observe that $H\_n(x)$ changes sign exactly $n$-times. _________________ [$\Leftarrow$](./S4.B.html) [$\Uparrow$](../contents.html) [$\Rightarrow$](../Chapter5/S5.1.html)