User:Bachargil

= Wavefunction = The normalized position wavefunctions, given in spherical coordinates are:
 * $$ \psi_{nlm}(r,\theta,\phi) = \sqrt {{\left ( \frac{2}{n a_0} \right )}^3\frac{(n-l-1)!}{2n[(n+l)!]} } e^{- \rho / 2} \rho^{l} L_{n-l-1}^{2l+1}(\rho) \cdot Y_{l,m}(\theta, \phi ) $$

where:
 * $$ \rho = {2r \over {na_0}} $$
 * $$ a_0 $$ is the Bohr radius.
 * $$ L_{n-l-1}^{2l+1}(\rho) $$ are the Generalized Laguerre polynomials of degree n-l-1.
 * $$ Y_{l,m}(\theta, \phi ) \,$$ is a spherical harmonic.

defenitions

 * $$x = r \sin\theta\cos\varphi\,$$
 * $$y = r \sin\theta\sin\varphi\,$$
 * $$z = r \cos\theta\,$$

= Spherical harmonics =

Spherical harmonics with l = 0

 * $$Y_{0}^{0}(\theta,\varphi)={1\over 2}\sqrt{1\over \pi}$$

Spherical harmonics with l = 1

 * $$Y_{1}^{-1}(\theta,\varphi)={1\over 2}\sqrt{3\over 2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\quad={1\over 2}\sqrt{3\over 2\pi}\cdot{(x-iy)\over r}$$
 * $$Y_{1}^{0}(\theta,\varphi)={1\over 2}\sqrt{3\over \pi}\cdot\cos\theta\quad={1\over 2}\sqrt{3\over \pi}\cdot{z\over r}$$
 * $$Y_{1}^{1}(\theta,\varphi)={-1\over 2}\sqrt{3\over 2\pi}\cdot e^{i\varphi}\cdot\sin\theta\quad={-1\over 2}\sqrt{3\over 2\pi}\cdot{(x+iy)\over r}$$

Spherical harmonics with l = 2

 * $$Y_{2}^{-2}(\theta,\varphi)={1\over 4}\sqrt{15\over 2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\quad={1\over 4}\sqrt{15\over 2\pi}\cdot{(x^{2}-2ixy-y^{2})\over r^{2}}$$
 * $$Y_{2}^{-1}(\theta,\varphi)={1\over 2}\sqrt{15\over 2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot\cos\theta\quad={1\over 2}\sqrt{15\over 2\pi}\cdot{(xz-iyz)\over r^{2}}$$
 * $$Y_{2}^{0}(\theta,\varphi)={1\over 4}\sqrt{5\over \pi}\cdot(3\cos^{2}\theta-1)\quad={1\over 4}\sqrt{5\over \pi}\cdot{(-x^{2}-y^{2}+2z^{2})\over r^{2}}$$
 * $$Y_{2}^{1}(\theta,\varphi)={-1\over 2}\sqrt{15\over 2\pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot\cos\theta\quad={-1\over 2}\sqrt{15\over 2\pi}\cdot{(xz+iyz)\over r^{2}}$$
 * $$Y_{2}^{2}(\theta,\varphi)={1\over 4}\sqrt{15\over 2\pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta\quad={1\over 4}\sqrt{15\over 2\pi}\cdot{(x^{2}+2ixy-y^{2})\over r^{2}}$$

Spherical harmonics with l = 3

 * $$Y_{3}^{-3}(\theta,\varphi)={1\over 8}\sqrt{35\over \pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\quad={1\over 8}\sqrt{35\over \pi}\cdot{(x^{3}-3ix^{2}y-3xy^{2}+iy^{3})\over r^{3}}$$
 * $$Y_{3}^{-2}(\theta,\varphi)={1\over 4}\sqrt{105\over 2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot\cos\theta\quad={1\over 4}\sqrt{105\over 2\pi}\cdot{(x^{2}z-2ixyz-y^{2}z)\over r^{3}}$$
 * $$Y_{3}^{-1}(\theta,\varphi)={1\over 8}\sqrt{21\over \pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot(5\cos^{2}\theta-1)\quad={1\over 8}\sqrt{21\over \pi}\cdot{(-x^{3}+ix^{2}y-xy^{2}+4xz^{2}+iy^{3}-4iyz^{2})\over r^{3}}$$
 * $$Y_{3}^{0}(\theta,\varphi)={1\over 4}\sqrt{7\over \pi}\cdot(5\cos^{3}\theta-3\cos\theta)\quad={1\over 4}\sqrt{7\over \pi}\cdot{(-3x^{2}z-3y^{2}z+2z^{3})\over r^{3}}$$
 * $$Y_{3}^{1}(\theta,\varphi)={-1\over 8}\sqrt{21\over \pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot(5\cos^{2}\theta-1)\quad={-1\over 8}\sqrt{21\over \pi}\cdot{(-x^{3}-ix^{2}y-xy^{2}+4xz^{2}-iy^{3}+4iyz^{2})\over r^{3}}$$
 * $$Y_{3}^{2}(\theta,\varphi)={1\over 4}\sqrt{105\over 2\pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta\cdot\cos\theta\quad={1\over 4}\sqrt{105\over 2\pi}\cdot{(x^{2}z+2ixyz-y^{2}z)\over r^{3}}$$
 * $$Y_{3}^{3}(\theta,\varphi)={-1\over 8}\sqrt{35\over \pi}\cdot e^{3i\varphi}\cdot\sin^{3}\theta\quad={-1\over 8}\sqrt{35\over \pi}\cdot{(x^{3}+3ix^{2}y-3xy^{2}-iy^{3})\over r^{3}}$$

Spherical harmonics with l = 4

 * $$Y_{4}^{-4}(\theta,\varphi)={3\over 16}\sqrt{35\over 2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta$$
 * $$Y_{4}^{-3}(\theta,\varphi)={3\over 8}\sqrt{35\over \pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot\cos\theta$$
 * $$Y_{4}^{-2}(\theta,\varphi)={3\over 8}\sqrt{5\over 2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(7\cos^{2}\theta-1)$$
 * $$Y_{4}^{-1}(\theta,\varphi)={3\over 8}\sqrt{5\over \pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot(7\cos^{3}\theta-3\cos\theta)$$
 * $$Y_{4}^{0}(\theta,\varphi)={3\over 16}\sqrt{1\over \pi}\cdot(35\cos^{4}\theta-30\cos^{2}\theta+3)$$
 * $$Y_{4}^{1}(\theta,\varphi)={-3\over 8}\sqrt{5\over \pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot(7\cos^{3}\theta-3\cos\theta)$$
 * $$Y_{4}^{2}(\theta,\varphi)={3\over 8}\sqrt{5\over 2\pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta\cdot(7\cos^{2}\theta-1)$$
 * $$Y_{4}^{3}(\theta,\varphi)={-3\over 8}\sqrt{35\over \pi}\cdot e^{3i\varphi}\cdot\sin^{3}\theta\cdot\cos\theta$$
 * $$Y_{4}^{4}(\theta,\varphi)={3\over 16}\sqrt{35\over 2\pi}\cdot e^{4i\varphi}\cdot\sin^{4}\theta$$

= Laguerre =
 * $$ L_0^{(\alpha)} (x) = 1 $$


 * $$ L_1^{(\alpha)}(x) = -x + \alpha +1$$


 * $$ L_2^{(\alpha)}(x) = \frac{x^2}{2} - (\alpha + 2)x + \frac{(\alpha+2)(\alpha+1)}{2}$$


 * $$ L_3^{(\alpha)}(x) = \frac{-x^3}{6} + \frac{(\alpha+3)x^2}{2} - \frac{(\alpha+2)(\alpha+3)x}{2}

+ \frac{(\alpha+1)(\alpha+2)(\alpha+3)}{6}$$


 * $$L_{n + 1}^\alpha(x) = \frac{1}{n + 1} \bigg( (2n + 1 + \alpha - x)L_n^\alpha(x) - (n + \alpha) L_{n - 1}^\alpha(x)\bigg).$$

= Harmonic ocillator =


 * $$\begin{matrix}

a &=& \sqrt{m\omega \over 2\hbar} \left(x + {i \over m \omega} p \right) \\ a^{\dagger} &=& \sqrt{m \omega \over 2\hbar} \left( x - {i \over m \omega} p \right) \end{matrix}$$


 * $$\begin{matrix}

a \left| \phi _n \right\rangle &=& \sqrt{n} \left| \phi _{n-1} \right\rangle \\ a^{\dagger} \left| \phi _n \right\rangle &=& \sqrt{n+1} \left| \phi _{n+1} \right\rangle \end{matrix}$$


 * $$\begin{matrix}

x &=& \sqrt{\hbar \over 2m\omega} \left( a^{\dagger} + a \right) \\ p &=& i \sqrt{{\hbar}m\omega \over 2} \left( a^{\dagger} - a \right) \end{matrix}$$

Hermit polynomials

 * $$H_0(x)=1\,$$
 * $$H_1(x)=x\,$$
 * $$H_2(x)=x^2-1\,$$
 * $$H_3(x)=x^3-3x\,$$
 * $$H_4(x)=x^4-6x^2+3\,$$
 * $$H_5(x)=x^5-10x^3+15x\,$$
 * $$H_6(x)=x^6-15x^4+45x^2-15\,$$
 * $$H_{n+1}(x) = xH_n(x) - 2nH_{n-1}(x)\ $$

Solution

 * $$e^{-{z^2/2}} H_n(z), z = \alpha x \,$$