Linear Operator

F(aΦ1+bΦ2)=a⋅FΦ1+b⋅FΦ2∀Φ1,Φ2F(a\bm\Phi_1 + b\bm\Phi_2) = a\cdot F\bm\Phi_1 + b\cdot F\bm\Phi_2\quad \forall \bm\Phi_1, \bm\Phi_2

Eigenvalue, Eigenfunction

Fun=fnunFu_n = f_nu_n
unu_n

is a eigen function to the operator

FF

with corresponding eigenvalue

fnf_n

.

Hermitian Operator

∫(Hu)∗v d3r=∫u∗Hv d3r,∀u,v\int (Hu)*v\: d^3r = \int u*Hv \: d^3r, \quad \forall u, v

A hermitian operator has real eigenvalues and corresponding eigenfunctions can be choosen to be orthonormal. Practically all operators in quantum mechanics are linear and hermitian.

Eigenfunction Expansion

ψ(r)=∑nanm⋅un(r),an=∫un∗⋅ψ⋅d3r\bm\psi(\bm r) = \sum_na_nm\cdot u_n(\bm r),\quad a_n = \int u_n*\cdot\bm\psi\cdot d^3r

Expansion Postulate

At a meassurement of an observable

FF

on a system described by a wavefunction

ψ\bm\psi

only eigenvalues of the operator

FF

can be found. The probability of the result

F=fnF = f_n

is given by

P(F=fn)=∣∫un∗ψ d3r∣2,Fun=fnunP(F = f_n) = \left|\int u_n*\bm\psi\: d^3r\right|^2, \quad Fu_n = f_nu_n

Momentum Operators

L2=−ℏ2[1sin⁡2θ∂2∂φ2+1sin⁡θ∂∂θ(sin⁡θ∂∂θ)]L^2 = -\hbar^2\left[\frac 1{\sin^2\theta}\frac{\partial^2}{\partial\varphi^2} + \frac 1{\sin\theta}\frac\partial{\partial\theta}\left(\sin\theta\frac\partial{\partial\theta}\right)\right]
Lz=ℏi∂∂φL_z = \frac \hbar i\frac\partial{\partial\varphi}
L2L^2

and

LzL_z

have normalized eigenfunctions

Υlm(θ,φ)\Upsilon_l^m(\theta, \varphi)

for which it holds that:

L2Υlm=ℏ2l(l+1)ΥlmL^2\Upsilon_l^m = \hbar^2l(l + 1)\Upsilon_l^m
LzΥlm=mℏΥlmL_z \Upsilon_l^m = m\hbar\Upsilon_l^m
lmΥlm(θ,φ)00Υ00=14π10Υ10=34πcos⁡θ1±1Υ1±1=±38πsin⁡θe±iφ20Υ20=516π(3cos⁡2θ−1)2±1Υ2±1=±158πsin⁡θcos⁡θe±iφ2±2Υ2±2=1532πsin⁡2θe±2iφ\def\arraystretch{2.5} \begin{array}{ccc} \hline l & m & \Upsilon_l^m(\theta, \varphi) \\ \hline 0 & 0 & \Upsilon_0^0 = \frac 1{\sqrt{4\pi}} \\ 1 & 0 & \Upsilon_1^0 = \sqrt{\frac 3{4\pi}}\cos\theta\\ 1 & \pm1 & \Upsilon_1^{\pm1} = \pm\sqrt{\frac 3{8\pi}}\sin\theta e^{\pm i\varphi}\\ 2 & 0 & \Upsilon_2^0 = \sqrt{\frac 5{16\pi}}\left(3\cos^2\theta - 1\right)\\ 2 & \pm1 & \Upsilon_2^{\pm1} = \pm\sqrt{\frac {15}{8\pi}}\sin\theta\cos\theta e^{\pm i\varphi}\\ 2 & \pm2 & \Upsilon_2^{\pm2} = \sqrt{\frac {15}{32\pi}}\sin^2\theta e^{\pm 2i\varphi}\\ \hline \end{array}

Commutators and Momentum Operators

ϵijk={1ijk  even−1ijk  odd0otherwise\epsilon_{ijk} = \begin{cases} 1 & ijk \;\textup{even} \\ -1 & ijk\; \textup{odd} \\ 0 & \textup{otherwise} \end{cases}
[xi,pj]=iℏ⋅δij[x_i, p_j] = i\hbar\cdot\delta_{ij}
[xi,Lj]=iℏ⋅ϵijk⋅xk[x_i, L_j] = i\hbar\cdot\epsilon_{ijk}\cdot x_k
[Li,Lj]=iℏ⋅ϵijk⋅Lk[L_i, L_j] = i\hbar\cdot\epsilon_{ijk}\cdot L_k
[xi,xj]=[pi,pj]=0[x_i, x_j] = [p_i, p_j] = 0
[pi,Lj]=iℏ⋅ϵijk⋅pk[p_i, L_j] = i\hbar\cdot\epsilon_{ijk}\cdot p_k
J+=Jx+iJyJ−=Jx−iJy\begin{gathered} J_+ = J_x + iJ_y\\ J_- = J_x - iJ_y \end{gathered}
J±J∓=J2−Jz2±ℏ⋅JzJ_\pm J_\mp = J^2 - J_z^2 \pm \hbar\cdot J_z
[J+,J−]=2ℏ⋅Jz[J_+, J_-] = 2\hbar\cdot J_z
[Jz,J±]=±ℏ⋅J±[J_z, J_\pm] = \pm\hbar\cdot J_\pm
J+ϕj,m=(j−m)(j+m+1⋅ℏ⋅ϕj,m+1J_+\phi_{j,m} = \sqrt{(j - m)(j + m + 1}\cdot\hbar\cdot\phi_{j, m + 1}
J−ϕj,m=(j+m)(j−m+1⋅ℏ⋅ϕj,m−1J_-\phi_{j,m} = \sqrt{(j + m)(j - m + 1}\cdot\hbar\cdot\phi_{j, m - 1}
Υll(θ,φ)=(−1)l2l+1)4π(2l)!22l(l!)2⋅sin⁡lθ⋅eilφ\Upsilon_l^l(\theta, \varphi) = (-1)^l\sqrt{\frac{2l + 1)}{4\pi}\frac{(2l)!}{2^{2l}(l!)^2}}\cdot\sin^l\theta\cdot e^{il\varphi}