Formulario generale

Forniamo infine un formulario generale del corso.

Commutatori

$${\begin{aligned}&[\mathbf {P} ,\mathbf {X} ]=-i\hbar \mathbb {I} \\&[\mathbf {X} ,\mathbf {P} ]=i\hbar \mathbb {I} \\&[\mathbf {X} ,H]=[\mathbf {P} ,H]=[\mathbf {L} ,H]=[\mathbf {L} ^{2},H]=0\\&[L_{i},V_{j}]=i\hbar \epsilon _{ijk}V_{k}\\&[\mathbf {L} ^{2},L_{i}]=0\\&[L_{i},L_{j}]=i\hbar \epsilon _{ijk}L_{k}\\{\text{trasf. di Galileo }}&[\mathbf {k} ,H]=-i\mathbf {p} \\{\text{trasf. di Galileo }}&[k_{i},P_{j,n}]=-i\delta _{ij}m_{n}\\{\frac {i}{\hbar }}&[J_{ij},V_{k}]=\delta _{kj}V_{j}-\delta _{ki}V_{i}\\{\frac {i}{\hbar }}&[J_{ij},J_{kl}]=\delta _{ik}J_{lj}+\delta _{jk}J_{il}-\delta _{il}J_{kj}-\delta _{lj}J_{ik}\\&[J_{i},V_{j}]=i\hbar \epsilon _{ijk}V_{k}&[S_{i},V_{k}]=0\\&[S_{i},S_{j}]=i\hbar epsilon_{ijk}S_{k}\\&[J_{k},H]=[\mathbf {J} ^{2},H]=0\\\end{aligned}}$$

$${\begin{aligned}&[\mathbf {P} ,\mathbf {X} ]=-i\hbar \mathbb {I} \\&[\mathbf {X} ,\mathbf {P} ]=i\hbar \mathbb {I} \\&[\mathbf {X} ,H]=[\mathbf {P} ,H]=[\mathbf {L} ,H]=[\mathbf {L} ^{2},H]=0\\&[L_{i},V_{j}]=i\hbar \epsilon _{ijk}V_{k}\\&[\mathbf {L} ^{2},L_{i}]=0\\&[L_{i},L_{j}]=i\hbar \epsilon _{ijk}L_{k}\\{\text{trasf. di Galileo }}&[\mathbf {k} ,H]=-i\mathbf {p} \\{\text{trasf. di Galileo }}&[k_{i},P_{j,n}]=-i\delta _{ij}m_{n}\\{\frac {i}{\hbar }}&[J_{ij},V_{k}]=\delta _{kj}V_{j}-\delta _{ki}V_{i}\\{\frac {i}{\hbar }}&[J_{ij},J_{kl}]=\delta _{ik}J_{lj}+\delta _{jk}J_{il}-\delta _{il}J_{kj}-\delta _{lj}J_{ik}\\&[J_{i},V_{j}]=i\hbar \epsilon _{ijk}V_{k}&[S_{i},V_{k}]=0\\&[S_{i},S_{j}]=i\hbar epsilon_{ijk}S_{k}\\&[J_{k},H]=[\mathbf {J} ^{2},H]=0\\\end{aligned}}$$

Principio di indeterminazione

Posti $A,B$$A,B$ due osservabili qualsiasi:

$$\Delta A\Delta B={\frac {1}{2}}|<[A,B]>|$$

$$\Delta A\Delta B={\frac {1}{2}}|<[A,B]>|$$

Formule di Ehrenfest

$${\begin{aligned}&{\frac {d}{dt}}<\mathbf {X} >={\frac {<\mathbf {P} >}{m}}\\&{\frac {d}{dt}}<\mathbf {P} >=-<{\boldsymbol {\nabla }}V>\end{aligned}}$$

$${\begin{aligned}&{\frac {d}{dt}}<\mathbf {X} >={\frac {<\mathbf {P} >}{m}}\\&{\frac {d}{dt}}<\mathbf {P} >=-<{\boldsymbol {\nabla }}V>\end{aligned}}$$

Atomo di Bohr

Raggio:

$$r_{n}={\frac {n^{2}\hbar ^{2}}{m_{e}e^{2}}}$$

$$r_{n}={\frac {n^{2}\hbar ^{2}}{m_{e}e^{2}}}$$

Livelli energetici:

$$E_{n}=-{\frac {1}{2}}{\frac {m^{2}Z^{2}e^{4}}{n^{2}\,\hbar ^{2}}}$$

$$E_{n}=-{\frac {1}{2}}{\frac {m^{2}Z^{2}e^{4}}{n^{2}\,\hbar ^{2}}}$$

Approssimazione semiclassica

Condizione affinché sia valida:

$${\frac {\hbar }{p^{2}}}\left|{\frac {dp}{dx}}\right|<<1$$

$${\frac {\hbar }{p^{2}}}\left|{\frac {dp}{dx}}\right|<<1$$

Funzione d'onda per stati non legati:

$$\psi (x)={\frac {c_{1}}{\sqrt {p(x)}}}e^{{\frac {i}{\hbar }}\int p(x)dx}+{\frac {c_{2}}{\sqrt {p(x)}}}e^{-{\frac {i}{\hbar }}\int p(x)dx}$$

$$\psi (x)={\frac {c_{1}}{\sqrt {p(x)}}}e^{{\frac {i}{\hbar }}\int p(x)dx}+{\frac {c_{2}}{\sqrt {p(x)}}}e^{-{\frac {i}{\hbar }}\int p(x)dx}$$

Funzioni d'onda in zone classicamente proibite:

$$\psi (x)={\frac {c}{\sqrt {|p(x)|}}}e^{-{\frac {1}{\hbar }}\int p(x)dx}$$

$$\psi (x)={\frac {c}{\sqrt {|p(x)|}}}e^{-{\frac {1}{\hbar }}\int p(x)dx}$$

Buche di potenziale

Livelli energetici della buca unidimensionale:

$$E_{n}={\frac {n^{2}\pi ^{2}\hbar ^{2}}{2ma^{2}}}$$

$$E_{n}={\frac {n^{2}\pi ^{2}\hbar ^{2}}{2ma^{2}}}$$

Funzioni d'onda per la buca unidimensionale:

$$\psi (x)={\sqrt {\frac {2}{a}}}\sin \left({\frac {n\pi }{a}}x\right)$$

$$\psi (x)={\sqrt {\frac {2}{a}}}\sin \left({\frac {n\pi }{a}}x\right)$$

Separazione energetica per la doppia buca di potenziale:

$$E_{2}-E_{1}={\frac {2\hbar ^{2}}{m}}\psi _{0}(0)\psi _{0}^{'}(0)$$

$$E_{2}-E_{1}={\frac {2\hbar ^{2}}{m}}\psi _{0}(0)\psi _{0}^{'}(0)$$

Funzione d'onda sotto il massimo locale:

$${\frac {c}{\sqrt {|p|}}}e^{-{\frac {1}{\hbar }}\int _{0}^{a}p(x)dx}$$

$${\frac {c}{\sqrt {|p|}}}e^{-{\frac {1}{\hbar }}\int _{0}^{a}p(x)dx}$$

Oscillatore armonico

Caos unidimensionale, livelli energetici:

$$E_{n}=\hbar \omega \left(n+{\frac {1}{2}}\right)$$

$$E_{n}=\hbar \omega \left(n+{\frac {1}{2}}\right)$$

Funzioni d'onda:

$${\begin{aligned}&\psi _{m}=\left({\frac {M\omega }{\hbar \pi }}\right)^{\frac {1}{4}}{\frac {1}{2^{\frac {m}{2}}{\sqrt {m!}}}}e^{-{\frac {M\omega }{2\hbar }}x^{2}}\cdot H_{m}\left(x{\sqrt {\frac {M\omega }{\hbar }}}\right)\\&H_{m}(\xi )=(-1)^{m}e^{\xi ^{2}}\,{\frac {d^{m}}{d\xi ^{m}}}e^{-\xi ^{2}}\end{aligned}}$$

$${\begin{aligned}&\psi _{m}=\left({\frac {M\omega }{\hbar \pi }}\right)^{\frac {1}{4}}{\frac {1}{2^{\frac {m}{2}}{\sqrt {m!}}}}e^{-{\frac {M\omega }{2\hbar }}x^{2}}\cdot H_{m}\left(x{\sqrt {\frac {M\omega }{\hbar }}}\right)\\&H_{m}(\xi )=(-1)^{m}e^{\xi ^{2}}\,{\frac {d^{m}}{d\xi ^{m}}}e^{-\xi ^{2}}\end{aligned}}$$

Caso tridimensionale, operatori di innalzamento e abbassamento:

$${\begin{aligned}&a_{i}={\frac {1}{\sqrt {2m\omega \hbar }}}\left(-i\hbar \partial _{i}-im\omega x_{i}\right)\\&a_{i}^{\dagger }={\frac {1}{\sqrt {2}}}(-i\hbar \partial _{i}+im\omega x_{i})\end{aligned}}$$

$${\begin{aligned}&a_{i}={\frac {1}{\sqrt {2m\omega \hbar }}}\left(-i\hbar \partial _{i}-im\omega x_{i}\right)\\&a_{i}^{\dagger }={\frac {1}{\sqrt {2}}}(-i\hbar \partial _{i}+im\omega x_{i})\end{aligned}}$$

Livelli energetici:

$$E_{n}=\hbar \omega \left(N+{\frac {3}{2}}\right)\quad N=n_{x}+n_{y}+n_{z}$$

$$E_{n}=\hbar \omega \left(N+{\frac {3}{2}}\right)\quad N=n_{x}+n_{y}+n_{z}$$

Hamiltoniana:

$$H=\hbar \omega \left(a_{i}^{\dagger }a_{i}+{\frac {1}{2}}\right)$$

$$H=\hbar \omega \left(a_{i}^{\dagger }a_{i}+{\frac {1}{2}}\right)$$

Degenerazione dei livelli:

$${\mathcal {N}}_{N}={\frac {(N+1)(N+2)}{2}}$$

$${\mathcal {N}}_{N}={\frac {(N+1)(N+2)}{2}}$$

Atomo di idrogeno

Funzioni d'onda:

$${\begin{aligned}&\psi =R_{nl}(r)Y_{l}^{m}\\&R_{nl}(r)\propto r^{l}e^{-{\frac {r}{na_{B}}}}F_{nl}\left({\frac {r}{na_{B}}}\right)\end{aligned}}$$

$${\begin{aligned}&\psi =R_{nl}(r)Y_{l}^{m}\\&R_{nl}(r)\propto r^{l}e^{-{\frac {r}{na_{B}}}}F_{nl}\left({\frac {r}{na_{B}}}\right)\end{aligned}}$$

Matrici di Pauli

$${\begin{aligned}&\sigma _{1}=\left({\begin{matrix}0&1\\1&0\end{matrix}}\right)\\&\sigma _{2}=\left({\begin{matrix}0&-i\\i&0\end{matrix}}\right)\\&\sigma _{3}=\left({\begin{matrix}1&0\\0&-1\end{matrix}}\right)\end{aligned}}$$

$${\begin{aligned}&\sigma _{1}=\left({\begin{matrix}0&1\\1&0\end{matrix}}\right)\\&\sigma _{2}=\left({\begin{matrix}0&-i\\i&0\end{matrix}}\right)\\&\sigma _{3}=\left({\begin{matrix}1&0\\0&-1\end{matrix}}\right)\end{aligned}}$$

Distribuzioni statistiche

Distribuzione di Fermi-Dirac:

$${\overline {N}}_{n}={\frac {1}{e^{\beta (\epsilon -\mu )}+1}}$$

$${\overline {N}}_{n}={\frac {1}{e^{\beta (\epsilon -\mu )}+1}}$$

Distribuzione di Bose-Einstein:

$${\overline {N}}_{n}={\frac {1}{e^{\beta (\epsilon -\mu )}-1}}$$

$${\overline {N}}_{n}={\frac {1}{e^{\beta (\epsilon -\mu )}-1}}$$

Teoria delle perturbazioni

Primo ordine:

$${\begin{aligned}&\delta E_{a}=({\boldsymbol {\Psi }}_{a},(\delta H){\boldsymbol {\Psi }}_{a})=<\delta H>_{{\boldsymbol {\Psi }}_{a}}\\&\delta {\boldsymbol {\Psi }}_{a}=\sum _{b\neq a}{\boldsymbol {\Psi }}_{b}{\frac {({\boldsymbol {\Psi }}_{b},(\delta H){\boldsymbol {\Psi }}_{a})}{E_{a}-E_{b}}}\end{aligned}}$$

$${\begin{aligned}&\delta E_{a}=({\boldsymbol {\Psi }}_{a},(\delta H){\boldsymbol {\Psi }}_{a})=<\delta H>_{{\boldsymbol {\Psi }}_{a}}\\&\delta {\boldsymbol {\Psi }}_{a}=\sum _{b\neq a}{\boldsymbol {\Psi }}_{b}{\frac {({\boldsymbol {\Psi }}_{b},(\delta H){\boldsymbol {\Psi }}_{a})}{E_{a}-E_{b}}}\end{aligned}}$$

Secondo ordine:

$$\delta _{2}E_{a}=({\boldsymbol {\Psi }}_{a},(\delta _{2}H){\boldsymbol {\Psi }}_{a})+\sum _{c:E_{c}\neq E_{a}}{\frac {|({\boldsymbol {\Psi }}_{a},(\delta _{1}H){\boldsymbol {\Psi }}_{c})|^{2}}{E_{a}-E_{c}}}$$

$$\delta _{2}E_{a}=({\boldsymbol {\Psi }}_{a},(\delta _{2}H){\boldsymbol {\Psi }}_{a})+\sum _{c:E_{c}\neq E_{a}}{\frac {|({\boldsymbol {\Psi }}_{a},(\delta _{1}H){\boldsymbol {\Psi }}_{c})|^{2}}{E_{a}-E_{c}}}$$

Dipendente dal tempo:

$$c_{m}(t)=c_{n}(0)-{\frac {i}{\hbar }}\sum _{n}c_{n}(0)\int _{0}^{t}dt'e^{i{\frac {(E_{m}-E_{n})}{\hbar }}t'}H_{mn}^{'}(t')$$

$$c_{m}(t)=c_{n}(0)-{\frac {i}{\hbar }}\sum _{n}c_{n}(0)\int _{0}^{t}dt'e^{i{\frac {(E_{m}-E_{n})}{\hbar }}t'}H_{mn}^{'}(t')$$

Regola d'oro di Fermi:

$$\Gamma (1\to n)={\frac {2\pi }{\hbar }}|{\mathcal {U}}_{n1}|^{2}\delta (E_{n}-E_{1}-\hbar \omega$$

$$\Gamma (1\to n)={\frac {2\pi }{\hbar }}|{\mathcal {U}}_{n1}|^{2}\delta (E_{n}-E_{1}-\hbar \omega$$

Regola d'oro per una perturbazione monocromatica:

$$\Gamma (1\to \mathbf {p} )={\frac {V}{h^{3}}}\int d^{3}p{\frac {2\pi }{\hbar }}|{\mathcal {U}}_{\mathbf {p} 1}|^{2}\delta (E_{\mathbf {p} }-E_{1}-\hbar \omega )$$

$$\Gamma (1\to \mathbf {p} )={\frac {V}{h^{3}}}\int d^{3}p{\frac {2\pi }{\hbar }}|{\mathcal {U}}_{\mathbf {p} 1}|^{2}\delta (E_{\mathbf {p} }-E_{1}-\hbar \omega )$$