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Expectation value:

<math> \left \langle \hat A \right \rangle  \equiv  \left \langle \Psi \mid \hat A \mid \Psi   \right \rangle              </math>

Probability that system is between <math>\overline{q} </math> and <math> \overline{q} + d \overline{q}</math> is <math>\Psi^*(\overline{q}, t) \Psi (\overline q, t) d \overline q</math>

Scalar product  (0 if orthogonal):

<math> \left \langle \Psi (x) \mid \phi (x) \right \rangle  \equiv \int_{}^{} \Psi (x)^* \phi (x)\, dx </math> 


Momentum:

<math> \hat P = \frac{\hbar}{i} \vec \nabla  </math>


Position:

<math> \hat R = \hat r  </math>



Energy:

<math> \hat H (\hat x, \hat p)=i\hbar\frac{\partial }{ \partial t} =\frac{\hat P ^2}{2m}+V(\hat x) =  \frac{\hat P ^2}{2m}=\frac {-\hbar^2}{2m} \frac{ \partial^2 }{ \partial x^2} +V(\hat x) </math>

<math> \frac{\hat P ^2}{2m}=\frac {-\hbar^2}{2m} \frac{ \partial^2 }{ \partial x^2} </math>


Time-dependent Schr��dinger's equation:


<math>\hat H  \Psi = i \hbar \frac{\partial \Psi}{\partial t}
</math>



General soluion to Schr��dinger's equation:

<math> \Psi(x,t)= \sum_{E} c_E u_E (x)  \mathrm{e}^{-\mathrm{i} Et/ \hbar} </math>

where <math> c_E </math> are the eigenfunctions of the Hamiltonian that has eigenvalues of E


Fermi function:
<math>f(\epsilon)=\frac{1}{e^{\frac{\epsilon_c - \mu}{k_B T}}+1}</math>

Density of states:
<math>g(\epsilon)</math>

<math>f(E) = \frac{1}{1 + e^{\frac{E-E_F}{kT}}}</math>




<math>[\hat H, \hat P] = 0</math> momentum is s constant of motion


Spherical harmonics:

<math>u(\vec r) = \phi (r) \Psi (\phi, \theta)</math>