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

Unknown macro: {q}

</math> and <math> \overline

+ d \overline

Unknown macro: {q}

</math> is <math>\Psi^*(\overline

, t) \Psi (\overline q, t) d \overline q</math>

Scalar product (0 if orthogonal):

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

Momentum:

<math> \hat P = \frac

Unknown macro: {hbar}
Unknown macro: {i}

\vec \nabla </math>

Position:

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

Energy:

<math> \hat H (\hat x, \hat p)=i\hbar\frac

Unknown macro: {partial }
Unknown macro: { partial t}

=\frac

Unknown macro: {hat P ^2}
Unknown macro: {2m}

+V(\hat x) = \frac

Unknown macro: {2m}

=\frac {-\hbar^2}

\frac

Unknown macro: { partial^2 }
Unknown macro: { partial x^2}

+V(\hat x) </math>

<math> \frac

Unknown macro: {hat P ^2}
Unknown macro: {2m}

=\frac {-\hbar^2}

\frac

Unknown macro: { partial x^2}

</math>

Time-dependent Schr��dinger's equation:

<math>\hat H \Psi = i \hbar \frac

Unknown macro: {partial Psi}
Unknown macro: {partial t}

</math>

General soluion to Schr��dinger's equation:

<math> \Psi(x,t)= \sum_

Unknown macro: {E}

c_E u_E \mathrm

Unknown macro: {e}

^{-\mathrm

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

Unknown macro: {1}

{e^{\frac

Unknown macro: {epsilon_c - mu}

{k_B T}}+1}</math>

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

<math>f(E) = \frac

{1 + e^{\frac

Unknown macro: {E-E_F}
Unknown macro: {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>

  • No labels