| Vorderseite | [latex]Calculate $[ \hat{x} , \hat{p}]$ with expressing $\hat{x}$ and $\hat{p}$ with creation and annihilation operators.[/latex] |
| Rückseite | [latex]$\hat{x}= \sqrt{\frac{\hbar}{2m\omega}}(\hat{a} + \hat{a}^+)$ \\ $\hat{p} = -i \sqrt{\frac{m\omega \hbar}{2}} (\hat{a} - \hat{a}^+)$\\ \\ \begin{eqnarray} \notag [ \hat{x} , \hat{p}] \notag &= -\frac{i \hbar}{2} [\hat{a}+\hat{a}^+, \hat{a}-\hat{a}^+] \\ \notag &= -\frac{i \hbar}{2} ( [\hat{a},\hat{a}] - [\hat{a}^+,\hat{a}^+] + [\hat{a}^+,\hat{a}] - [\hat{a}, \hat{a}^+]) \\ \notag &= -\frac{i \hbar}{2} (0 - 0 - 1 - 1) \\ \notag &= -\frac{i \hbar}{2} (-2) \\ \notag &= i \hbar \end{eqnarray}[/latex] |
| Tags | QLast |
| Vorderseite | [latex]What is the definition of the probability current density $\vec{j}(\vec{x},t)$ and the probability density $\rho(\vec{x},t)?$[/latex] |
| Rückseite | [latex]$\vec{j}(\vec{x},t) = \frac{\hbar}{i2m} [\Psi^* (\vec{\nabla} \Psi) - (\vec{\nabla} \Psi^*) \Psi ] \\ \rho = | \Psi (\vec{x},t) |^2$[/latex] |
| Tags | Q2 |
| Vorderseite | [latex]What is spin? What is it's relation to external degrees of freedom?[/latex] |
| Rückseite | [latex]Spin is the angular momentum of a point-like particle. It is an internal degree of freedom, and therefore commutes with external degrees of freedom.\\ \\$[\hat{S}, \hat{x}] = [\hat{S}, \hat{p}] = [\hat{S}, \hat{L}] = 0$[/latex] |
| Tags | Q8 |