# Tag Info

### How to take derivative of density operator?

This is just a product rule. Take $\rho(t) = e^{-iHt/\hbar} \rho(0) e^{iHt/\hbar}$ and use the product rule: $$\frac{d}{dt}\rho(t) = \frac{d}{dt}[e^{-iHt/\hbar} \rho(0) e^{iHt/\hbar}]$$ This is just:...

### Why does $\delta \vec{r} = \delta \vec{ \theta} \times \vec{r}$?

$\delta \vec{r} = \delta \vec{ \theta} \times \vec{r} \tag{1}\;,$ ...but I don't quite catch where that particular equation (1) It is easiest to start in two dimensions. Recall the equation for the ...

### Why is the regular Taylor expansion so similar to the construction of the Hamiltonian from a Lagrangian/Legendre transform?

No, Legendre transforms are not the same as Taylor expansions. Well ok, so derivatives are by definition linear approximations, and by definition of Taylor polynomials, the first-order term is by ...
1 vote

### Why does $\delta \vec{r} = \delta \vec{ \theta} \times \vec{r}$?

starting with $$\mathbf r=\mathbf R\,\mathbf r_0\tag 1$$ the rotation matrix $~\mathbf R~$ is described with three Euler angles $~\theta_i(t)~$ for a small angles \mathbf R\mapsto I+\left[ \begin {...

### Why does $\delta \vec{r} = \delta \vec{ \theta} \times \vec{r}$?

Let us actually discuss more generally, because your picture and argument are very limited to 3D (especially with the invocation of cross products), and make too much appeal to vague intuition (which ...
1 vote

### Why does $\delta \vec{r} = \delta \vec{ \theta} \times \vec{r}$?

Suppose you have a vector $\vec{r}_0$ that is rotating counter clockwise with constant angular velocity $\vec{\omega}$. Our task is to determine the tangential velocity $\vec{v}$. We can do this in ...

### Why does $\delta \vec{r} = \delta \vec{ \theta} \times \vec{r}$?

The length of an arc that subtends an angle $\delta\theta$ is just $\text{arc length} =(\text{radius})(\delta\theta)$. This radius is the horizontal distance from the vertical axis in your picture ...
1 vote
Accepted

### "Why is $n$ held constant when taking the time derivative in the course of the Van Kampen's system size expansion?"

When you have a master equation for $P(n,t)$ is describes how the probability of a certain state (in this case the state corresponds to the particle number n) changes over time. It is important to ...