# Recent Posts

### Approximating the Geometric Mean

In my last post, I discussed the geometric mean and how it relates to the more familiar arithmetic mean. I mentioned that the geometric mean is often useful for estimation in physics. Lawrence Weinstein, in his book Guesstimation 2.0, gives a mental algorithm for approximating the geometric mean. Given two numbers in scientific notation $a \times 10^x \quad \text{and}\quad b \times 10^y,$ where the coefficients $a$ and $b$ are both between 1 and 9.

### The Probability Current

Probability Current in Quantum Mechanics In nonrelativistic 1D quantum mechanics, consider the probability of finding a particle in the interval $[x_1, x_2]$: $P_{x_1,x_2} = \int_{x_1}^{x_2} |\Psi(x,t)|^2\, \text{d}x.$ It is interesting to look at how this probability changes in time. Taking a time derivative, and using the Schrodinger equation $i \hbar \partial_t \Psi = \frac{-\hbar^2}{2m} \partial_x^2 \Psi + V \Psi$ to simplify things, we get (assuming a real potential) $\frac{d P_{x_1, x_2}}{dt} = \frac{\hbar}{2mi}\int_{x_1}^{x_2} \left[ (\partial_x^2 \Psi^*) \Psi - (\partial_x^2 \Psi) \Psi^* \right] \, \text{d}x$ where since $\left[ (\partial_x^2 \Psi^*) \Psi - (\partial_x^2 \Psi) \Psi^* \right] = \partial_x \left[ (\partial_x \Psi^*) \Psi - (\partial_x \Psi) \Psi^*\right],$ we can write this as $\frac{d P_{x_1, x_2}}{dt} = J(x_1, t) - J(x_2, t),$ where $J(x,t) = \frac{\hbar}{2mi} (\Psi^* \partial_x \Psi - \Psi \partial_x \Psi^*)$ is called the “probability current.

### Purgatory for Physicists

I once read, though I cannot now recall where, that an appropriate purgatory for theologians would be forcing them to read all of the words ever written on the Filioque.

If that is so, then an appropriate purgatory for physicists would be forcing them to read all of the words ever written on the interpretation of quantum mechanics.