Recent Posts

More Posts

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.

CONTINUE READING

Given two positive numbers \(a\) and \(b\), the most well-known way to average them is to take the sum divided by two. This is often called the average or the mean of \(a\) and \(b\), but to distinguish it from other means it is called the arithmetic mean: \[ \text{arithmetic mean} = \frac{a + b}{2}. \] Another common, and useful, type of mean is the geometric mean. For two positive numbers \(a\) and \(b\), \[ \text{geometric mean} = \sqrt{ab}.

CONTINUE READING

I was recently flipping through a real analysis textbook, and came across a notation that I had not seen before. First, the set difference of two sets \(A\) and \(B\) (which I had seen before) was defined as \[ A \setminus B = A \cap B^c. \] Here \(B^c\) is the set compliment of \(B\), or all elements not in \(B\). You can equivalently write the set difference as \[ A \setminus B = \{x \, |\, x \in A\, \text{and}\, x \notin B\} \]

CONTINUE READING

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.

CONTINUE READING

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.

CONTINUE READING

Contact