Neutron Phase and Gravity

In my research this morning, I came across a collection of interesting articles from the 1970s where researchers observed a shift in the phase of a neutron's quantum wave function because of a change in the gravitational potential. This is similar to the Aharanov-Bohm effect, but using a gravitational rather than an electromagnetic potential. If you're a university student like myself, all the journal links below should be accessible via your university's library.

The experiment design is described in
Experimental Test of Gravitationally Induced Quantum Interference

The results of the experiment are presented in:
Observation of Gravitationally Induced Quantum Interference

Another similar experiment by the same researchers was described in:
Observation of a Phase Shift of a Neutron Due to Precession in a Magnetic Field

For a nice primer of all things Aharanov-Bohm with reference to the above mentioned experiments, see Reviews of Modern Physics:
The quantum effects of electromagnetic fluxes

One of the experimenters, Overhauser, discusses the experiment and its relation to the Aharanov-Bohm effect at length in:
Coherence effects in neutron diffraction and gravity experiments
Greenberger D. & Overhauser A. (1979). Coherence effects in neutron diffraction and gravity experiments, Reviews of Modern Physics, 51 (1) 43-78. DOI: A

review of the great things that can be done with a neutron interferometer including the experiments mentioned above can be found in Reviews of Modern Physics.
The neutron interferometer as a device for illustrating the strange behavior of quantum systems
Greenberger D. (1983). The neutron interferometer as a device for illustrating the strange behavior of quantum systems, Reviews of Modern Physics, 55 (4) 875-905. DOI:

The author, Greenberger, of the above-mentioned article thanks John Wheeler, Overhauser, Werner, and Herman Feshbach as well as other famous physicists.

Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla

to the nabla for another coordinate system, say… cylindrical coordinates.

What we’ll need:

1. The Cartesian Nabla:

2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:

3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:

How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differential operator in terms of one variable into a series of differential operators in terms of othe…

The Valentine's Day Magnetic Monopole

There's an assymetry to the form of the two Maxwell's equations shown in picture 1.  While the divergence of the electric field is proportional to the electric charge density at a given point, the divergence of the magnetic field is equal to zero.  This is typically explained in the following way.  While we know that electrons, the fundamental electric charge carriers exist, evidence seems to indicate that magnetic monopoles, the particles that would carry magnetic 'charge', either don't exist, or, the energies required to create them are so high that they are exceedingly rare.  That doesn't stop us from looking for them though!

Keeping with the theme of Fairbank[1] and his academic progeny over the semester break, today's post is about the discovery of a magnetic monopole candidate event by one of the Fairbank's graduate students, Blas Cabrera[2].  Cabrera was utilizing a loop type of magnetic monopole detector.  Its operation is in concept very simpl…

Unschooling Math Jams: Squaring Numbers in their own Base

Some of the most fun I have working on math with seven year-old No. 1 is discovering new things about math myself.  Last week, we discovered that square of any number in its own base is 100!  Pretty cool!  As usual we figured it out by talking rather than by writing things down, and as usual it was sheer happenstance that we figured it out at all.  Here’s how it went.

I've really been looking forward to working through multiplication ala binary numbers with seven year-old No. 1.  She kind of beat me to the punch though: in the last few weeks she's been learning her multiplication tables in base 10 on her own.  This became apparent when five year-old No. 2 decided he wanted to do some 'schoolwork' a few days back.

"I can sing that song... about the letters? all by myself now!"  2 meant the alphabet song.  His attitude towards academics is the ultimate in not retaining unnecessary facts, not even the name of the song :)

After 2 had worked his way through the so…