What evidence is there that electrons move around in definite pathways around the nucleus?

The electrons moving around the nucleus do not move in the define pathways. They rather exhibit their cloud nature.

Further Explanation:

The Danish physicist, Niels Bohr gave the model of an atom whose most important postulate is that only those electrons can move around in circular orbits whose angular momentum is given by the integral multiple of .

It can be represented mathematically as:

This equation was also verified by French physicist Louis de Broglie as his hypothesis stated the dual nature of matter and radiation which is also now been proved experimentally by the quantum physicists.

So, an electron is both a particle and a wave and by wave nature of electron, it travels approximately in circular paths called as energy levels like a wave as shown in the figures given below:

In this theory, electron moves approximately circularly and thus, the circumference of the circle is equated with the integral multiple of the wavelength of matter wave associated with the electron.

It can be represented as:

Here, is the wavelength of the moving particle and is the radius of the circular path.

Also according to de-Broglie equation,  

Here, is the mass of the electron and is the velocity of the electron.

Substituting this in above equation,

We have,

It can also we expressed as:

Therefore, the above expressions shows that an electron moves in the form of an electronic cloud but it also exhibits the wave nature as it travels in circular paths.

Learn more:

1.  The threshold frequency Vo of cesium

2.  Calculate the broadcast wavelength of the radiostation 88.70 fm

Answer Details:

Grade: High school

Subject: Physics

Chapter: Modern physics

Keywords:

Electrons, circular path, wave nature, electronic cloud, bohr postulates, de broglie, nh/2pi, angular momentum, nucleus, pathways.