## Development of Quantum Theory

Bohr’s model explained the experimental data for the hydrogen atom and was widely accepted, but it also raised many questions. Why did electrons orbit at only fixed distances defined by a single quantum number *n* = 1, 2, 3, and so on, but never in between? Why did the model work so well describing hydrogen and one-electron ions, but could not correctly predict the emission spectrum for helium or any larger atoms? To answer these questions, scientists needed to completely revise the way they thought about matter.

## Behavior in the Microscopic World

We know how matter behaves in the macroscopic world—objects that are large enough to be seen by the naked eye follow the rules of classical physics. A billiard ball moving on a table will behave like a particle: It will continue in a straight line unless it collides with another ball or the table cushion, or is acted on by some other force (such as friction). The ball has a well-defined position and velocity (or a well-defined momentum, *p* = *mv,* defined by mass *m* and velocity *v*) at any given moment. In other words, the ball is moving in a classical trajectory. This is the typical behavior of a classical object.

When waves interact with each other, they show interference patterns that are not displayed by macroscopic particles such as the billiard ball. For example, interacting waves on the surface of water can produce interference patterns similar to those shown on the figure below. This is a case of wave behavior on the macroscopic scale, and it is clear that particles and waves are very different phenomena in the macroscopic realm.

As technological improvements allowed scientists to probe the microscopic world in greater detail, it became increasingly clear by the 1920s that very small pieces of matter follow a different set of rules from those we observe for large objects. The unquestionable separation of waves and particles was no longer the case for the microscopic world.

One of the first people to pay attention to the special behavior of the microscopic world was Louis de Broglie. He asked the question: If electromagnetic radiation can have particle-like character, can electrons and other submicroscopic particles exhibit wavelike character? In his 1925 doctoral dissertation, de Broglie extended the wave–particle duality of light that Einstein used to resolve the photoelectric-effect paradox to material particles. He predicted that a particle with mass *m* and velocity *v* (that is, with linear momentum *p*) should also exhibit the behavior of a wave with a wavelength value *λ*, given by this expression in which *h* is the familiar Planck’s constant:

\(\lambda =\phantom{\rule{0.2em}{0ex}}\cfrac{h}{mv}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\cfrac{h}{p}\)

This is called the *de Broglie wavelength*. Unlike the other values of *λ* discussed in this chapter, the de Broglie wavelength is a characteristic of particles and other bodies, not electromagnetic radiation (note that this equation involves velocity [*v*, m/s], not frequency [*ν*, Hz]. Although these two symbols appear nearly identical, they mean very different things).

Where Bohr had postulated the electron as being a particle orbiting the nucleus in quantized orbits, de Broglie argued that Bohr’s assumption of quantization can be explained if the electron is considered not as a particle, but rather as a circular standing wave such that only an integer number of wavelengths could fit exactly within the orbit (see figure below).

For a circular orbit of radius *r*, the circumference is 2*πr*, and so de Broglie’s condition is:

\(2\pi r=n\lambda ,\phantom{\rule{0.2em}{0ex}}n=1,\phantom{\rule{0.2em}{0ex}}2,\phantom{\rule{0.2em}{0ex}}3,\phantom{\rule{0.2em}{0ex}}\dots \)

Since the de Broglie expression relates the wavelength to the momentum and, hence, velocity, this implies:

\(2\pi r=n\lambda =\phantom{\rule{0.2em}{0ex}}\cfrac{nh}{p}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\cfrac{nh}{mv}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\cfrac{nhr}{mvr}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\cfrac{nhr}{L}\)

This expression can be rearranged to give Bohr’s formula for the quantization of the angular momentum:

\(L\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\cfrac{nh}{2\pi }\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}n\hslash \)

Classical angular momentum *L* for a circular motion is equal to the product of the radius of the circle and the momentum of the moving particle *p*.

\(L=rp=rmv\phantom{\rule{0.2em}{0ex}}\text{(for a circular motion)}\)

## Optional Video: TED-Ed Particles and Waves

One of the most amazing facts in physics is that everything in the universe, from light to electrons to atoms, behaves like both a particle and a wave at the same time. But how did physicists arrive at this mind-boggling conclusion? Chad Orzel recounts the string of scientists who built on each other’s discoveries to arrive at this ‘central mystery’ of quantum mechanics.