FIGURE 38.17 shows a Rutherford hydrogen atom, with a single electron orbiting a nucleus that consists of a single proton. We will assume a circular orbit of radius $r$ and speed v. We will also assume, to keep the analysis manageable, that the proton remains stationary while the electron revolves around it. This is a reasonable assumption because the proton is roughly 1800 times as massive as the electron. With these assumptions, the atom’s energy is the kinetic energy of the electron plus the potential energy of the electron-proton interaction. This is

where we used $q_{\text{elec}}=-e$ and $q_{\text{proton}}=+e$.

FIGURE 38.17 A Rutherford hydrogen atom. The size of the nucleus is greatly exaggerated.

Now, the electron, as we are coming to understand it, has both particle-like and wave-like properties. First, let us treat the electron as a charged particle. The proton exerts a Coulomb electric force on the electron:

This force gives the electron an acceleration $\vec{a}{}_{\text{elec}}=\vec{F}{}_{\text{elec}}\text{/}m$ that also points to the center. This is a centripetal acceleration, causing the particle to move in its circular orbit. The centripetal acceleration of a particle moving in a circle of radius $r$ at speed v must be $v^2\text{/}r$, thus

Rearranging, we find

Equation 38.25 is a constraint on the motion. The speed $v$ and radius $r$ must satisfy Equation 38.25 if the electron is to move in a circular orbit. This constraint is not unique to atoms; we earlier found a similar relationship between $v$ and $r$ for orbiting satellites.

Now let’s treat the electron as a de Broglie wave. In Section 38.4 we found that a particle confined to a one-dimensional box sets up a standing wave as it reflects back and forth. A standing wave, you will recall, consists of two traveling waves moving in opposite directions. When the round-trip distance in the box is equal to an integer number of wavelengths $(2L=n\lambda )$, the two oppositely traveling waves interfere constructively to set up the standing wave.

Suppose that, instead of traveling back and forth along a line, our wave-like particle travels around the circumference of a circle. The particle will set up a standing wave, just like the particle in the box, if there are waves traveling in both directions and if the round-trip distance is an integer number of wavelengths. This is the idea we want to carry over from the particle in a box. As an example, FIGURE 38.18 shows a standing wave around a circle with $n=10$ wavelengths.

FIGURE 38.18 An $n=10$ electron standing wave around the orbit’s circumference.

The mathematical condition for a circular standing wave is found by replacing the round-trip distance $2L$ in a box with the round-trip distance $2\pi r$ on a circle. Thus a circular standing wave will occur when

But the de Broglie wavelength for a particle has to be $\lambda =h\text{/}p=h\text{/}mv$. Thus the standing-wave condition for a de Broglie wave is

This condition is true only if the electron’s speed is

The quantity $h\text{/}2\pi$ occurs so often in quantum physics that it is customary to give it a special name. We define the quantity $\hslash$, pronounced “h bar,” as

With this definition, we can write Equation 38.27 as

This, like Equation 38.25, is another relationship between $v$ and r. This is the constraint that arises from treating the electron as a wave.

Now if the electron can act as both a particle and a wave, then both the Equation 38.25 and Equation 38.28 constraints have to be obeyed. That is, $v^2$ as given by the Equation 38.25 particle constraint has to equal $v^2$ of the Equation 38.28 wave constraint. Equating these gives

We can solve this equation to find that the radius $r$ is

where we have added a subscript $n$ to the radius $r$ to indicate that it depends on the integer $n$.

The right-hand side of Equation 38.29, except for the $n^2$, is just a collection of constants. Let’s group them all together and define the Bohr radius $a_{\text{B}}$ as

With this definition, Equation 38.29 for the radius of the electron’s orbit becomes

For example, $r_1=0.053\text{nm,}{r}_2=0.212\text{nm}$, and $r_3=0.476\text{nm}$.

We have discovered stationary states! That is, a hydrogen atom can exist only if the radius of the electron’s orbit is one of the values given by Equation 38.30. Intermediate values of the radius, such as $r=0.100\text{nm,}$ cannot exist because the electron cannot set up a standing wave around the circumference. The possible orbits are quantized, with only certain orbits allowed.

The key step leading to Equation 38.30 was the requirement that the electron have wave-like properties in addition to particle-like properties. This requirement leads to quantized orbits, or what Bohr called stationary states. The integer $n$ is thus the quantum number that numbers the various stationary states.