Two sets of positions of an electron at regular time intervals, starting from 𝑡 zero, are shown in the following figure. Which set of positions is more consistent with the change in the wavelength of the electron over the time shown? (A) Set I, (B) set II, (C) the sets of positions are equally consistent.
In our diagram, we see an initial time 𝑡 zero. And starting at this time, there are two sets of electron positions along with a graph of the wavelength of that electron. We’re told that our two sets of electron positions are shown at regular time intervals. We can say then that starting at the initial time 𝑡 zero, the electron position in set I moves from that initial position to here, while the electron position over that same time interval for the second set changes just in this way. Then, over the next equal time interval, the electron position in set I changes like this, while that in set II changes like this, and so on all across these sets of positions.
We want to know which set of positions is more consistent with the change in the wavelength of the electron shown in our graph. The key to understanding the connection between the wavelength of the electron and the change in the electron’s position over time is to recall what’s called the de Broglie relationship. This relationship says that the wavelength of a particle, such as an electron, equals the Planck constant ℎ divided by the mass of that particle multiplied by its velocity. What this equation is saying then is that wavelength is inversely proportional to particle velocity.
In our diagram, we’re not shown the velocity of the electron in either of these two sets. But we are shown the way that the electron’s position changes with time. We can recall that, in general, velocity 𝑉 equals the change in position of a particle divided by the change in time. Considering, for example, the time interval that starts at the time 𝑡 zero, we saw that over that interval, the electron in set I traveled this relatively larger distance, while that in set II traveled this relatively smaller distance. Since these changes in distance happened over the same amount of time, we can say that the velocity of the electron in set I was greater than that of the electron in set II. What we discover then is that when our electron positions are relatively far apart over these regular time intervals, that indicates a relatively greater electron velocity. The electron moves faster.
On the other hand, when the electron positions are relatively closer together over these time intervals, that tells us that over these intervals of time, the electron’s velocity is relatively small. It moves more slowly. If we consider set I and set II overall, we can say that over this part of the motion of the electron in these two sets, the electron in set I has a higher velocity, while that in set II has a lower velocity. And then, if we consider the two sets for the other part of their motion, the electron in set I now has a relatively lower velocity, while that in set II has a relatively higher velocity.
The question now becomes, which of these two sets indicates a better match for the electron’s wavelength? Looking at this half of the wavelength graph, we see that the overall wavelength is relatively longer here, while the electron wavelength becomes shorter to the left of our dashed line. We found that electron wavelength and electron velocity are inversely proportional. This means that when the electron wavelength is small or shorter, we would expect a larger or a higher electron velocity. Similarly, when the electron wavelength is longer, then we would expect a relatively lower velocity for the electron.
In both cases, the electron positions indicated in set II are a better match for our electron wavelength. This then is our answer. Set II is more consistent with the change in the wavelength of the electron over the time shown.