which in this case is in its lowest energy state. We then construct a probability density
graph. It shows that the particle is more likely to be found in the center of the box. The
walls of the box are at 0 and A, and the particle never escapes the box. This means the
particle is always present within the box, that is, the probability of finding it outside this
region is zero.
The probability density that describes the position of an actual particle is rarely as
simple as the one shown in Concept 3. In Concept 4, you see the graph of the
probability density function for the electron in the ground-state in a hydrogen atom (the
electron has the lowest energy that it can have). You can see that the electron is most
likely to be found at the Bohr radius, but that it is possible for the electron to be found at
radii other than this value. For large values of the radius, the probability density
approaches zero. The very low probability density at large radii means it is possible, but
highly, highly unlikely, that a bound electron can be found, say, one meter from the
nucleus. Quantum physicists can use the Schrödinger equation to create a
wavefunction that yields the graph you see in Concept 4. This is one of the fundamental
equations in quantum mechanics.
The graph in Concept 4 tells us about the behavior of an electron in the ground-state of
a hydrogen atom. If you measured the distance of the electron from the nucleus, and then did so again and again, each time charting its
location, you would create a graph (a radial probability density) similar to that. You would note that the largest number of your observations
have the electron at a distance equal to the Bohr radius, but you would find it at other locations as well.
Along with Einstein’s theories of relativity, quantum mechanics has changed the world’s conception of reality. Einstein showed that time and
length cannot be treated as absolutes, but that the motion of the observer affects these properties. Physicists have now shown that concepts
as common as the location of a particle must be stated in terms of probabilities. The work of these physicists has set the direction of physics for
the last 100 years or so.
Position of bound hydrogen
electron
Height of graph reflects likelihood of
finding particle in given region37.5 - Interactive exercise: observing the probabilities of a particle
The simulation on the right is used to show how an interference pattern emerges as
more and more electron waves pass through a pair of slits.
You are asked to observe two things.
First, press the FIRE button a few times and observe the location of the first five or
six electrons. Then press RESET, and do this again. Do you observe each electron
at the same location every time you run the simulation?
At the risk of ruining the punch line, the answer is no. If this is not clear, just fire one
electron, press RESET, and fire one electron again.
You cannot predict in advance where a given electron will land on the screen. You
can predict the probability of it landing near a certain spot, but not where it will land
for sure.
Although you cannot predict the location of any given electron, after enough
electrons pass through, the overall pattern is visible as predicted by the intensity
function.
After you fire 200 or so electrons, look at the graph. (Depress and hold down the FIRE button.) It will very closely resemble the graph of the
intensity function for the interference of two electromagnetic waves passing through a pair of slits. Where the function is greater, you will see
more photons accumulate over time. The smaller the function at a point, the fewer the number of particles (be they photons or electrons) that
will accumulate nearby. Try firing 500 electrons, and you will see that the graph resembles the intensity function even more as the number of
particles increases.
What you are observing here is in contrast to what one would expect using classical physics. If you toss a ball with a speed of exactly 25.0 m/s
at an angle of 30.0 degrees from a height of exactly 1.50 m in a vacuum, classical physics states that you can exactly predict its trajectory and
where it will land. If you again throw it with the same velocity from the same height, it will always land at the same location. Classical physics
would claim the same certainty and reproducibility for an electron launched in the same way.
No such luck with quantum physics í or maybe one should say “it’s all luck” with quantum physics? If you fire two consecutive electrons at the
double-slit system, under identical conditions, you do not know for sure that the particles will land even remotely near the same spot. You can
only observe where they land this time.
The more electrons that are fired, the more accurately you can predict the overall pattern. But you can no more predict the outcome of a single
spin of a roulette wheel than the landing point of any one photon or electron.