37.4 - Matter waves are probabilistic
De Broglie correctly asserted that electrons and other particles can be described as
matter waves. Physicists, being quantitative types, wanted to know: How could they
mathematically describe the location of a wave-particle like an electron?
In classical mechanics, when the net force exerted on a particle of a certain mass is
known, the equation F = ma is used to calculate the acceleration of the particle. If the
initial position and velocity are also known, then other equations can determine the
particle’s location and velocity at any time.
Quantum physicists found that applying classical equations to atomic-sized particles led
to paradoxes. For instance, the equations could be interpreted to predict that a quarter
photon, or half an electron, should be present at a given location. This was in
contradiction not only to the tenets of quantum physics, but to experiments, which
showed that photon and electrons were indivisible.
Another conundrum can be discussed using the double-slit experiment. Electrons are
fired through a pair of slits toward a barrier, as shown in Concept 1. A photographic
plate records where the electrons strike the barrier. The pattern recorded on the plate
(and shown in Concept 1) looks the same as that created by shining light through the
slits.
Although physicists can state in advance what the overall pattern will look like based on
factors such as how wide the slits are, they cannot predict in advance where any given
electron will land on the photographic plate. They can only state the probability that it
will land near a given location í for instance, the probability is much higher in the areas
of the plate that contain the most dots.
You can liken this to a game of cards. You know that on average, one out of four cards
will be a spade. However, given a shuffled deck and asked to state for sure whether a
spade will be the first card you draw, or the second, you cannot. You can only say that
there is a one in four chance that it will be a spade.
If 12 cards are dealt from a shuffled deck of cards, the probability is greatest that 3
cards will be spades, but from observing the repetition of many such trials, you could
conclude that it is fairly likely that the actual number will be 2 or 4, and it is possible that
there will be 1 or 5 spades, or even 0 or 12. All you can do is describe the probability of
a given number of spades that will emerge when you deal 12 cards.
The section started with a question: How can the location of a particle like an electron
be described? Quantum physicists describe such a particle with a wavefunction. The
value of a wavefunction at a particular point and time is related to the probability of
finding the particle near that point, at that time. To be more precise, the absolute square
of the wavefunction is a probability density, a concept we discuss next.
Quantum physicists use the idea of probability density to describe the likelihood of
finding a particle in a given region of space. Probability density is the probability per unit
volume and is the absolute square of the wavefunction. (In one-dimensional problems,
probability density is the probability per unit length, and in two-dimensional problems, it
is the probability per unit area.)The term “absolute square” covers the case when the wavefunction has complex values
(containing the imaginary number i). The absolute square equals the wavefunction
multiplied by its complex conjugate. If the wavefunction contains only real values, so
that its complex conjugate is the wavefunction itself, then the absolute square of the
wavefunction is the same as the “ordinary” square of the wavefunction.
What is meant by probability density? It is a concept that can be applied to objects
typically described with classical mechanics. For instance, someone could create a
graph of your probability density at midnight. It is relatively likely that if someone is looking for you at that time, they would find you in bed. This
means that the probability density function has a relatively high value there. The probability density function at midnight would also have
smaller peaks at other locations in space, such as the chair in front of your computer, or in front of the refrigerator. Since the probability is zero
to find you on Mars at midnight (or at any other time), the probability density function would be zero there.
The same idea can be applied to the position of a mass on a spring that is moving back and forth in simple harmonic motion with a certain total
energy (or, physicists say that the system is in a particular state). You could graph the probability density function for the location of the mass.
In fact, we did so, and that graph is shown in Concept 2.
How do you interpret this graph? Again, the locations where the graph is higher are the locations where it is more likely you would observe the
mass. If you were only allowed to take photographs of the mass at random times, you would create a graph like this. (This is the situation in
which quantum physicists find themselves.) The graph is higher near the endpoints of the mass’s motion because it is moving more slowly
there, which means it spends more time there and you are more likely to observe it there. Taking a snapshot at a random time, you are least
likely to find it near the center because it moves the most quickly through that region.
In Concept 3, we move more to the realm of quantum mechanics. One intellectual construct that quantum physicists use to describe particles is
the concept of a particle moving in a one-dimensional rigid box. We use the concept of waves and a wavefunction to describe that particle,Electron fired through slits
Create interference pattern
Where a single electron will land can
only be stated as likelihoodPosition
Can be stated as a probability for
“classical” objectsParticle in a box
Probability density describes likelihood
that particle will be found in a given
region(^692) Copyright 2000-2007 Kinetic Books Co. Chapter 37