nodes | symmetry | overlap | observables | energy | position
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The previous page defined a one-dimensional box, and showed how to calculate the wavefunctions of an electron confined to this box. This page describes some of the mathematical properties of these wavefunctions, and shows how to use them to calculate the energy and electron distribution.
The following figure shows the shapes of the three lowest energy "box" wavefunctions, ø1-3 (only the portion of the wavefunction INSIDE the box is shown).
Notice how each wavefunction contains a unique number of "nodes", or places where ø equals zero (the nodes are marked by arrows). As a general rule, the number of nodes in a wavefunction increases with its quantum number. For the "box" wavefunctions, we find #nodes = n - 1.
The potential energy function for the box is symmetric about its center point (x = a/2). If we go a certain distance "y" from the center point, then we find the same potential regardless of the direction we travel. In other words: V(a/2 + y) = V(a/2 - y).
Inspection of ø1-3 shows that these functions have the same symmetry properties as V(x). ø1 and ø3 are symmetric about the center point, i.e.,
ø(a/2 + y) = ø(a/2 - y)
while ø2 is antisymmetric, i.e.,
ø(a/2 + y) = -ø(a/2 - y)
In general, the "symmetry" of the box wavefunctions can be inferred from their quantum number: symmetric if n = odd, antisymmetric if n = even.
The usefulness of symmetry will become apparent later when we examine integrals involving ø, and functions derived from ø. Inspection of the above graphs shows that:
Symmetry can also be used to calculate wavefunctions. A valid wavefunction must have the same symmetry (symmetric or antisymmetric) as the molecule or system that it describes. This must be so because applying a symmetry operation to a molecule must not change its energy, so applying the operation to the wavefunction must give back the wavefunction multiplied by +1 or -1.
Another important wavefunction property is its "overlap" with other wavefunctions. Overlap, S, is defined by the following integral:
You can easily show that the overlap of different wavefunctions is necessarily zero. Consider, for example, if øm and øn have different symmetries their product will be antisymmetric and the integral will vanish. Different arguments are required for wavefunctions of the same symmetry type, but the result is the same. S = 0 if "m" and "n" are different.
We indicate the "no overlap" property of wavefunctions by saying they are "orthogonal". Valid wavefunctions with different quantum numbers must always be orthogonal.
Note that a wavefunction can never be orthogonal to itself. Setting m = n makes the overlap integral identical to the normalization integral, i.e., Smm = 1.
(see especially eq. 3.29-3.32)
Our next task is to calculate some of the measurable properties of the electron. Physicists refer to measurable quantities, such as energy, position, and momentum, as "observables". The wavefunction is NOT an observable.
According to quantum mechanics, there is an operator X for every observable x., and the value of x that will be measured experimentally is called its "expectation value" or <x >. If the electron has been "prepared" to be in a particular state ø, then the expectation value will be given by:
where the integrals run over all space. Note that X, because it is an operator, will usually change the function that it operates on. Therefore, the order things are written in is important. As a rule, ø*X ø is not equal to Xø*ø (the former means "operate on ø and multiply the result by ø*", while the latter means "operate on the product of ø* and ø").
We can use the rule given above to calculate the energy of øn(x). We will call this energy, En, to show that it is associated with state "n".
The operator corresponding to energy is the Hamiltonian operator. Therefore, we simply plug H and øn into the formula given above (the denominator equals one because ø is normalized for all "n"):
Evaluating the derivative of ø, and then the integral gives:
This result shows that each state has a unique energy, En. Since "n" > 0, the energy of the electron can never be zero. Furthermore, this energy is entirely kinetic energy (V = 0 inside the box, so H = T ), so the kinetic energy of the electron is always greater than zero, and the electron is always moving around inside the box.
Note that both the oscillatory behavior of the wavefunction and the kinetic energy increase with quantum number. This result is due to the fact that the kinetic energy operator is a second derivative that describes the "curvature" of the wavefunction. Forcing more oscillations into a function of fixed length necessarily increases its curvature.
Also note that kinetic energy and box size, "a", are inversely related. If we try to confine an electron to smaller and smaller regions of space (make "a" approach zero), the kinetic energy will become larger and larger. Conversely, an electron confined in a really large box can have a very small kinetic energy (and the energy gap between states can be smaller as well, i.e., the electron can start to behave more like a classical particle and vary its kinetic energy more or less continuously).
The last question I want to consider is "where is the electron inside the box?" Recall that this question cannot be answered with certainty. We can only describe the probability of finding the electron at different locations. Also, recall that:
Since the probability and probability density differ only by a constant factor, "dx", we can use the probability density to make qualitative arguments about the probability. For example, the probability will be largest where the probability density is largest, and the probability will be zero wherever the probability density vanishes.
Graphs of probability density versus position (also known as "probability density distributions") are shown below for the three lowest energy states.
The probability distributions tell us several interesting things about the electron. First, although the "average" position of the electron is always "a/2", this may not be the most probable place to find the electron. For the lowest energy state, the most likely place to find the electron is indeed "a/2", however, for the second energy state, the probability of finding the electron at "a/2" is zero!
It is interesting to compare kinetic energy and the probability density distribution. When the kinetic energy is lowest, there is only one maximum in the probability density - it appears that the electron tends to stay near the middle of the box. However, as the kinetic energy increases, so do the number of places where one has a high probability of finding the electron. This is consistent with the classical idea that a particle with high kinetic energy moves around more.
Finally, we can also ask about finding the electron in particular regions of the box. For example, we might ask what the probability is of finding the electron within 10% of a box length of a WALL (i.e., finding the electron between "0-0.1a", or between "0.9a-a"). The answer to this question is obtained by integrating the probability density from "0-0.1a" and from "0.9a-a", and adding the two numbers together. The results are shown in the following graph:
The probability for finding the electron near the wall is less than 10% when the electron is in a low energy state (n = 1 or 2), but it rises with increasing "n" and approaches 20% as a limiting value at high "n".
We can calculate the probability of finding the electron near the CENTER of the box in the same way, i.e., by integrating the probability density from "0.4a-0.6a" (this covers 10% of the box on each side of the center). The results are shown below:
The probability is almost 40% for the lowest energy state (n = 1), but falls sharply to 5% (n = 2). As "n" increases, the probability rises and falls almost erratically, but eventually approaches 20% as a high energy limit.
Did you notice that both of these probabilities, near a WALL and near the CENTER, approach 20% as "n" increases? This is easy to understand. The electron's kinetic energy and mobility increase as "n" increases. Therefore, the probability of finding the electron within a given section of the box should be proportional to the size of the section. In the examples considered above, each section contained 20% of the box, so the limiting value of the probability should also be 20% at high "n".
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(last updated 6/8/97)