1-D box | trial wavefn | normalize | quant number & state
The previous sections described some of the conditions that an acceptable wavefunction must satisfy, and I would like to demonstrate how these conditions are used to calculate actual wavefunctions.
Unfortunately, it is either difficult or impossible to calculate the wavefunction of an electron inside the average atom or molecule. Therefore, we will have to make do with a simple system known as a "one-dimensional box". This box (or line segment) consists of a set of points along the x-axis in which the electron is free to move.
A number of useful conclusions will emerge from this study including: 1) how to calculate a wavefunction, 2) what is meant by quantum numbers and quantum states, 3) wavefunction normalization and orthogonality, and 4) how one can relate customary physical quantities like position, kinetic energy, and potential energy, to quantum mechanical quantities like wavefunctions, probability densities, and expectation values.
The presentation is broken into two sections. This page describes calculation of the wavefunction and energy eigenvalues, and the next page describes important observations about this system.
An one-dimensional "box" on the x-axis can be defined by picking appropriate values for the potential energy, V(x). For example, the following definition of the potential energy operator, V,
gives a "box" that stretches from x = 0 to x = "a" (we assume "a" to be a positive number).
The electron's potential energy is infinitely lower inside this box than it is outside the box, therefore the electron is effectively confined to the box.
In order to calculate the wavefunction, we must decide whether to find a general wavefunction, ø(x,t), that satisfies the time-dependent Schrödinger equation, or a spatial wavefunction, ø(x), that satisfies the time-independent Schrödinger equation. The latter choice is appropriate if the energy of the system remains constant, and that is indeed the case here (this conclusion is justified if 1) the electron collides elastically with the box "walls", and 2) that there are no electromagnetic fields to accelerate the electron). So we need only find ø(x) using the time-independent equation.
Outside The Box. The first step is to find the value of the wavefunction, ø(x), for x outside the box, i.e., x < 0 or x > a. This is easy because we know that the probability of detecting an electron outside the box is zero. Therefore, ø(x) must also be zero everywhere outside the box. That is,
Boundary Conditions. The next step is to introduce two new conditions for an acceptable wavefunction: we demand that ø(x) be continuous and single-valued at all x. These constraints require that ø(x) inside the box blend smoothly with ø(x) outside the box, and this gives the following "boundary conditions" at each wall:
Inside The Box. The next step is to use the fact that V(x) = 0 inside the box. This means that ø(x) must satisfy the following Schrödinger equation INSIDE the box:
[Note: the partial derivative has been written as a full derivative because ø(x) is only a function of x.]
Two functions that might satisfy the Schrödinger equation are cos(kx) and sin(kx). This is because their second-derivatives are simply the same functions multiplied by a constant, i.e.,
Checking these possibilities against our boundary conditions shows that ø(x) = cos(kx) is not a useful answer; cos(k0) = cos(0) = 1, whereas we require ø(0) = 0. On the other hand, we see that ø(x) = sin(kx) is a useful answer because sin(k0) = sin(0) = 0.
Our trial wavefunction, sin(kx), must also meet the other boundary condition, ø(a) = 0. We can make sin(ka) = 0 if we choose "k" so that "ka" will be an integer multiple of 180o (or pi radians). That is, we require:
or
We are nearly done. We have found a family of trial wavefunctions that satisfy both the boundary conditions and the Schrödinger equation inside the box. The only remaining step is to scale the wavefunctions so that they give the right absolute values for the probability of finding an electron. This process is called "normalization".
So far we have said that the probability of detecting an electron at a given point, P(x), is proportional to [ø(x)]2 (assuming our wavefunction is real). We now need to give an exact relationship between the two quantities so that we can scale our trial wavefunctions properly.
The square of the wavefunction gives a number that is referred to variously as the "probability density", the "electron density", and the "charge density". The Greek letter "rho" is used to represent this quantity, and the relationship between P, ø, and "rho" is given by:
These equations say that the probability of detecting the electron between "x" and "x+dx" is the electron density at "x" times the length of the interval, dx. This relationship is only valid in the limit that "dx" approaches zero, however, since p(x) will not be constant over the interval otherwise.
(see postulate #1)
Also, note the similarity between electron density and "mass" density. To calculate the mass of an object you would multiply its mass density by its volume. Similarly, to calculate the probability of detecting an electron in an interval, you must multiply the electron density by the size of the interval.
As we said above, an acceptable wavefunction must be normalized, i.e., it must predict a probability of one (absolute certainty) that the electron is somewhere in the box. This probability is obtained by summing up probabilities for all of the small intervals separating x = 0 and and x = a, i.e.,
The normalization requirement is met by introducing a scaling factor, A, into the trial wavefunction such that ø(x) = Asin(kx). We need only choose the value for A that makes P(0 to a) = 1. The calculation of A proceeds as follows:
Once we apply the correct normalization constant, A, to sin(kx), our calculation is finished. The acceptable wavefunctions for the electron in the one-dimensional box are:
The above equations define all of the acceptable wavefunctions for an electron in an one-dimensional box. These wavefunctions satisfy the boundary conditions and the Schrödinger equation, and are continuous, single-valued, and normalized. Interestingly, there are an infinite number of acceptable solutions, a result that we might not have anticipated.
The various wavefunctions have different integer values for "n". "n" is called a "quantum number" and emerges naturally during the boundary condition part of the calculation. We did not need to assume that a quantum number would exist. Many other systems also have quantum numbers, and these numbers usually emerge in the same way, i.e., as a natural part of calculating the wavefunction.
Finally, we note that each wavefunction corresponds to a different "quantum state" of the electron. The properties of these states (energy, distribution of electron density) do not change in time, but unless the electron has been "prepared" in a particular state, its experimental properties cannot be predicted. An electron may, with varying probabilities, behave in way that is consistent with any number of states. What is paradoxical is that an experimental measurement will yield a value that is consistent with only one state, i.e., the measurement will seem to suggest that the electron had been in a particular state all along.
(see postulates #3
and #4)
(see eq. 3.23 to end.
Note: The above text describes what happens in a single measurement, while
CCQC text describes a measured value that is an "average" of many
measurements.)
(last updated 6/8/97)