Experiment: Particle in a One-Dimensional Box


    This Java applet provides a visual representation of the "Particle in a One-Dimensional Box" problem. In this problem a particle of mass m is constrained to move along a line of length a. At the ends of the line are large potential energy barriers that keep the particle within the "box." In the quantum-mechanical treatment of the problem the Schrödinger wave equation is used to calculate the allowed energies of the particle. Since the potential energy inside the "box" is zero, the Schrödinger wave equation takes the form (eq 1)



The term is a differential operator and h is Planck's constant. The solution to this second-order, homogeneous, linear differential equation (eq 1) is



where n called the quantum number is a dimensionless integer (1, 2, 3, 4, ). The solution, yn, is called a wave function. Substitution of the yn (eq 2) into the Schrödinger wave equation (eq 1) yields the following expression for the energy of the particle.



The applet below illustrates some important properties of the quantum-mechanical treatment of particles.

    Select the mass (1.0e-27 or 2.0e-27 g) of the particle from the menu labeled "Mass" and the length of the line (7 or 14 Å) from the "Width of Box" menu. Click the "Start" button to initiate the animation. An Energy Level Diagram will appear to the right of the animation. The allowed energies, En, for values of n up to 10 are plotted as horizontal lines in the diagram. These lines are called energy levels. The energy of a level is displayed to the left of the line and the value of the quantum number n is found to the right. Note: En • 1012 erg-1 = 4.48 means En= 4.48 • 10-12 erg. The energy of the particle, En, in the "Box" is indicated by the presence of a plot of yn versus x on the Enenergy level in the Energy Level Diagram. When the "Probability" button in the lower right corner of the "Energy Level Diagram" window is clicked, a plot of (yn)2 versus x appears on the diagram. (yn)2 is called the probability density and is proportional to the probability of finding the particle between x and x + dx.

    The "Lamp" emits monochromatic light at 3941, 4054, 4434, 4730, 5068, 5457, 5912, 6249, 6450, and 6999 Å. To radiate the particle in the "Box" with monochromatic light, select the wavelength of the light from the pull-down menu labeled "Wavelength." With the selection of the wavelength, the particle is illuminated with a two-second burst of monochromatic light.

   Complete the following exercises and answer the questions. See the Tutorial for Experiment: Particle in a One-Dimensional Box for an example and help.

  1. Select 2.0e-27 g from the "Mass" menu and pick 7 Å from the menu labeled "Width of Box." Click the "Start" button.
    1. Can the particle have any energy from zero to infinity?
    2. What are the values of n and En for the particle?
    3. Does the speed (magnitude of the velocity) of the particle change with time?
    4. Does the energy of the particle change with time?
  2. Select 4434 Å from the pull-down menu labeled "Wavelength" and illuminate the particle in exercise #1.
    1. Did the energy of the particle change? If your answer is yes, what is the new energy of the particle?
    2. Is the speed of the particle different from the speed observed in the previous exercise? To return to the animation in exercise #1, click the "Stop" button, set the "Wavelength" to 0, and click the "Start" button. Return to the animation in this exercise by selecting 4434 Å from the "Wavelength" menu.
    3. Is yn for the new energy En the same as y1? If your answer is no, how do the wave functions differ? To revisit the plot of y1 versus x, click the "Stop" button, set the "Wavelength" to 0, and click the "Start" button. Please return to the En energy level by selecting 4434 Å from the "Wavelength" menu.
    4. Does the energy of the particle change with time?
    5. Does the speed of the particle change with time?
  3. Select 6999 Å from the pull-down menu labeled "Wavelength" and illuminate the particle in exercise #2.
    1. Did the energy of the particle change? If your answer is yes, what is the new energy of the particle?
    2. Does the energy of the particle change with time?
    3. Does the speed of the particle change with time?
  4. Select 5068 Å from the pull-down menu labeled "Wavelength" and illuminate the particle in exercise #3.
    1. Did the energy of the particle change? If your answer is yes, what is the new energy En of the particle and what is the value of the quantum number n?
    2. Is yn for the new energy En the same as yn for the energy En in exercise #2? If your answer is no, how do the wave functions differ? To revisit the plot of yn in exercise #2, click the "Stop" button, set the "Wavelength" to 4434 Å, and click the "Start" button. Return to the En energy level in this exercise by selecting 5068 Å from the "Wavelength" menu.
    3. Does the energy of the particle change with time?
    4. Does the speed of the particle change with time?
    5. In order for the particle to gain energy what must happen?

    If you have preformed all of the exercises, answered all of the questions, and completed the report to be submitted for credit, then you may check the Answers to Particle in a One-Dimensional Box Questions



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