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6.
Alpha Decay |
One-Dimensional
Applications of Quantum Mechanics
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One-dimensional applications of quantum mechanics may not appear to be very practical at first glance. But they illustrate many of the fundamental properties of quantum systems without getting bogged down with unnecessary mathematics. For example, the one-dimensional quantum particle in a box is mathematically no more challenging than the solution to the classical problem of a vibrating string that is covered in a typical sophomore physics class. Indeed, the solution to this simple problem illustrates why the energy levels of all bounded systems must ultimately be quantized.
The solution to the problem of a particle in a well is conceptually only slightly different than the particle in a box. However, the mathematics becomes noticeably more difficult. Nevertheless, it illustrates how the boundary conditions of quantum states must be carefully satisfied and how the energy levels are modified by the finite potential well. It also shows how the particle can escape from the well and how its energy levels become continuous when it is no longer bounded.
The quantum oscillator is an excellent approximation to real world situations. And it shows how symmetry arguments can be used to solve certain aspects of a quantum problem without delving deeply into more complicated mathematics. It is also a good example to show how quantum solutions approach classical solutions when the quantum numbers become large.
The square barrier problem illustrates what happens when an energetic quantum particle strikes a barrier. Classically, the particle penetrates the barrier if it has enough energy and bounces back if it does not. But a quantum particle is both transmitted through the barrier and reflected backward. The amount of transmission and reflection depend upon the barrier height and the energy of the particle. Even if the particle does not have enough energy to penetrate through the barrier, there is a finite probability that it can tunnel through the barrier and emerge on the other side.
One example of the quantum tunneling phenomena is that of field emission for an electron. In order to remove an electron from the surface of a metal, a certain amount of energy (called the work function) is required. High-energy light photons can supply this energy thereby ejecting photoelectrons. Thermal energy can also do the job if the metal is heated hot enough. But if there is no light and no heat, the electrons simply do not have enough energy to overcome the potential barrier at the surface of the metal. Even if a strong external electric field is applied to the metal the electrons should not be able to escape. Nevertheless, they do escape because they tunnel through the potential barrier far enough to be pulled the rest of the way out by the electric field.
Another example of tunneling occurs in alpha decay of certain radioactive elements. Again, there is a strong nuclear force producing a barrier that the alpha particles should not be able to penetrate. But again, the quantum tunneling allows them to tunnel through the barrier far enough for the Coulomb repulsion to send them on their way.
A third example of tunneling is associated with the ammonia molecule. This molecule consists of one nitrogen atom bonded to three hydrogen atoms. A stable configuration of this atom occurs when each atom is near one of the four corners of a tetrahedron. In this configuration the three hydrogen atoms constitute a plane with the nitrogen atom either above or below that plane. If the nitrogen atom happens to be located above the plane, it does not have the energy to penetrate the potential barrier of the plane to get below, and vice versa. Nevertheless, it does. In fact, it penetrates back and forth time and again at a definite frequency precise enough to be used in some of the first atomic clocks. This multiple tunneling phenomenon is called ammonia inversion.
A fourth example of quantum tunneling occurs in the scanning tunneling microscope. In this application, the tunneling effect is so sensitive that the microscope can actually image the individual atoms located on the surface of a crystal.
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