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9/22/03 |
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4.
Applications
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Unit 3One-Dimensional Quantum Mechanics |
The concepts of quantum mechanics run so counter to common sense and everyday experience that many people experience great difficulty trying to understand and accept them. Even Albert Einstein, who made significant contributions to the development of quantum mechanics, could never quite come to believe that “God plays dice with the universe.” Although quantum theory, its mathematical framework, and its applications to microscopic phenomena are well developed, thoroughly explored, and verified experimentally, the philosophical implications of the theory, the interpretation of the mathematics, and the true physical significance of the applications are still being debated today by authorities in the field.
Yet as far back as 1925 Max Born proposed an interpretation
of quantum mechanics that is still widely accepted today. The Born interpretation
asserts that there exists a wavefunction Y
that completely describes the state of a quantum particle. This wavefunction
contains all the information that can possibly be known about the particle. In
fact, it often contains information that cannot be known. Nevertheless, the
square of this wavefunction is the probability density of the particle, i.e.
the probability per unit volume of finding the particle at a certain region of
space. Therefore, in one-dimensional quantum mechanics, the probability of
finding the particle within a distance dx of position x is P(x)
dx = |Y|2 dx
= Y*Y dx.
Some of the fundamental principles of quantum mechanics assert that physical states are represented by wavefunctions, that observable properties of those states are represented by operators that operate on the wavefunctions, that these observables can take on only specific eigenvalues each associated with a particular eigenstate, that the observables have only expectation values with an accompanying uncertainty whenever the state is not an eigenstate, and that an equation of motion, called Schrödinger’s equation, describes the evolution of a quantum state between periods of observation.
Once these principles are understood, they can be applied to numerous situations. Examples of one-dimensional applications of quantum mechanics include a particle in a box, a particle in a well, a quantum oscillator, reflection off a square barrier, and quantum tunneling through a potential barrier.
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