Visual Quantum Mechanics

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Chapter Summary:

3. Free Particles

We start our exposition of quantum mechanics with a "derivation" of the Schrödinger equation for free particles. This is just a first step. In realistic situations, particles interact with force fields and other particles and can only be detected through their interaction with some measurement device. Nevertheless, a good understanding of the free motion is important, e.g., for the asymptotic description of interacting particles in scattering experiments.

The (time-dependent) free Schrödinger equation will be obtained as the differential equation for deBroglie's plane waves. The plane waves have been introduced to describe the wave-like behavior of a beam of particles, thereby relating the wave number to the momentum and the frequency to the energy of the particles. However, the property of being localized in some region -- which is typical for particles -- cannot be described by plane waves. Therefore, we exploit the linearity of the Schrödinger equation to form (continuous) superpositions of plane waves. This process can be described as a Fourier transform and leads to wave packets corresponding to fairly localized phenomena. In this way, the problem of solving the Schrödinger equation with an arbitrary initial function can be reduced to the calculation of a Fourier integral.

Unlike classical particles, wave packets can neither have a sharp position nor a sharp momentum. The extension of the wave packets in position and momentum space can be described by the uncertainties of position and momentum which satisfy Heisenberg's uncertainty relation.

The wave function is usually interpreted statistically. The square of the absolute value of the complex-valued wave packet describes a position probability distribution. The Fourier transform of the wave packet is related in the same way to the distribution of momenta. According to the statistical interpretation, the predictions of quantum mechanics concern the probability distributions of measurement results. This is rather a theory of statistical ensembles than of individual particles. The status of individual systems within quantum theory depends on the interpretation of the measuring process. In this context we mention the paradox with Schrödinger's cat which occurs whenever quantum systems are in a superposition of rather distinct wave functions. The chapter ends with a description of the asymptotic form of the free time evolution and a discussion of the energy representation. Both will be important for the formulation of scattering theory in Chapter 9.




< Previous \| Index \| Next > Chapter Summary: 3. Free Particles We start our exposition of quantum mechanics with a "derivation" of the Schrödinger equation for free particles. This is just a first step. In realistic situations, particles interact with force fields and other particles and can only be detected through their interaction with some measurement device. Nevertheless, a good understanding of the free motion is important, e.g., for the asymptotic description of interacting particles in scattering experiments. The (time-dependent) free Schrödinger equation will be obtained as the differential equation for deBroglie's plane waves. The plane waves have been introduced to describe the wave-like behavior of a beam of particles, thereby relating the wave number to the momentum and the frequency to the energy of the particles. However, the property of being localized in some region -- which is typical for particles -- cannot be described by plane waves. Therefore, we exploit the linearity of the Schrödinger equation to form (continuous) superpositions of plane waves. This process can be described as a Fourier transform and leads to wave packets corresponding to fairly localized phenomena. In this way, the problem of solving the Schrödinger equation with an arbitrary initial function can be reduced to the calculation of a Fourier integral. Unlike classical particles, wave packets can neither have a sharp position nor a sharp momentum. The extension of the wave packets in position and momentum space can be described by the uncertainties of position and momentum which satisfy Heisenberg's uncertainty relation. The wave function is usually interpreted statistically. The square of the absolute value of the complex-valued wave packet describes a position probability distribution. The Fourier transform of the wave packet is related in the same way to the distribution of momenta. According to the statistical interpretation, the predictions of quantum mechanics concern the probability distributions of measurement results. This is rather a theory of statistical ensembles than of individual particles. The status of individual systems within quantum theory depends on the interpretation of the measuring process. In this context we mention the paradox with Schrödinger's cat which occurs whenever quantum systems are in a superposition of rather distinct wave functions. The chapter ends with a description of the asymptotic form of the free time evolution and a discussion of the energy representation. Both will be important for the formulation of scattering theory in Chapter 9.

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