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Chapter Summary:

8. Special Systems

In some rare cases we can find a complete and explicit solution of the Schrödinger equation without having to use numerical methods. Our collection of analytically solvable problems so far includes only the free particle, the particle in a box, and the harmonic oscillator. In this chapter we add the free fall in a linear potential and the motion in a constant magnetic field.

The linear potential describes a constant force field, like the homogeneous gravitational field near earth's surface. Whenever the free time evolution of the initial wave packet is known, the free fall can also be calculated exactly. The solution describes a uniformly accelerated motion and the expectation values of position and momentum behave in a classical way. Next we add a reflecting boundary condition in order to describe the quantum analog of a steel ball dancing on a horizontal pane of glass. In quantum mechanics this system has only discrete energies. The motion of wave packets in this situation is not periodic, because the frequencies of the eigenfunctions are incommensurable.

A charged particle in a constant magnetic field "feels" a force which is always perpendicular to its velocity. It is sufficient to treat this problem in the two-dimensional plane orthogonal to the direction of the magnetic field. The trajectories of classical particles are circles in that plane and the solutions of the two-dimensional Schr\"odinger equation all describe bound states. We determine the eigenvalues of the Hamiltonian operator by exploiting an analogy between the components of the velocity operator (which do not commute with each other) and the position and momentum observables for a harmonic oscillator. By this analogy we obtain results about the motion of arbitrary Gaussian wave packets and derive the integral kernel of the time evolution.

In quantum mechanics, the magnetic field has to be described by a (gauge dependent) vector potential, but we expect that physical predictions should be independent of the chosen gauge. Thus the interpretation of wave packets in magnetic fields becomes a tricky business, and one has to take into account the fact that the canonical momentum has no simple (gauge invariant) relation with the velocity. For example, the Schrödinger equation in a constant magnetic field is not invariant under translations. Nevertheless, any translation can be compensated by a gauge transformation and therefore the system has a translational symmetry. As a consequence, the energy eigenvalues have an infinite multiplicity.

The constant magnetic field has a rotational symmetry, because the Hamiltonian commutes with the angular momentum operator. This leads us to an investigation of systems with sperical symmetry in two dimensions, which -- after a transition to polar coordinates -- can be solved in a basis of angular momentum eigenstates. The Schrödinger equation for the eigenvalues is thus reduced to an ordinary differential equation in the radial variable. This treatment prepares the ground for the more complicated reduction to angular momentum eigenspaces for three-dimensional systems in Book Two.




< Previous \| Index \| Next > Chapter Summary: 8. Special Systems In some rare cases we can find a complete and explicit solution of the Schrödinger equation without having to use numerical methods. Our collection of analytically solvable problems so far includes only the free particle, the particle in a box, and the harmonic oscillator. In this chapter we add the free fall in a linear potential and the motion in a constant magnetic field. The linear potential describes a constant force field, like the homogeneous gravitational field near earth's surface. Whenever the free time evolution of the initial wave packet is known, the free fall can also be calculated exactly. The solution describes a uniformly accelerated motion and the expectation values of position and momentum behave in a classical way. Next we add a reflecting boundary condition in order to describe the quantum analog of a steel ball dancing on a horizontal pane of glass. In quantum mechanics this system has only discrete energies. The motion of wave packets in this situation is not periodic, because the frequencies of the eigenfunctions are incommensurable. A charged particle in a constant magnetic field "feels" a force which is always perpendicular to its velocity. It is sufficient to treat this problem in the two-dimensional plane orthogonal to the direction of the magnetic field. The trajectories of classical particles are circles in that plane and the solutions of the two-dimensional Schr\"odinger equation all describe bound states. We determine the eigenvalues of the Hamiltonian operator by exploiting an analogy between the components of the velocity operator (which do not commute with each other) and the position and momentum observables for a harmonic oscillator. By this analogy we obtain results about the motion of arbitrary Gaussian wave packets and derive the integral kernel of the time evolution. In quantum mechanics, the magnetic field has to be described by a (gauge dependent) vector potential, but we expect that physical predictions should be independent of the chosen gauge. Thus the interpretation of wave packets in magnetic fields becomes a tricky business, and one has to take into account the fact that the canonical momentum has no simple (gauge invariant) relation with the velocity. For example, the Schrödinger equation in a constant magnetic field is not invariant under translations. Nevertheless, any translation can be compensated by a gauge transformation and therefore the system has a translational symmetry. As a consequence, the energy eigenvalues have an infinite multiplicity. The constant magnetic field has a rotational symmetry, because the Hamiltonian commutes with the angular momentum operator. This leads us to an investigation of systems with sperical symmetry in two dimensions, which -- after a transition to polar coordinates -- can be solved in a basis of angular momentum eigenstates. The Schrödinger equation for the eigenvalues is thus reduced to an ordinary differential equation in the radial variable. This treatment prepares the ground for the more complicated reduction to angular momentum eigenspaces for three-dimensional systems in Book Two.

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