Once Before Time - Martin Bojowald [31]
Provided that the corrections correspond to small shifts of the trajectory (in comparison, for instance, with the radius of an atom in such a state), one can apply mathematical approximations called perturbation theories: The deviation is seen as a small disturbance of the classical trajectory and can be computed with more ease. Instead of computing such perturbations directly from a semiclassical wave function, solving the rather complicated Schrödinger equation, the classical equations themselves can be modified. If the mathematical procedure is followed correctly, solutions to the new equations describe the motion of a wave function better than do solutions of the purely classical equations. Corrected equations can, for instance, incorporate a force unknown to the classical theory. Such a quantum force would constitute the effective description of implications of the changing wave mountain on its peak position.
Modifications to the classical equations are called quantum corrections, and in contrast to the classical terms, they depend on Planck’s constant. When this constant is set to zero, quantum corrections vanish—another example of how the classical theory appears as a limit of the quantum theory. Quantum-corrected equations are often very efficient and not only allow simpler computations of certain quantities, but often provide an intuitive understanding of quantum phenomena. While wave functions are indirectly used in the determination of quantum corrections, their properties, such as difficulties in simultaneous measurements of position and velocity arising from the uncertainty relation, need not be taken explicitly into account. All this is, after all, already provided for in the derivation of correction terms. Such equations—called effective—have played a large role ever since the early stages of quantum mechanics.
Especially in semiclassical applications of quantum theory, one can often find effects for which, at least to some degree, visualization by analogy with water waves on a lake applies. A wave’s maximum corresponds to the position of a classical particle, but the wave must always be spread out: The particle’s position is not determined precisely. Because of the spreading, a wave can stretch out its sentinels and become sensitive to possible unevenness in different areas of the lakebed, even at places where the particle would never tread in classical physics. Tails of the wave can, for instance, extend into more shallow parts of the lake even if the maximum lies far away. A wave propagates differently in shallow areas than in deep ones: The speed of waves depends on the depths of the lake where they are traveling. Accordingly, the direction of the whole wave as well as its form change depending on how the lakebed is formed. A well-known example is the fact that ocean waves always move toward the beach because they are redirected by the depth decreasing toward the land. Changes in the shape of a wave due to the ground profile can be observed in an impressive way by the piling up and eventual breaking of waves near the beach.
Breaking waves show strong unparticlelike effects and can no longer be considered to result from mere perturbations of simple motion. But before it comes to this, the seabed’s profile causes the peak position to shift only slightly, which can be taken into account by a correction to motion over level ground. Moreover, the wave will spread out ever more, sending its sentinels farther in new areas: Owing to their different propagation speeds at different depths, separate components of an extended wave experience slightly distinct shifts over the course of time, and the wave becomes more dispersed. (In exceptional cases a wave may be concentrated into a small region, but this would require a fine-tuning of all constituent wave contributions. Moreover, even then the wave would, after having been focused at one time, disperse again.)
In addition to the need for corrections to a particle trajectory when a wave symbolizes the