Physical Basis of Quantum Mechanics
In this chapter, we will discuss the experimental evidence that led to the development of quantum mechanics, as well as some of the basic principles of the theory. In quantum mechanics, systems are described in terms of wave functions, which specify the probability of finding a system in a given state. This is in contrast to classical physics, which describes systems in terms of well-defined positions and velocities of particles. While many classical systems, such as fluids, are described in terms of statistical theories, said theories are only convenient approximations to the underlying deterministic theories that describe the individual particles that make up the system. Quantum mechaincs, on the other hand, is not a statistical approximation to some underlying deterministic theory. It proposes an inherent randomness to the behavior of physical systems, and all of its laws are written in terms of wave functions rather than definite properties of particles.
Due to this probabilistic nature, one can no longer think of particles as having definite positions and momenta. These become observables, which are linear operators that act on wave functions, in quantum mechanics. These operators allow one to extract from the wave function all of the possible outcomes of a given measurement and the probabilities of said outcomes. We will see, in fact, that quantities represented by non-commuting observables, such as position and momentum, cannot, even in principle, be simultaneously measured to infinite precision; this is known as the Heisenberg Uncertainty Principle.
The concept of a wave function leads to some of the basic ideas of quantum mechanics, such as wave-particle duality and the Principle of Complementarity. Wave-particle duality is the idea that particles possess a wave-like nature, and vice versa. The Principle of Complementarity states that it is impossible to observe both particle-like and wave-like behaviors in the same experiment.
While quantum mechaincs is fundamentally different from classical mechanics, we know that classical mechanics describes sufficiently large objects very well; it is only at very small length scales (on the order of a nanometer) that classical mechanics begins to break down. This leads us to the Correspondence Principle, which states that quantum mechanics must reduce to classical mechanics at sufficiently large length and energy scales. The fact that the theory does, in fact, reduce to classical mechanics in the appropriate limit will become more clear in later chapters.
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