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2 Principles of Quantum Mechanics

Quantum mechanics presents an approach to describe the behavior of microscopic systems. Whereas in classical mechanics the position and momentum of a moving particle can be established simultaneously, Heisenberg's uncertainty principle prohibits the simultaneous determination of those two quantities. This is manifested by Eq. (2.1):

(2.1)

      which implies that the uncertainty in the momentum and position always exceeds ħ/2, where ħ is Planck's constant divided by 2π. Mathematically, Eq. (2.1) follows from the fact that the operators responsible for defining position and momentum, and , do not commutate; that is, . (This aspect will be discussed in more detail at the end of Section 2.1.) As we shall see later (Chapter 5), the uncertainty principle also can be rewritten in terms of the uncertainty in energy and lifetime of a spectroscopic state or in frequency and time of a wave.

      The incorporation of this uncertainty into the picture of the motion of microscopic particles leads to discrepancies between classical and quantum mechanics: classical physics has a deterministic outcome, which implies that if the position and velocity (trajectory) of a moving body are established, it is possible to predict with certainty where it is going to be found in the future. This principle certainly holds at the macroscopic scale: if the position and trajectory of a macroscopic body, for example, the moon, are known, it is certainly possible to calculate its position six days from now and to send a spaceship to this predicted position.

Quantum mechanical systems, on the other hand, obey a probabilistic behavior. Since the position and momentum can never be determined simultaneously at any point in time, the position (or momentum) in the future cannot be precisely predicted, only the probability of either of them. This is manifested in the postulate that all properties, present or future, of a particle are contained in a quantity known as the wavefunction Ψ of a system. This function, in general, depends on spatial coordinates and time; thus, for a one‐dimensional motion (to be discussed first), the wavefunction is written as Ψ(x, t). The probability of finding a quantum mechanical system at any time is given by the integral of the square of this wavefunction: ∫Ψ(x, t)² dx. This is, in fact, one of the “postulates” on which quantum mechanics is based to be discussed next. Different authors list these postulates in different orders and include different postulates necessary for the description of quantum mechanical systems [1]. Quantum mechanics is unusual in that it is based on postulates, whereas science, in general, is axiom‐based.

2.1 Postulates of Quantum Mechanics

      Postulate 1: The state of a quantum mechanical system is completely defined by a wavefunction Ψ(x, t). The square of this function, or in the case of complex wavefunction, the product Ψ*(x, t) Ψ(x, t), integrated over a volume element dτ (= dx dy dz in Cartesian coordinates or sin²θ dθ dφ in spherical polar coordinates) gives the probability of finding a system in the volume element dτ. Here, Ψ*(x, t) is the complex conjugate of the function Ψ(x, t). This postulate contains the transition from a deterministic to probabilistic description of a quantum mechanical system. The wavefunctions must be mathematically well behaved, that is, they must be single‐valued, continuous, having a continuous first derivative, and integratable (so they can be normalized).

      Postulate

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