Abstract
The wave-particle duality asserts that every entity, in spite of being of particle character in its very fabric, nevertheless moves as a wave. This dual nature of the World, on which Quantum Mechanics is based, provides the required kinetic energy to counterbalance the attractive nature of the interactions and to stabilize both the microscopic and the macroscopic structures of matter. Proof of this is the presence of Planck’s constant in formulae referring to properties of all these structures. The core of Quantum Mechanics can be condensed to the following three fundamental principles possessing amazing quantitative explanatory power: Those of Heisenberg, Pauli, and Schrödinger. The content of this chapter is ‘sine qua non’ for what follows.
So reasonable the incomprehensible.
O. Elytis
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Notes
- 1.
A vivid picture of interference is provided by throwing two stones in a quiet pond and observe the resulting two circular waves cancelling each other at some points and being reinforced at some other.
- 2.
The magnitude k of the wavevector is related to the wavelength \(\lambda\) by the relation, \(k = 2\pi /\lambda\), and its direction gives the direction of propagation of the wave.
- 3.
- 4.
\(\Delta x\) is the so-called standard deviation defined by the relation: \(\Delta x^{2} \equiv \, < (x - < x > )^{2} > \, = \, < x^{2} > - < x >^{2} .\) A similar definition applies to \(\Delta p_{x}\) with x replaced by \(p_{x} .\) The symbol <f>, for any random quantity f, denotes its average value.
- 5.
However, the average value of \(p_{x}\) can be zero, if the contributions of positive and negative values cancel each other.
- 6.
At this point it is not useless to clarify the relation between the confinement length \(l_{x} \equiv 2a\) along the direction x and the standard deviation Δx. For a waveparticle confined in a fixed region of space we can always choose the coordinate system in such a way that \(< x > = 0\); since this region is fixed, i.e. not moving, \(< p_{x} >\) is also necessarily equal to zero. By the general definition of the standard deviation given in the footnote 5 and taking into account that \(< x > = 0\) and \(< p_{x} >\) = 0, we obtain the relations \(\Delta x^{2} \equiv < x^{2} >\) and \(\Delta p_{x}^{2} = < p_{x}^{2} >\). Hence, it follows from (3.8) that \(< p_{x}^{2} > \,\, \ge \hbar^{2} /(4 < x^{2} >)\), where by definition (and the choice \(< x > = 0\)),\(< x^{2} > \, \equiv \int_{{l_{x} }}^{{}} {dx\left| {\psi (x)} \right|}^{2} x^{2} ;\) for a uniform probability density where \(\left| {\psi (x,t)} \right|^{2} = 1/l_{x}\), we have \(< x^{2} > \, = \varDelta x^{2} = l_{x}^{2} /12\). In general \(\Delta x\) is proportional to and smaller than the length of confinement. The proportionality factor depends on the probability density. It is important to keep in mind that the probability density in Quantum Mechanics (actually the wave function \(\psi\) for the lowest energy state to be precise) is everywhere as smooth as possible within the restrictions imposed by the potential. This feature of QM is implicitly assumed here and in the rest of this book as a supplement of the uncertainty principle. Otherwise, as it was pointed out by Lieb [11], the uncertainty principle per se cannot guarantee the stabilization of the structures of the world.
- 7.
Equation (3.10) was obtained by using the symmetry of the sphere and by assuming constant probability density for any value of r inside the sphere.
- 8.
The correct relativistic relation between kinetic energy and momentum is: \(\varepsilon_{K} = (m_{o}^{2} c^{4} + c^{2} p^{2} )^{1/2} - m_{o} c^{2}\). This relation in the non-relativistic limit, \(m_{o} c^{2} \gg c\,p,\) becomes \(\varepsilon_{K} \approx {\mathbf{p}}^{2} /2m_{o} ,\) while in the extreme relativistic limit, \(m_{o} c^{2} \ll c\,p\), becomes \(\varepsilon_{K} = c\,p;\) \(m_{o}\) is the rest mass of the particle.
- 9.
For the numerical values of all universal constants see Table I.1 in Appendix I .
- 10.
Problems indicated by an asterisk have broader physical implications, but require more than a simple application of a formula.
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Economou, E.N. (2016). The Wave-Particle Duality. In: From Quarks to the Universe. Springer, Cham. https://doi.org/10.1007/978-3-319-20654-7_3
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DOI: https://doi.org/10.1007/978-3-319-20654-7_3
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