By E. Elizalde, S. D. Odintsov, A. Romeo, Andrei A. Bytsenko, S. Zerbini
Discussing Zeta regularization ideas, this ebook comprises chapters on: the Reimann Zeta functionality; the Casimir impact in flat space-time; and houses of the chemical strength in higher-dimensional manifolds.
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Additional resources for Zeta regularization techniques with applications
And t,,P = Ae-'"". 19) It follows that the particle is equally likely to be found anywhere along the x axis, which is equivalent to stating that its position at any instant is unknown. From this we conclude that a particle with wavefunction y = Aeiks has a definite momentum but an undefined position. We will come back to this subject in Chapter 3, when the Heisenberg Uncertainty Principle is discussed. 4 Time-independent Wavefunctions All the wavefunctions that will be considered in this book are functions only of the spatial coordinates of the system; they do not contain time as a variable.
8 The superposition of an infinite number of waves which are in phase at only one point Since the delta function is made up of an infinite number of sinusoidal waves with differing wavelengths, and therefore differing k values, there is an infinite uncertainty in the momentum of the particle, with each sinusoidal wave representing a different momentum according to the formula p = hk. 46 Quantum Mechanics for Chemists Conclusion: the position of the particle is known, but there is an infinite uncertainty as to its momentum.
Use of the diffraction relation nil = dsine, with n = 1, leads to Apx = h/d. This is only an approximate formula because the other four first-order diffraction peaks have been ignored, as well as higher-order peaks, but it gives a rough measure of the overall uncertainty in the momentum. 1) It is significant that the particular properties of the system have cancelled out, and we are left simply with a universal constant. It is also worth noting that the “uncertainty” or “fuzziness” that we have discussed does not arise from any defect in our knowledge of the system, or the method of measurement, but from the inherent nature of the diffraction process.
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