# A garden of quanta: Essays in honor of Hiroshi Ezawa by A Tonomura, T Nakamura, I Ojima By A Tonomura, T Nakamura, I Ojima

This e-book is a set of studies and essays concerning the fresh wide-ranging advancements within the parts of quantum physics. The articles have quite often been written on the graduate point, yet a few are obtainable to complex undergraduates. they're going to function solid introductions for starting graduate scholars in quantum physics who're trying to find instructions. elements of mathematical physics, quantum box theories and statistical physics are emphasised.

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34) that p p|q = i ∂ p|q . ∂q 29 2 Reminder: Classical and quantum mechanics Similarly, by considering p| qˆ |q we find q p|q = i ∂ p|q . ∂p Integrating these identities over q and p respectively, we obtain p|q = C1 (p) exp − ipq , p|q = C2 (q) exp − ipq , where C1 (p) and C2 (q) are arbitrary functions. The last two equations are compatible only if C1 (p) = C2 (q) = const, therefore p|q = C exp − ipq . 35) The constant C is determined (up to an irrelevant phase factor) by the normalization condition to be C = (2π )−1/2 .

In that case, we need to replace e−iωk t by mode functions vk (t) which are certain complex-valued solutions of the equation v¨k + ωk2 (t)vk = 0. The mode expansion is written more generally as d3 k 1 −ik·x √ a . ) From Eq. 17) we can read off the mode functions of a free field in flat space, 1 vk (t) = √ eiωk t , ωk ωk = k 2 + m2 . 19) In this case the mode functions depend only on the magnitude of the wave number k, so we write vk and not vk . Remark: quantitative meaning of mode functions. 18) relates φˆ to a ˆ± k and vk .

The Green’s functions Gret (t, t′ ) and GF (t, t′ ) are defined by Eqs. 17). For t1,2 ≥ T , show that: (a) The expectation value of qˆ(t1 )ˆ q (t2 ) in the “in” state is 0in | qˆ (t1 ) qˆ (t2 ) |0in = 1 iω(t2 −t1 ) e + 2ω Z 0 T dt′1 Z T 0 ` ´ ` ´ ` ´ ` ´ dt′2 J t′1 J t′2 Gret t1 , t′1 Gret t2 , t′2 . (b) The in-out matrix element of qˆ(t1 )ˆ q (t2 ) is 0out | qˆ (t1 ) qˆ (t2 ) |0in 0out | 0in Z T Z T ` ´ ` ´ ` ´ ` ´ 1 iω(t2 −t1 ) e + dt′1 dt′2 J t′1 J t′2 GF t1 , t′1 GF t2 , t′2 . = 2ω 0 0 40 4 From harmonic oscillators to fields Summary: Collections of quantum oscillators.