By R. Guardiola, J. Ros
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Extra resources for The Many-Body Problem Jastrow Correlations Versus Brueckner Theory
Here, the number of quasi-particles decreases with increasing energy ωk . (the operator a ˆ† does not carry any quantum number labels) of energy ω. e. within the new picture, a ˆ† is an operator that creates particles. The total energy of these states is given by ω × (occupation number). e. the Hamiltonian basically counts the number of particles. While, at ﬁrst sight, this may look unfamiliar, the new interpretation is internally consistent. e. it allows an interpretation of the HO states as a superposition of independent structureless entities.
Next quantize the theory by promoting ﬁelds to operators αk → α ˆ k and πk → π ˆk , and declaring [ˆ πk , α ˆ k ] = −iδkk . The quantum Hamiltonian operator, again of harmonic oscillator type, then reads ˆ =1 H 2 ˆk + ωk2 α ˆk α ˆk , π ˆk π k ωk2 where = λk . 4, we then deﬁne ladder ωk ωk operators ak ≡ αk + ωik π ˆk ), a†k ≡ αk − ωik π ˆk ), whereupon the Hamiltonian 2 (ˆ 2 (ˆ assumes the now familiar form 1 ˆ = H . 39) ωk a†k ak + 2 k For the speciﬁc problem of the ﬁrst excited mode in a waveguide of width Ly , ωk = 1/2 k 2 + (π/Ly )2 .
M. Lifshitz, Classical Mechanics (Pergamon, 1960). 6 Noether’s theorem 31 What happens when one advances from point to continuum mechanAmalie E. 27 Clearly, multidimensional confor her groundbreaking contritinuum theories leave more room for butions to abstract algebra and the emergence of complex symmetheoretical physics. Alive at a tries; even more so than in classical time when women were not supposed to attend college preparatory schools, she was often forbidden mechanics, we are in need of a tool from lecturing under her own name.