Generic example of algebraic bosonisation

Abstract

Two identical non-interacting fermions in a three-dimensional harmonic oscillator well are bosonised exactly according to a recently developed general algebraic scheme. Rotational invariance is taken into account within the scheme for the first time. The example is generic for the excitation spectra of finite systems, in particular for the appearance of bands in spectra. A connection to the formalism of the fractional quantum Hall effect is pointed out.PACS. 71.10.-w Theories and models of many-electron systems -73.21.La Quantum dots -03.65.-w Quantum mechanics

Entropy of pure states: not all wave functions are born equal

“…One way to find it is to look for states of low energy which have a different configuration than states of even lower energy, so that both the thermodynamic forces and the Hamiltonian matrix elements along the deexcitation path are small. As already discussed elsewhere [10], the shape wave functions are band-heads in the spectra of finite systems. They fit both these properties, because band-heads are the lowest excited states which are of a different configuration than all lower states, which belong to other bands in the spectrum.…”

“…In order to construct a physically realizable state from them, one must consider rotational invariance next. Shapes either form a rotational multiplet by themselves, or else components of individual shapes are found embedded in rotational multiplets, which include bosonic excitations [10]. Combining the above considerations, a robust rotationally invariant state is expected to be a pure-shape multiplet.…”

“…For example, it must be checked what the angular momentum operators look like in Bargmann space. Luckily they look the same as in real space, so that solid harmonics in Bargmann space can be obtained from those in real space simply by replacing Cartesian coordinates (x, y, z) with (t, u, v) [10]. In particular, the triplet (t i , u i , v i ) transforms as a vector, which supports the above grouping of variables into points from a physical viewpoint, marking the first step in developing a geometric visualization of wave-function space.…”

“…The rotationally invariant formulation, including center-of-mass excitations, is described in Ref. [10]. Wait, one could ask, isn't tuv = tÁuÁv a syzygy?…”

“…In this way, a kinematic reason is found for the existence of bands in spectra of finite systems. The spectrum of two fermions has two bands because there are exactly two distinct rotationally invariant ways to comply with the Pauli principle [10].…”