54.7386 115.299 523.1o2 1o54.54 1419.47

o.422o o.13o5 o.o635 o.o375 o.o131 o.oo81

o oo67

o.625o o.175o o.o822 o.o477

o.o162 o.o1oo o oo81

the one-particle Hamiltonian, Eq. (318), with the electron mass replaced by the total mass M = nem*e and the electron charge replaced by the total charge Q = nee. The relative Hamiltonian part, which involves only the relative coordinates and momenta, has a cylindrically symmetric potential and can be handled by the 1/N-expansion technique. When the confining potential is quadratic, FIR spectroscopy is insensitive to the interaction effects because of CM and relative motions. The radiation dipole operator Y1 i e^i = QR, being a pure CM variable, then it does not couple to Hr which contains all the electron-electron interactions. The dipole operator then induces transitions between the states of the CM but does not affect the states of the relative Hamiltonian. The eigenenergies for the CM Hamil-tonian, Eq. (320), do not change because wc in the energy expression remains the same; namely, QB/Mc = eB/m*ec. Consequently, the FIR absorption experiments see only the feature of the single electron energies. There are only two allowed dipole transitions (Am = ±1) and the FIR resonance occurs at frequencies with

Many different experiments on quantum dots have proved the validity of Kohn's theorem and that the observed resonance frequency of an electron system in a parabolic potential is independent of electron-electron interactions and thus the actual number of electrons in the well, as reported by Wixforth et al. in a very recent review article [188].

In conclusion we have obtained the energy spectra of two interacting electrons as a function of confinement energies and magnetic field strength. The method has shown good agreement with the numerical results of Merkt et al. [178], Taut [187], and Wagner et al. [185]. Our calculations have also shown the effect of the electron-electron interaction term on the ground state energy and its significance on the energy level crossings in states with different azimuthal quantum numbers. The shifted 1/N expansion method yields quick results without putting restrictions on the Hamiltonian of the system.

4.4.3. Calculation of Parameters y^ and y2

The explicit forms of the parameters y1 and y2 are given in the following. Here R*y and a*B are used as units of energy and length, respectively y1 = c1e2 + 3c2e4 - a>-1 [e^ + 6c1e1e3 + c4e:^J (333)

T7 = Ti - ä>-1[T2 + T3 + T4 + T5 + T6] Ti2 = ¿-2 [T8 + T9 + T10 + T11] T16 = ¿Ö-2 [Ti3 + Ti4 + T15]

T1 = c1 d2 + 3c2 d4 + c3d6 T2 = c1e^ + 12c2e2e4 T3 = 2e1d1 + 2c5e4 T4 = 6c1e1d3 + 30c2e1d5 T5 = + T6 = 10c6 e3d5

T8 = 4e2 e2 + 36c1e1e2 e3 T10 = 24ce2e4 + 8c7e1e3e4 T13 = 8ej e3 + 108c1e1e3 T15 = 30c9e3

T14 = 48c4e1e3

where c, d, and e are parameters given as c1 = 1 + 2nr c4 = 11 + 30nr + 30n2

c8 = 57 + 189nr + 225n2 + 150n3 c2 = 1 + 2nr + 2n2r c5 = 21 + 59nr + 51n2 + 34n3 c3 = 3 + 8nr + 6n2r + 4n3

(1 - a)(3 - a) 3(1 - a)(3 - a) 81 =--Ö- °2 =--Â-

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