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An analytic expression for the acoustoelectric current (AC) <i>j</i><sup>ac</sup> induced by electron-external acoustic wave interactions and electron-internal acoustic wave (internal phonons) scattering in a quantum well (QW) is calculated by using the quantum kinetic equation for electrons. The physical problem is investigated in the region <i>ql</i> ≫ 1 (where q is the acoustic wave number and <i>l</i> is the electrons mean free path). The dependence of the AC <i>j</i><sup>ac</sup> on the external acoustic wave frequency ω<sub>q</sub>, the width of the QW <i>L</i> and the temperature <i>T</i> for a specific QW of AlGaAs/GaAs/AlGaAs is achieved by using a numerical method. The computational results show that the dependence of the AC <i>j</i><sup>ac</sup> on the temperature <i>T</i>, the external acoustic wave frequency ω<sub>q</sub>, the width of the QW <i>L</i> is non-monotonic and that the peaks can be attributed to transitions between mini-bands <i>n</i> → <i>n</i>0. The dependence of the AC <i>j</i><sup>ac</sup> on the temperature <i>T</i> and the Fermi energy ε<sub>F</sub> is obtained, and a maximum of the AC <i>j</i><sup>ac</sup> for ε<sub>F</sub> = 0.038 eV and ω<sub>q</sub> = 3 × 10<sup>11</sup> s<sup>−1</sup> seen at <i>T</i> = 50 K, which agrees with the experimental results for AlGaAs/GaAs/AlGaAs QWs. All these results are compared with those for normal bulk semiconductors and superlattices to show the differences. Finally, the quantum theory of the acoustoelectric effect in a quantum well is newly developed.