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The proposed potentials give $\nabla \times A=(0,0,B)$ and $-\nabla \varphi =(0,E,0)$. Substituting $\psi =e^{ikx}{e}^{-i\mathcal{E}t/\hslash }\phi (y)$ givesThis is a harmonic oscillator with cyclotron frequency and centrefor $eB>0$ as written. Completing the square yieldsThus the Landau levels in crossed electric and magnetic fields retain their oscillator spacing but acquire a linear dependence on $k$, lifting their guiding-centre degeneracy.