By Levi A.F.J.

ISBN-10: 0521860962

ISBN-13: 9780521860963

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C) The momentum probability density, ψ ( p x , t ) 2 (d) The average momentum, 〈 px ( t ) 〉 . (e) The current flux, J ( x, t ) . PROBLEM 2 (a) Show that the density of states for a free-particle of mass m in two-dimensions is m D2 ( E ) = ------------2 2πh (b) At low temperature, electrons in two electrodes occupy states up to the Fermi energy, EF. The two electrodes are connected by a two dimensional conductance region. Derive an expression for the conductance of electrons flowing between the two electrodes as a function of applied voltage V, assuming the transmission coefficient through the two-dimensional region is unity.

07 × m 0 , where m 0 is the bare electron mass, and the potential well is of width L = 10 n m , calculate the value of E 0 for which P 12 = 1 . Comment on your result. PROBLEM 3 An electron is initially in the ground state of a one-dimensional rectangular potential well for which V ( x ) = 0 in the range 0 < x < L and V ( x ) = ∞ elsewhere. The ground state energy is E1 and the first excited state energy is E 2 . At time t = 0 the sys2 –t ⁄ τ tem is subject to a perturbation V ( x , t ) = V0 x e .

3 . Compare with the result you obtain when t′ = 0 . PROBLEM 5 A hydrogen atom is in its ground state with electron wave function 1⁄2 –r ⁄ a 1 2 ⋅ e B ⋅ ------ φ 100 = ----------3⁄2 4 π aB In this expression aB is the Bohr radius and r is a radial coordinate. Use spherical coordinates to find the expectation value of position r and momentum p r for the electron in this state. You should use the fact that in radial coordinates the Her1∂ mitian momentum operator is pˆ r = – ih -- r . , xˆ † = xˆ ).

### Applied quantum mechanics: Solutions manual by Levi A.F.J.

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