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Problem sets and Midterm Solutions

Problem sets and Midterm Solutions - Chem 163A Solutions to...

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1 Chem. 163A Solutions to Midterm Fall 2006 e ax 2 0 dx = 1 2 π a xe ax 2 0 dx = 1 2 a x 2 e ax 2 0 dx = 1 4 a π a x 3 e ax 2 0 dx = 1 2 a 2 x n e ax 0 dx = 1 × 2 × 3... × n a n + 1 (sin 2 ax ) dx = x 2 sin2 ax 4 a + C (sin ax )(cos ax ) dx = sin 2 ax 2 a + C x (sin 2 ax ) dx = x 2 4 x sin2 ax 4 a cos2 ax 8 a 2 + C x (cos 2 ax ) dx = x 2 4 + x sin2 ax 4 a + cos2 ax 8 a 2 + C Physical Constants Name symbol value (SI units) Avogadro’s number N A 6.022 × 10 23 Speed of light in vacuum c 2.9979 × 10 8 m/s Plank constant h 6.626 × 10 -34 J s “hbar” h = h 2 π 1.5457 × 10 -34 J s electron charge e 1.6021 × 10 -19 C electron mass m e 9.1094 × 10 -31 kg Bohr Radius a 0 5.29177 × 10 -11 m Vaccum permittivity ε 0 8.85419 × 10 -12 C 2 /(J m)
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2 Problem 1 (25 points) An electron is restricted to be in the positive x direction 0 x ≤ ∞ under the influence of some potential energy function which is infinite at x = 0. i . Although all the functions given below are finite, single-valued, continues with continues 1 st derivatives, only one of them is an acceptable wave function for this system. ( N is the normalization constant) Which one? Explain! (1) ψ ( x ) = N cos x ; (2) ψ ( x ) = Nxe x 2 ; (3) ψ ( x ) = Ne x 2 . (5 points) Because of the infinite jump in potential energy at x = 0, ψ (0) must be zero and only function (2) satisfy this condition.
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