p40_021 - 21. (a) and (b) Schrdingers equation for the...

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21. (a) and (b) Schr¨ odinger’s equation for the region x>L is d 2 ψ dx 2 + 8 π 2 m h 2 [ E U 0 ] ψ =0 , where E U 0 < 0. If ψ 2 ( x )= Ce 2 kx ,then ψ ( x )= C 0 e kx ,where C 0 is another constant satisfying C 0 2 = C .Thu s d 2 ψ/dx 2 =4 k 2 C 0 e kx =4 k 2 ψ and d 2 ψ dx 2 + 8 π 2 m h 2 [ E U 0 ] ψ = k 2 ψ + 8 π 2 m h 2 [ E U 0 ] ψ. This is zero provided that k 2 = 8 π 2 m h 2 [ U 0 E ] . The quantity on the right-hand side is positive, so k is real and the proposed function satisFes Schr¨ odinger’s equation. If k
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