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Chapter 10: Quantum physics 47 10.7 The tunnel effect The wavefunction of a particle in an high potential step from x =0 to x = a is given by ψ ( x )= a - 1 / 2 sin( kx ) . The energylevels are given by E n = n 2 h 2 / 8 a 2 m . If the wavefunction with energy W meets a potential well of W 0 >W the wavefunction will, unlike the classical case, be non-zero within the potential well. If 1, 2 and 3 are the areas in front, within and behind the potential well, holds: ψ 1 = A e ikx + B e - ikx , ψ 2 = C e ik ± x + D e - ik ± x , ψ 3 = A ± e ikx with k ± 2 =2 m ( W - W 0 ) / ¯ h 2 and k 2 =2 mW . Using the boundary conditions requiring continuity: ψ = continuous and ∂ψ / x = continuous at x =0 and x = a gives B , C and D and A ± expressed in A . The amplitude T of the transmitted wave is deFned by T = | A ± | 2 / | A | 2 . If W>W 0 and 2 a = n λ ± =2 π n/k ± holds: T =1 . 10.8 The harmonic oscillator ±or a harmonic oscillator holds: U = 1 2 bx 2 and ω 2 0 = b/m . The Hamiltonian H is then given by:
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This note was uploaded on 01/30/2011 for the course PHYSICS 208 taught by Professor Ye during the Spring '10 term at Blinn College.

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