{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

phys documents (dragged) 53

# phys documents (dragged) 53 - Chapter 10 Quantum physics 47...

This preview shows page 1. Sign up to view the full content.

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 defined 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 For a harmonic oscillator holds: U = 1 2 bx 2 and ω 2 0 = b/m . The Hamiltonian H is then given by: H = p 2 2 m + 1 2 m ω 2 x 2 = 1 2 ¯ h ω + ω A A with A = 1 2 m ω x + ip 2 m ω and A = 1 2 m ω x - ip
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Ask a homework question - tutors are online