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Quantum Mechanics: Tunneling in Finite Potential Wells

Explore wave functions, particle probability, and tunneling effects in finite potential wells through Schrödinger equation analysis. Discover unique properties of wave functions and the implications of energy conservation in quantum physics.

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Quantum Mechanics: Tunneling in Finite Potential Wells

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  1. Quantum Mechanics:Tunneling Physics 123 Lecture XVI

  2. Wave Function • (Wave function Y of matter wave)2dV=probability to find particle in volume dV . In 1-dimentional case probability P to find particle between x1 and x2 is • Unitarity condition (probability to find particle somewhere is one): • Schrödinger equation predicts wave function for a system • System is defined by potential energy, boundary conditions Lecture XVI

  3. Properties of Wave function • Wave function respects the symmetry of the system • For example if the system is symmetric around zero • x-x • then wave function is either symmetric or antisymmetric around zero: Lecture XVI

  4. Count knots 0 knots in the box symmetric 1 knot in the box antisymmetric 2 knots in the box Symmetric N-th state: (N-1) knots in the box N-odd – symmetric N-even - antisymmetric Lecture XVI

  5. Particle in a finite potential well U0 I II III x 0 L • Particle mass m in a finite potential well: • U(x)=0, if 0<x<L, • U(x)=U0, if x<0-or-x>L • Boundary conditions: Lecture XVI

  6. Particle in a finite potential well U0 I II III x 0 L • Inside the box (region II) • Possible solutions: sin(kx) and cos(kx) Lecture XVI

  7. Particle in a finite potential well U0 I II III x 0 L • Outside the box (regions I and III) • Possible solutions: exp(Gx) and exp(-Gx) Lecture XVI

  8. Wave functions 2 knots in the box Symmetric 1 knot in the box antisymmetric 0 knots in the box symmetric Lecture XVI

  9. Probability to find particle at x • Particle can be found outside the box!!! • E=U0+KE • KE must be positive • KE=E-U0, but U0>E • Energy not conserved?! • Fine print: Heisenberg uncertainty principle • Time spent outside the box is less than h/2p divided by energy misbalance, then energy non-conservation is “virtual”=undetectable Lecture XVI

  10. Probability to find particle at x Consider electron mc2=0.5 MeV with U0=2eV, E=1eV How much time does it spend outside the box? Characteristic depth of penetration x0: exp(-Gx)=exp(-x/x0) Lecture XVI

  11. You can go through the wall!!! • It’s called tunneling effect • Probability of tunneling P=|y|2=exp(-2GL), L-width of the barrier • Transmission coefficient T~P=exp(-2GL) Lecture XVI

  12. Problem 39-34 • A 1.0 mA current of 1.0 MeV protons strike 2.0 MeV high barrier of 2.0x10-13m thick. Estimate the current beyond the barrier. p Lecture XVI

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