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Free Particle

Free Particle.  (x) = A cos(kx) or  (x) = A sin(kx)  (x)= A e ikx = A cos(kx) + i A sin(kx)  (x)= B e -ikx = B cos(kx) - i B sin(kx). Travelling wave to left. Travelling wave to right. Free Particle.  (x)= A e ikx +B e -ikx is a solution A and B are constants

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Free Particle

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  1. Free Particle • (x) = A cos(kx) or (x) = A sin(kx) • (x)= A eikx = A cos(kx) + i A sin(kx) • (x)= B e-ikx = B cos(kx) - i B sin(kx)

  2. Travelling wave to left Travelling wave to right Free Particle • (x)= A eikx +B e-ikx is a solution • A and B are constants • hence (x,t)= (x)e-it • = A ei(kx- t) +B e-i(kx+ t)

  3. Free Particle • (x,t)= A ei(kx- t) is matter wave travelling to the right(along the positive x-axis) •  *(x,t)= A* e-i(kx- t) • | (x,t)|2 = (x,t)  *(x,t)= AA* =|A|2 • intensity of wave is constant! • Probability is the same everywhere • a free particle is equally likely to be found anywhere

  4. Free Particle • P(x,t)= |(x,t)|2 is probability of finding a particle at position x at time t • total probability of finding it somewhere is • consider a classical point particle moving back and forth with constant speed between two walls located at x=0 and x=8cm • particle spends same amount of time everywhere • P(x)=P0 if 0< x < 8 cm • P(x)=0 if x< 0 or x> 8cm

  5. Free Particle • Since • hence P0 = (1/8) cm-1 ===> probability/unit length is 1/8 • probability of finding particle in length dx is (1/8)dx • probability of finding it at x=2cm is zero! (dx=0) • Probability of finding it in some range 1.9 to 2.1 is (1/8)x = (1/8)(2.1-1.9)= .025

  6. Free Particle • Probability of finding it between x=0 and x=8cm is (1/8)(8-0) = 1 • intensity of wave is constant! • Probability is the same everywhere • a free particle is equally likely to be found anywhere • free particle has definite energy E=(1/2)mv2 and momentum p=mv but uncertain position

  7. Barrier Tunneling U0 • consider a barrier E < U0 R+T=1

  8. U0 Schrodinger Solution • Consider the three regions : left of barrier, right of barrier and in the barrier • left: • right: • inside:

  9. Barrier Tunneling • Solution inside barrier has form • since • P(x) is smaller as U0 increases Tunneling

  10. Tunneling • Transmission coefficient T ~ e-2kL • k={82m(U0-E)/h2}1/2 Note: E < U0 • if T=.02 then for every 1000 electrons hitting the barrier, about 20 will tunnel • extremely sensitive to L and k • width and height of barrier

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