1. 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)

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

3. Free Particle  (x,t)= A e i(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. 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 Free Particle 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)=P 0 if 0< x < 8 cm P(x)=0 if x 8cm

5. Free Particle Since hence P 0 = (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)mv 2 and momentum p=mv but uncertain position

7. R+T=1 Barrier Tunneling consider a barrier E < U 0 U0U0

8. Schrodinger Solution Consider the three regions : left of barrier, right of barrier and in the barrier left: right: inside: U0U0

9. Barrier Tunneling Solution inside barrier has form since P(x) is smaller as U 0 increases Tunneling

10. Transmission coefficient T ~ e -2kL k={8  2 m(U 0 -E)/h 2 } 1/2 Note: E < U 0 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