De Broglie and experiment have told us the wavelength of a particle with momentum p, ?=h/p but what if the particle is rolling up a hill: it will eventually turn around and roll down again. Suppose the hill has a very gentle slope, so that the

1

• De Broglie and experiment have told us the
  wavelength of a particle with momentum p, ?h/p
  but what if the particle is rolling up a hill
  it will eventually turn around and roll down
  again. Suppose the hill has a very gentle slope,
  so that the momentum is not changing much in a
  distance h/p, then the de Broglie formula must
  give the wavelength pretty accurately. As the
  particle rolls up the hill, p goes down and the
  wavelength goes up. We see that the wavelength
  depends on the local potential energy--- we are
  going to need a wave equation that contains the
  potential energy U(x)

2

• But let us put off worrying about that equation
  and see what we can do with our knowledge of the
  wavelength of a free particle-- one moving in a
  region where U(x) is constant.
• If the particle has to be held in the box, it has
  to have a rigid wall, so U(x) has to go to
  infinity outside, then we should keep it inside...

box
U(x)
L
0
x
3

• Inside the box, the particle is free, so the
  wavelength is h/p, ?(x) sin kx, but the
  probability has to be zero outside the box-a
  boundary condition!! Prob (x) ?(x)2 0 at
  edges of the box, and our practice with strings
  stands us in good stead here, and we see that sin
  k0 0 (no information) and at the other end sin
  kL 0 -gt kL n? -gt
• k n?/L or 2 ?/? n ?/L or ? 2L/n
• much like the string. But this now gives us the
  momentum!! From ? h/p, p (nh)/(2L)
• and there is a smallest momentum h/(2L) which we
  call the zero point energy Ep²/2m(n²h²)/(8mL²)

4

• There are lots of other problems we can do by the
  locally free approximation, such as the ball
  rolling up the hill we mentioned,
• We do have to face up to the problem of the wave
  equation for the general U(x) cant quite be
  deduced but Schrödinger didnt have much trouble
  guessing it,

5

• Try separation of space and time variables,
  because we want at first the stationary states
  which do not change in time

6

• Example of use of the wave equation tunneling
• the bump in the middle, of thickness a, we call a
  barrier. When 0ltxlta the energy E is less that
  U(x) U? and the time-independent Schrödinger
  equation reads

U(x)
U?
a
7

• Match the wavefunctions at the boundaries
• and after the amplitudes are calculated, there is
  found a transmission coefficient