PH 401

1
PH 401

• Dr. Cecilia Vogel
• Lecture 19

2
Review

• Finite square well
• tunneling

Outline

• Tunneling

3
Energy Level of Finite Box

• Solving
• This is a transcendental eqn for energy
• Let y kL/2, and
• Solving for y means solving for E

4
Energy Levels

• Energy levels found graphically
• Compare to infinite well
• finite well energies are lower
• lowest energy levels are similar
• higher energy levels differ more and more
• Finite number of bound energy levels
• Also compare wavefunctions qualitatively.

5

• y-values where the two plots coincide correspond
  to energy levels of a finite square well
• y-values of asymptotes correspond to energy
  levels of an infinite square well

6
What is EgtVo?

• All regions are CA
• oscillatory functions
• compare wavelength in box and outside

7
Apply Strategy to Finite Barrier

• Suppose particle is incident from left
• and EltVo
• Three regions
• 1st and 3rd are CA, center region is CF
• CA regions include xinfinity

8
CA Regions

• Left CA region
• r for reflection amplitude
• not normalizable, so call incident amplitude 1
  all will be compared to it
• Right CA region
• can only be moving right
• t for transmission amplitude

9
Finite Barrier Wavefunction

• CF region
• Pieced together

10
Continuity at 0

• Continuity of y and dy/dx at 0

11
Continuity at L

• Continuity of y and dy/dx at L

12
Continuity Eqns

• Solving

13
Probability

• Probabilities are amplitudes squared
• Probability of transmission
• aka tunneling
• T(E)t2
• Probability of reflection
• R(E)r2
• R(E)1-T(E)

14
Continuum Energy Levels

• This time we have 4 eqns and five unknowns
• normalization is no help
• There is one unknown we cant solve for
• that is the energy
• There are no restrictions on the energy for this
  unbound system
• energy levels are not quantized
• called continuum states

15
Classical Physics

• Classically
• If EltV
• that region is forbidden
• and the particle will bounce off the barrier
• 100
• Classical waves, however
• can tunnel
• water waves, light waves

16

• Classical waves, however
• can tunnel
• http//mkat.iwf.de/index.asp?SignaturC2014861

17
Non-tunneling Barrier

• This problem can be solved also for EgtVo
• Classically allowed everywhere.
• The wavefunction then is oscillatory everywhere.
• CcoskxDsinkx within barrier

18
Non-tunneling Barrier

• A classical particle would just glide on by the
  barrier,
• 100 transmitted
• But for a wave, there is a probability of
  reflection and of transmission
• at each surface
• e.g. light on glass
• Consider graph from text (or this one)
• transmission is much more likely if EgtVo
• but not usually 100

19
Thin Film Interference?

• Consider graph from text
• Very unlikely to be transmitted if EltVo
• For some values of EgtVo, the transmission is 100
  no reflection from barrier at all.
• the waves reflected from two edges of barrier
  destructively interfere
• the amplitude for reflection is zero
• For some values of EgtVo, theres a local max in
  reflection from barrier
• the waves reflected from two edges of barrier
  constructively interfere
• the amplitude for reflection is larger

20
Thin Film Interference?

• Consider graph from text
• Very unlikely to be transmitted if EltVo
• For some values of EgtVo, the transmission is 100
  no reflection from barrier at all.
• the waves reflected from two edges of barrier
  destructively interfere
• the amplitude for reflection is zero
• For some values of EgtVo, theres a local max in
  reflection from barrier
• the waves reflected from two edges of barrier
  constructively interfere
• the amplitude for reflection is larger