tunneling e.g consider an electron of energy 5.1 ev approaching an energy barrier of height 6.8 ev...
Post on 18-Dec-2015
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Tunneling• e.g consider an electron of energy 5.1 eV
approaching an energy barrier of height 6.8 eV and thickness L= 750 pm What is T?
0 L
• Classically => reflected
• quantum mechanics => non-zero probability of penetration
Solution• k={82m(U0-E)/h2}1/2
• ={8(3.14)2(9.11x10-31)(6.8eV-5.1eV)1.6x10-19/(6.63x10-34)2}1/2
• = 6.67x109 m-1
• 2kL = 2(6.67x109 m-1)(750x10-12 m) =10.
• T = e-2kL = e-10 = 45x10-6
• how would this change for a proton?
• m is larger by a factor of 2000 => k larger by ~45
• T ~ e-186
• what about a baseball? => zero!
Problem
• A 1500 kg car moving at 20 m/s approaches a hill 24 m high and 30 m long. What is
• the probability that the car will tunnel quantum mechanically through the hill?
30m
24m U0 = mgh
Solution
• k={82m(U0-E)/h2}1/2
• T~ e-2kL • k={8(3.14)2(1500)(1500x9.8x24 - .5x1500x(20)2/(6.63x10-34)2}1/2
• = 1.2x1038 m-1
• 2kL ~ 72x 1038
• T ~ e-2kL => 0
Tunneling
• a possible application is as a switch
• change U0 =>larger k => smaller transmission
• exponential dependence!
• STM scanning tunneling microscope
Electrons tunnel from tip to atoms
Adjust L to keep currentconstant
L
Gives image of electrondensity
Atoms of iron arranged on a copper surface Quantum Corral ~ 14 nm diameter
Notice ripples of charge density inside the box
Semiconductor acts as a potential well which can trap electronsLower insulating layer is thin enough to allow tunneling into the well
Quantum Dots - designer atoms
Atoms
• Confined matter waves are standing waves
• 1926 Quantum mechanics was developed
• explained the structure of atoms and molecules
• electrons, protons treated as matter waves
• produce standing wave patterns
Standing Waves
• y(x,t) = A sin(kx -t)
• travelling wave of any frequency is possible on an infinite string
• finite length = confined wave
• y(x,t)=0 at ends !
• only certain frequencies allowed
• standing wave patterns
Need to adjust f so that =2L
f=v/ = v/2L ‘fundamental mode’
v2= F/ => F = 4L2f2
=L for next stable pattern second harmonic
=2L/n in generaln is the number of loops
fn = n v/2L = n f1 ‘harmonics’
natural frequencies of the wire
Atoms• For standing waves on a string fn= n f1
• matter waves confined to atoms =>discrete energies
• E = hf
• atom has a set of natural frequencies
0
E1=hf1
E2=hf2
E3=hf3
Atoms• For standing waves y(x,t)=Asin(kx)cos(t)
• y(x,t)=Asin(2x/) cos(t)
• y(x=0,t)= 0 = y(x=L,t) => 2L/ = n =>=2L/n
• => only certain frequencies f=v/= nv/2L=nf1
• Matter waves: P(x,t) =0 outside box
• P(x,t)= |A`|2 sin2(kx) => need P(L,t)=0
• kL =(2/)L = n => = (2L)/n “standing waves”
• only certain => only certain p => discrete energies
Atoms• Standing matter waves: = (2L)/n
• p=h/ = nh/2L
• En=p2/2m = h2n2/8mL2 n=1,2,3,…
• only certain energies allowed!
Correspondence Principle
• As n => large, the particle is found with equal probability at all points inside
• large n => large k => large p => large E
• this is the classical result
• large n => classical limit
• small n => quantum limit