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

Graphite surface

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!

P(x,t)= (2/L) sin2(nx/L) = |n(x)|2

En=n2h2/8mL2

“Standing matter waves” As n, P(x)=1/L

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

Normalization

P(x,t)=|n(x)|2 = |A`|2 sin2(kx) • Choose area under each curve to be unity

• total probability should be unity

n

L

x dx

AL

2

0

1

2

( )

z n

n

xL

n x

L

Eh

mLn

( ) sin( )

FHG

IKJ

2

8

2

22

Note: n = 1,2,3,... 0(x)=0 for all x!