Chapter 6: Solutions of the Schrödinger Equation for Unbound

(Free) Statesq We will continue to study solutions of the time-independent Schrödinger Equation, but for cases where the particles are free, not bound

q This has the following implications:- the objects are not described by standing waves- energies (states, levels) are not quantized

Þ a continuum of levels

)()()()(2 2

22

xExxUdxxd

myyy

=+-!

We will consider four different cases

E

E

1

2

3

4

III

Classically, (2) and (3) are the same – from the particle picture, the object just passes over the “wall”

(1) and (4) are also the same, classically – the object, viewed as a particle, just “bounces” off of the wall and is never in regions II and III

However, consider an EM wave at a surface (or interface of different materials), some of the wave is transmitted into the surface and some is reflected

The same effect occurs for a matter wave- the particle is not “split”, there is just a probability

of it being transmitted or reflected

For the free particle, we found previously that the solution to the time-dependent Schrödinger equation is the complex exponential

this represents a plane-wave moving in the positive x-direction (right-moving)

tiikxAetx w-+=Y ),(

Wave function when E<U0 for a potential step (Case 1)

Reflection and Transmission Probabilities for a Potential Step

R

TCase 1

Case 2

E/U0

Same as Fig. 6.4

Wave function when E<U0 for a potential barrier (Case 4)

Reflection and Transmission Probabilities for a Potential Barrier

E/U0

T

R

Case 4

Case 3

3.3220 =LmU

!

Same as Fig. 6.8

Reflection and Transmission Probabilities for a Potential Barrier

E/U0

6.6220 =LmU

!R

T

Case 3Case 4

Wave packet propagation for a potential step with E>U0

Wave packet propagation for a potential barrier with E<<U0

Wave packet propagation for a potential barrier with E<U0

Applications of Tunneling: Fusion and Alpha Decay (6.3 and 11.6)

Tunneling is extremely important in nuclear physics (and atomic and molecular)Fusion: a fusion

reaction occurs when two nuclei tunnel through a barrier caused by their mutual electrostatic repulsion and approach each other close enough to fuse.

The Coulomb force

acts to repel them, but once they are able to tunnel to within about 10-15 m (the edge of the nucleus), the strong nuclear force binds themThe potential energy for the Strong Nuclear force, whose form is unknown, can be approximated by the finite potential wellThe process occurs in the sun, for example

Classically, the proton and deuteron cannot overcome, or penetrate, the Coulomb barrierTreated as waves, they tunnel through the barrier and fuse to make helium-3

221

0

))((41

reZeZFC pe

=

g+®+ HeHH 32

21

11 XAZ

The main process studied by the Fusion Energy program is the D-T reaction

Alpha Decay: The reverse process in which an unstable heavy nucleus emits an alpha particle

for example

MeV18 n HeHH 10

42

31

21 ++®+

He)(42He42

42 +® -

- YX AZ

AZ

HeThU 42

23490

23892 +®

Exponential decay curve

Scanning Tunneling Microscope

The Quantum Stadium

Probability density of electrons of iron atoms on a copper surface (measured with a STM)

Quantum Dot

Wave packet propagation on two potential barriers with E<U0