Section 5: Solutions of Schrodinger Equations in 1D - Scattering States

Question 1

scattering states - E

Answer 1

> U

Question 2

scattering off infinite steps is rather boring

Answer 2

the incoming wavepacket just bounces back — and just like with bound states, this means the solutions are plane waves or decaying exponentials through each distinct U(x)=const region

Question 3

Unlike the bound states, though, the boundary conditions no longer force the solutions to be

Answer 3

real-valued

meaning probability flux will be non-zero

Question 4

Scattering states are not localised in space, so the wavefunctions are not

Answer 4

normalisable

Question 5

the probability fluxes must satisfy

Answer 5

the flux conservation
JI=JR+JT

Question 6

the flux conservation can be normalised to JI to give the condition

Answer 6

R+T=1

Question 7

scattering from forbidden potential step (E<V)

Answer 7

The transmitted component is
0 in the sense that there is not a probability flux in region 2, but there is still a probability density in region II due to tunneling.

Question 8

steps for potential step calculations

Answer 8

1. write the wavefunctions for the two regions
2. impose continuity conditions and its derivative
3. rearrange

Question 9

scattering from potential well (double step)

Answer 9

similar but an algebraic pain as now 5 wave components

Question 10

scattering from potential well (double step) - interesting takeaway is

Answer 10

the interference of incoming and reflected waves, with the same wavenumber, gives rise to interference patterns in the form of standing waves on the “incident” side of the step. This interference is then seen again within the region between two steps.

Question 11

Ramsauer effect - limit E»|V|

Answer 11

the finite well is barely visible

the transmission is perfect

Question 12

Ramsauer-Townsend effect

Answer 12

does not have a classical explanation

if the incident wavelength is a multiple of the width of the well, then there is perfect transmission,

Question 13

quantum tunnelling

Answer 13

a finite, albeit exponentially suppressed, wavefunction manages to “leak through” the classically forbidden region and resume its oscillatory behaviour on reaching the classically allowed region on the other side.

Question 14

tunneling - the procedure to find the ampltidues

Answer 14

still the same

impose continuity conditions for wavefunction and its derivative

Question 15

How can we have flux on the left- and right-hand sides of the barrier, but no flux within?

Answer 15

we can’t

C has a different overall complex phase than D

the phase-interaction of the two decaying waves within the barrier that creates a non-zero flux within the barrier

Question 16

The tunnelling phenomenon is of phenomenal importance, both in human affairs via quantum tunnelling technologies such as

Answer 16

scanning tunnelling microscopes

Question 17

tunneling deeply important in natural processes such as

Answer 17

alpha decay
alpha capture in nuclear fusion