Finite square well

Question 1

Why do we get even and odd solutions here and not some other random shapes (symmetric and antisymettric channels)

Answer 1

V(x) is symmetric in our problem setup (Around x=0)

Question 2

Besides sym vs anti sym what other 2 classes of solutions is the FSW divided into and why

Answer 2

Bound and scattering solutions

defined by whether |E| is larger than Vo (can get out of well -> scattering state)
or when |E| is less than V0 (cannot get out of well adn thus bound state)

Question 3

What kinds of solutions do you get for sym and antysm channels and when do you see these

Answer 3

When you fix boundary and mathcing condiitons (and normalisability)
you see that sym channels have cos
anti sym channels have sin

Question 4

Which comes first even or odd solns to FSW

Answer 4

Even
corr to shape of tan(z) and that for even the narrowest shallow well there still excisits at least 1 bound state that is EVEN

odd will require some min well depth and width

Question 5

Where into the fomrulas are the well deoth and width baked in

Answer 5

Inot the l and k numebrs

The energy is also in there

Question 6

Where do new bound states form

Answer 6

top of the well (smallest E needed E~0 there) adn then as we add more energy we mve down the well

Question 7

Difference between energies for Bound and scattering states

Answer 7

Bound energies : negative, discrete

Scattering : positive, continuous

Question 8

eigstates for bound states

Answer 8

Bound: normalisable, barrier makes WF ->0 at infiinity, real WF not exp(ix) here, number of WF determined by well width and V0(depth)

Question 9

eigstates for scattering states

Answer 9

Scattering: plane waves, not normalisable, generally complex (left adn right mover with e^- left and e^+ the right mover)

Question 10

-Cot first intercept

Answer 10

ANTISYM channel
first intercept > 0 is at pi/2
-> there exists no antisym bound wf forsufff narrow shallow well

Question 11

tan first intercept

Answer 11

SYM channel
firts ontercept at 0
-> there always exsists at least one even bound state

Question 12

What happens outside for bound and scattering states

Answer 12

Bound -> decays exponentiall (not totally 0 liek classsical due to tunnelling)

Scattering -> Plane waves

Question 13

Scattering state boundar/matching condition

Answer 13

lhs have both incoming (right mover) and reflected (left mover ) at lhs boundary

RHS have only right mover (transmitted) plane wave,

Question 14

Tranmission and refl amplitude

Answer 14

Prob that particle will be reflected or trans derived from continuity relations

Question 15

Coeff of relf and trans add up to ? why?

Answer 15

1

since they are porobabilities of particle being relf or trans theymust add to 1

Question 16

bohrs correspondene principle here?

Answer 16

is we make the enegies very large
As E -. infty , |T| - > 1
As E->0 |T| -> 0 (not to do with high energy corresp tho….) and we get total reflection

Question 17

List some examples which only have bound state WF solns

Answer 17

ISW
HO

these WF are always normalisable

Question 18

Some examples which have both bound and scattering states

Answer 18

ISW
Step function
Delta function potential

Question 19

Some examples which only have scatttering satets

Answer 19

Free particle

these wf never normalisable, have ot make wavepackets