lect17.ppt_notesit is for sure man man man

lect17.ppt_notesit is for sure man man man

1. finite potential well
“I don't like it, and I'm sorry I ever had anything to do with it.”—Erwin
Schrödinger, on quantum mechanics.

2. 5.9 Finite Potential Well
Our new toy worked very well for the
particle-in-a-box.
Where do you find infinite square wells in
real life? Answer—nowhere that I know
of.*
Our next test of Schrödinger’s equation ought to be a step up
in realism. In real life, particles do exist inside of potentials,
only the walls are not infinitely high.
So our next problem will be a finite potential well.
*A black hole might come close, except its potential is not square.

3. E=U
x
E
x=0 x=L
L
Our finite potential well has
length L and walls of
height U.
There are three regions:
I, II, and III.
I II III
In regions I and III,
Schrödinger’s equation is2
22
2m
+ (E - U) = 0 ,
x



which we can re-write as2
2ψ
- ψ
2
a = 0 ,
x

 2m (U - E)
α = .
where
- +
On the next slide we will
assume E<U so that a will
be real.

4. E=U
x
E
x=0 x=L
L
I II III
- +
Consider a particle with
E < U. Solutions in region I
and III areax -ax ax -ax
I III
= Ce + De and = Fe + Ge .
Why call the coefficients C, D, F, G? You’ll find out later.2
2ψ
- ψ
2
a = 0 ,
x

 2m (U - E)
α = .

5. E=U
x
E
x=0 x=L
L
I II IIIψ
ax -ax
I
= Ce + De ψ
ax -ax
III
= Fe + Ge
These solutions are real, and
not complex exponentials, so
they are not “wiggly” waves.
C, D, F, and G are coefficients
to be determined using the
boundary conditions of the
problem.
Regions I and III extend to x = -and x = +.
Because the wavefunction must be finite everywhere, the
coefficients D and F must be 0, so that ψψ
ax -ax
I III
= Ce and = Ge .
Both solutions decrease exponentially as we move away from
the barrier walls.
- +

6. ψ
II
2mE 2mE
= A sin x + B cos x ,
   
   
   
    E=U
x
E
x=0 x=L
L
I II III
In region II, Schrödinger’s
equation is2
22
2m
+ (E - U) = 0 ,
x



This equation has the same
kind of solutions as we had
for the particle in the
(infinite) box:
except that now B 0 because has an amplitude at each
barrier.
Now you see why we called the region I and III coefficients C, D, F, G. A
and B were reserved for region II, and E is reserved for energy.
- +

7. ax -ax
I II III
2mE 2mE
= Ce = A sin x + B cos x = Ge
   
      
   
    There are 5 “pieces” of information we want: the coefficients
A, B, C, and G, and the energy E.
We have 5 conditions: continuous at x = 0 and x = L (2
boundary conditions), continuous at x = 0 and x = L (2
more boundary conditions), and normalization of . These 5
conditions give 5 equations.
Five equations, five unknowns, the rest is “just” mathematics.
Hereis a Mathcad document that explores the finite square
well.

8. It’s a bit difficult to see the extent of the well in the Mathcad plots, so I
have reproduced some of them here. (The well goes from -1 x 1.)
for n=1
for n=2
for n=3
Notice how the wave function
tails extend inside the barrier-
-there is a finite probability of
finding the particle there.
Know this for the test!
Longer wave function tails
mean longer wavelengths and
therefore lower momenta and
energies.
A particle inside a finite box
can have a lower energy than
a particle in an infinite box.

9. Here’s a comparison of
wave functions for infinite
and finite square wells,
n=1, 2, and 3.
Be able to tell me which wave
function goes with which well.
How can you tell?

10. Here’s a comparison of the
probability density
functions for infinite and
finite square wells, n=1, 2,
and 3. (X-axis scale is
changed from previous
slide.)
Which probability goes with
which well. Why?
Which plot corresponds to n=2?
How can you tell?
What is the meaning of the red
shaded areas?

11. Are you trying to tell me there is a
probability of finding this brick
stuck halfway through an
impenetrable wall?
Well…not exactly…
…but I am telling you there is a
probability of finding the brick
somewhere inside the impenetrable
wall! (You’re supposed to imagine the
brick inside the wall, please.)