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Lecture 10 - Finite Square Well.pdf
Lecture_10_-_Finite_Square_Well.pdf
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Lecture_10_-_Finite_Square_Well.pdf-Today (1) Review of time dependent
Page 6
6
Nanotechnology: how small does a wire have to be
before movement of electrons starts to depend on size
and shape due to quantum effects?
How to start?
Need to look at
a. size of wire compared to size of atom
b. size of wire compared to size of electron wave function
c. Energy level spacing compared to thermal energy, kT.
d. Energy level spacing compared to some other quantity
(what?)
e. something else (what?)
Page 7
7
Nanotechnology: how small does a wire have to be
before movement of electrons starts to depend on size
and shape due to quantum effects?
How to start?
Need to look at
c. Energy level spacing compared to thermal energy, kT.
to move much have higher energy/momentum
Almost always focus on energies in QM.
Electrons, atoms, etc. hopping around with random energy kT.
Larger than spacing, spacing irrelevant. Smaller, spacing big deal
.
spacing of levels in
pit compared to kT?
Main parameter in
behavior of solids
Page 8
8
“quantumticity” parameter,
ratio of spacing between energy levels of the electron
and thermal energy kT
E/kT
-- small, no quantum effects-- spacing of levels
does not matter.
Big-- quantum important.
E=.06 eV/(L/4nm)
2
,
kT = 0.025 eV (T/300 kelvin)
but I just made up
E’s for this problem.
Now we need to calculate what they really are.
Page 9
9
Solving the Schrodinger equation for electron wave in 1 D
1. Figure out what V(x) is, for situation given.
2. Guess or look up functional form of solution.
3. Plug in to check if
’s, and all x’s drop out, leaving
equation involving only bunch of constants; showing
that trial solution is correct functional form.
4. Figure out what boundary conditions must be to make
sense physically.
5. Figure out values of constants to meet boundary conditions
and normalization
6. Multiply by time dependence
(t) =exp(-iEt/h)
to have full solution if needed.
STILL TIME DEPENDENCE!
|
(x)|
2
dx =1
-
∞
∞
time independent
eq.
)
(
)
(
)
(
)
(
2
2
2
2
x
E
x
x
V
x
x
m
Page 10
10
+
PE
+
+
+
+
+
+
+
+
1 atom
many atoms
but lot of e’s
move around
to lowest PE
repel other electrons = potential energy near that spot higher.
as more electrons fill in, potential energy for later ones gets
flatter and flatter.
For top ones, is VERY flat.
+
Page 11
11
PE for electrons with most PE. “On top”
as more electrons fill in, potential energy for later ones gets
flatter and flatter.
For top ones, is VERY flat.
+
+
+
+
+
+
+
+ +
+
+
+
+
+
+ +
How could you find out how deep the pit is for the top
electrons in copper wire?
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