Some more waveforms

1

• Some more waveforms

2
The Time-dependent Schrödinger equation
Total energy Kinetic energy Potential
energy
The kinetic energy operator is
The potential energy operator is V
3
Note that the left side is a function of time and
the right side is a function of position
time space
Therefore, I will assume any solution is a
product of a time-dependent portion and a space
dependent protion
4
If I put this separated form into the TDSE
5
And the separate them
The entire right side has no time-dependent
terms. So if I am solving for
6
The entire right side has no time-dependent
terms. So if I am solving for
Where is just a constant
7
I will rewrite this to get
I know that solutions to this are of the form
8
It turns out that is related to the
energy, so I might as well write
Now I can write
9
And when I put this back in the Time-dependent
Schrödinger equation
I get
This is the Time-independent Schrödinger equation
10
Sometime it is more appropriate to write it as
Notice that now the energy looks more like an
independent variable that is part of the problem.
This is the case for the infinite well.
11
If I were to rewrite this
with
It starts to look more like the type of wave
equations Im used to dealing with. If Egt V, I
will have positive values of k.
12
Assuming one dimension
Solutions are of the form
or
i.e, propagating solutions
13
(No Transcript)
14
(No Transcript)
15
(No Transcript)
16
(No Transcript)
17
(No Transcript)
18
(No Transcript)
19
(No Transcript)
20
(No Transcript)
21
One the other hand, if EltV
then
22
Which has solutions are of the form
i.e, decaying solutions
23
(No Transcript)
24
(No Transcript)
25
(No Transcript)
26
(No Transcript)
27
(No Transcript)
28
(No Transcript)
29
(No Transcript)
30
(No Transcript)
31
(No Transcript)
32
Note that the total energy remains the same!
33
The following simulation is a pulse passing
through a slab of material. Notice that the
waveform changes wavelength as is goes through,
but then returns to its original form.
34
(No Transcript)
35
(No Transcript)
36
(No Transcript)
37
(No Transcript)
38
(No Transcript)
39
(No Transcript)
40
(No Transcript)
41
(No Transcript)
42
(No Transcript)
43
(No Transcript)
44
The following is similar, except EltV
45
(No Transcript)
46
(No Transcript)
47
(No Transcript)
48
(No Transcript)
49
(No Transcript)
50
(No Transcript)
51
What do you think will happen with the following
simulation?
52
(No Transcript)
53
(No Transcript)
54
(No Transcript)
55
(No Transcript)
56
(No Transcript)
57
(No Transcript)
58
(No Transcript)
59
(No Transcript)
60
(No Transcript)
61
(No Transcript)