Physics Friday 8

Finite Potential Barrier in an Infinite Potential Well (Part Two):

Part one is here
First, let us look at limiting behavior for certain conditions.

I.

Here, we have for even solutions: ;
and for odd solutions
These are combined with and to give an equation to solve numerically for energy.

First, we examine the case :
Then
For even wavefunctions, our equation thus approaches


, n a non-negative integer.


For odd wavefunctions, our equation becomes


, n a positive integer.

These can be combined:
, n a positive integer
Note that these are, as expected, the energy levels of the infinite potential well of width b. The energy levels of our well with barrier should approach these levels when .

Now, let us examine
In this case, we see our even and odd equations become and , respectively. These are the same as the equations: the energy levels approach those of the well without the barrier, again as expected.

Lastly, we examine these levels for :
The even equation becomes , and the odd equation becomes . These combine to give:
, n a positive integer


Which are the levels for an infinite potential well with potential in the well.

I.

Here we have the equations setting energy levels as for even wavefunctions, and for odd wavefunctions, along with and

First, let us examine the case :
For even solutions, we have

As , , and the above becomes

For odd solutions, we have

Which, as becomes

So both odd and even wavefunctions approach the same energy levels. These levels are:
, n a positive integer


These are the energy levels of an infinite potential well of width . Notice that as , our potential approaches a pair of infinite wells each of width , thus explaining our result. Wavefunctions of the same energy in the two wells can be combined as , corresponding to the even case, or as , corresponding to the odd case.

Now, let us examine
Our even and odd equations become and
These give rise to energies , n a positive integer, just as expected.

Lastly, we examine :
The odd and even equations become and . Neither of these have solutions for , as one should expect.

Numerical solutions:
First, let us set a scale. We shall measure our length in units of b (i.e b=1, 0<a<1). Next, we shall measure energy in terms of . Now, I will display tables of some numerical solutions for different values of the parameters a and

First, for comparison, the infinite potential well energies, which, in these energy units, will be for a well of width w. Thus:

Infinite potential well of width 1
E Parity of wavefunction
9.86960 Even
39.47842 Odd
88.82644 Even
157.91367 Odd
246.74011 Even
355.30576 Odd
483.61062 Even
631.65468 Odd
799.43796 Even
986.96044 Odd

Now we examine the energy levels for our problem:

a=0.5, V0=1
E Parity of wavefunction
10.68654 Even
39.97683 Odd
89.22095 Even
158.41486 Odd
247.30401 Even
355.80558 Odd
484.06572 Even
632.15498 Odd
799.97340 Even
987.46038 Odd

a=0.5, V0=40
E Parity of wavefunction
39.82546 Even
56.99077 Odd
106.23989 Even
179.76397 Odd
269.54433 Even
375.03024 Odd
502.04121 Even
652.12727 Odd
820.95373 Even
1006.85994 Odd

a=0.5, V0=500
E Parity of wavefunction
112.19568 Even
112.20190 Odd
419.12541 Even
421.02768 Odd
546.61094 Even
652.85406 Odd
785.40333 Even
918.25807 Odd
1058.35349 Even
1238.39109 Odd

a=0.5, V0=10000
E Parity of wavefunction
145.972564854340 Even
145.972564854340 Odd
583.551714726197 Even
583.551714726197 Odd
1311.652495230419 Even
1311.652495230419 Odd
2328.196839461453 Even
2328.196839461453 Odd
3629.527092598046 Even
3629.527092598046 Odd

(This is to as many digits as my math software could compute)

a=0.1, V0=1
E Parity of wavefunction
10.06727 Even
39.48486 Odd
89.01244 Even
157.93797 Odd
246.90395 Even
355.35527 Odd
483.74754 Even
631.73126 Odd
799.54897 Even
987.06042 Odd

a=0.1, V0=40
E Parity of wavefunction
16.76173 Even
39.72807 Odd
96.44796 Even
158.85835 Odd
253.56658 Even
357.24162 Odd
489.30627 Even
634.66903 Odd
804.02214 Even
990.92340 Odd

a=0.1, V0=500
E Parity of wavefunction
38.48330 Even
41.74965 Odd
152.03468 Even
166.74022 Odd
334.72735 Even
374.18652 Odd
578.42506 Even
662.76113 Odd
881.99049 Even
1030.62483 Odd

a=0.1, V0=10000
E Parity of wavefunction
46.64098 Even
46.64135 Odd
186.54474 Even
186.54634 Odd
419.65258 Even
419.65659 Odd
745.86240 Even
745.87068 Odd
1165.02147 Even
1165.03722 Odd

a=0.9, V0=1
E Parity of wavefunction
10.86797 Even
40.46196 Odd
89.81227 Even
158.88934 Odd
247.70375 Even
356.25618 Odd
484.54737 Even
632.57804 Odd
800.34886 Even
987.86042 Odd

a=0.9, V0=40
E Parity of wavefunction
49.80178 Even
79.21141 Odd
128.24138 Even
196.91140 Odd
285.24676 Even
393.27641 Odd
521.03079 Even
668.53963 Odd
835.82998 Even
1022.92458 Odd

a=0.9, V0=500
E Parity of wavefunction
508.42437 Even
533.95610 Odd
577.26145 Even
639.17071 Odd
720.45888 Even
821.73374 Odd
943.41752 Even
1085.77402 Odd
1248.94611 Even
1432.98811 Odd

a=0.9, V0=10000
E Parity of wavefunction
2695.144547 Even
2695.144547 Odd
9628.692822 Even
9628.692864 Odd
10014.201215 Even
10056.338162 Odd
10125.312825 Even
10219.944832 Odd
10339.293701 Even
10482.711672 Odd

You can observe in these some of the limiting cases noted earlier.

Lastly, we indicate some of the values of for which there is an state for different values of a:

a=0.1
V0 Parity of wavefunction
12.18469 Even
39.72642 Odd
109.66227 Even
161.41698 Odd
304.61742 Even
370.23786 Odd
597.05014 Even
670.60369 Odd
986.96044 Even
1065.21546 Odd

a=0.5
V0 Parity of wavefunction
39.47842 Even
65.85373 Odd
355.30576 Even
386.22947 Odd
986.96044 Even
1018.54570 Odd
1934.44246 Even
1966.22659 Odd
3197.75183 Even
3229.62013 Odd

a=0.9
V0 Parity of wavefunction
986.96044 Even
1073.87966 Odd
8882.64396 Even
8971.29645 Odd
24674.01100 Even
24762.81422 Odd
48361.06157 Even
48449.90667 Odd
79943.79565 Even
80032.65803 Odd

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