**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:

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:

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 |

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 |

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 |

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)

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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:

V_{0} |
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 |

V_{0} |
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 |

V_{0} |
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 |

Tags: Friday Physics, Science

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