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These are the wave function solutions ( eigenfunctions ) for the finite square well. PowerPoint PPT Presentation


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These are the wave function solutions ( eigenfunctions ) for the finite square well. These are the 4 equations from applying the continuity of psi and its first derivative at the well boundaries. Now add (1) and (3). S ubtract (3) from (1). Now add (2) and (4). S ubtract (4) from (2).

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These are the wave function solutions ( eigenfunctions ) for the finite square well.

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These are the wave function solutions eigenfunctions for the finite square well

These are the wave function solutions (eigenfunctions) for the finite square well.


These are the wave function solutions eigenfunctions for the finite square well

These are the 4 equations from applying the continuity of psi and its first derivative at the well boundaries.


These are the wave function solutions eigenfunctions for the finite square well

Now add (1) and (3)

Subtract (3) from (1)

Now add (2) and (4)

Subtract (4) from (2)


These are the wave function solutions eigenfunctions for the finite square well

As long as and divide (8) by (5)

As long as and divide (6) by (7)


These are the wave function solutions eigenfunctions for the finite square well

Both (9) and (10) cannot both be true at the same time.

Proof: add the two equations

Which cannot be since both k and a are real.


These are the wave function solutions eigenfunctions for the finite square well

Solutions of the second kind: odd solutions

Solutions of the first kind: Even solutions

From the even solutions, we have that:

So it must be the case for the even solutions that:


These are the wave function solutions eigenfunctions for the finite square well

Using these relations below we can write the wave functions for the square well – even solutions.


These are the wave function solutions eigenfunctions for the finite square well

Using one of the boundary conditions we can solve for B

And the wave functions (Eigenfunctions) in each region are (where B is determined by normalization in each region so to match the solutions at the boundaries.)


These are the wave function solutions eigenfunctions for the finite square well

Let’s look at the even solutions and determine the allowed energies


These are the wave function solutions eigenfunctions for the finite square well

The tangent term as an a/2 factor. Let’s put that in everything.

Define alpha and r as:


These are the wave function solutions eigenfunctions for the finite square well

A special case: the infinite square well:

Which are half of the allowed energy states of the infinite square well (n odd).


These are the wave function solutions eigenfunctions for the finite square well

Let’s go back and find the solutions for the allowed energies by graphing

P(a)

Q(a)

This cannot be solved analytically. So, where the two functions these will be related to the allowed energies for the case of r = 2 as an example.


These are the wave function solutions eigenfunctions for the finite square well

From Mathematica we can find the places where these two functions cross. They are given by x below which is what we’ve called alpha:

FindRoot[p[x] == q[x], {x, 1.2}]

{x -> 1.25235}

FindRoot[p[x] == q[x], {x, 3.6}]

{x -> 3.5953}

Now to calculate the values of the allowed energies.


These are the wave function solutions eigenfunctions for the finite square well

continuum

V0

E3=0.808V0

E3=0.383V0

E3=0.098V0


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