       Re: Exact Solution to the double well potential problem in

• To: mathgroup at smc.vnet.net
• Subject: [mg132473] Re: Exact Solution to the double well potential problem in
• From: Narasimham <mathma18 at gmail.com>
• Date: Wed, 26 Mar 2014 03:22:49 -0400 (EDT)
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```> A little background. This is a Schrodinger equation
> problem to find the energy stages e in the double
> well where there is a step potential inside a
> infinite rectangular well. After solving the boundary
> conditions analytically, I tried to solve the
> following program in mathematica
>
> hbar := 1.05*10^-34;
> m := 0.096*9.11*10^-31;
> a := 5*10^-9;
> b := 0.5*10^-9;
> V := 0.3*1.6*10^-19;
> k = Sqrt[(2*m*e)/((hbar)^2)];
> l = Sqrt[(2*m*(V - e))/(hbar)^2];
> Block[{e}, e /. First@Solve[k*Cot[(-k)*a] -
> l*Tanh[(-l)*b/2] == 0, e]]
>
> As you can see since the values of the constants are
> very small, I am getting a lot of errors and I am not
> sure how to get the solution.
>
> Please let me know if there is a way to solve this
> equation using Mathematica. I have to submit this
> assignment in a couple of days, so I wold be really
> really grateful if you guys can show me the way to
> solve this.

Hint: Always deal with non-dimensional quantities. Next, multiply quantities by 10^10 to be comparable to unity.The relations

(k/l) = +/- Sqrt[e /(V -e)] = Tan[10 b k] Tanh [b l/2] may be useful.

Narasimham

```

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