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This is problem 2.51 from Griffiths Introduction to Quantum Mechanics, 2nd ed. p89.

The problem statement, all variables and given/known data

Consider the potential

[tex]V(x) = -\frac{\hbar^2 a^2}{m}sech^2(a x)[/tex]

where [itex]a[/itex] is a positive constant, and "sech" stands for the hyperbolic secant.

a) Graph this potential.

b) Check that this potential has the ground state

[tex]\psi_0(x) = A sech(a x)[/tex]

and find its energy. Normalize [itex]\psi_0[/itex], and sketch its graph.

c) Show that the function

[tex] \psi_k(x) = A\left(\frac{i k - a tanh(a x)}{i k + a}\right)e^{i k x} [/tex]

(where [itex] k = \sqrt{2 m E}/\hbar [/itex] as usual) solves the Schr\ddot{o}dinger equation for any (positive) energy [itex] E [/itex]. Since [itex] tanh z \rightarrow -1 [/itex] as [itex] z \rightarrow -\infty[/itex],

[tex] \psi_k(x) \approx A e^{i k x} [/tex], for large negative x.

This represents, then, a wave coming in from the left with no accompanying reflected wave (i.e., no term exp(-ikx)). What is the asymptotic form of [itex]\psi_k(x)[/itex] at large positive [itex]x[/itex]? What are R and T, for this potential? Comment: This is a famous example of a reflectionless potential - every incident particle, regardless of its energy, passes right through.

The attempt at a solution

Part a is easy, just draw the graph, intersecting the y axis at -\frac{\hbar^2 a^2}{m}.

Part b is a little more difficult. I can show that it is a solution, but I'm not sure how to guarantee that it's the ground state. With that potential, I get that

[tex] \hat H\psi_0 = -\frac{\hbar^2 a^2}{2 m} \psi_0[/tex]

If this result was the same as the minimum potential, I could say for certain that it's the ground state.

And for the normalization constant I get

[tex] A = \sqrt{\frac{a}{2}} [/tex]

Part c I don't have much clue for. I tried just putting the hamiltonian into mathematica with that wave function, to see what it gives me, but it didn't give me anything that I can see looks like the RHS of the equation, any suggestions on how to go about part c would be appreciated.

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# Homework Help: Griffiths Quantum 2.51

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