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Two-mode variational bound for a linearly tilted square well

For the first two sine modes in a width-

a

infinite well with

V (x) = 9 ℏ^{2} x / (m a^{3})

, the Hamiltonian in units

ℏ^{2} / (m a^{2})

is

((π^{2} + 9) /2 - 16/ π^{2} - 16/ π^{2} (4 π^{2} + 9) /2) .

Its lower eigenvalue gives the upper bound

E_{0} \leq \frac{ℏ ^{2}}{m a ^{2}} [\frac{5 π ^{2} + 18}{4} - \frac{9 π ^{4}}{16} + \frac{256}{π ^{4}}] .

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