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The normalized wavefunction $\psi (x)=\sqrt{3}x/L^{3/2}$ on $(0,L)$ has sine coefficients $c_n=(-1{)}^{n+1}\sqrt{6}/(n\pi )$. Its ground-state probability is $6/{\pi }^2$, while its position probability in $(0,L/3)$ is $1/27$. The constant values of $E_n|c_n{|}^2$ make its energy expectation divergent. It is a valid Hilbert space vector but fails the finite-energy Dirichlet boundary-trace condition.

Apply the position version of the Born rule to the normalized wavefunction:The result is independent of $L$ because the state and the measurement interval scale together. Most position probability lies towards the right wall, since the probability density grows as $x^2$.