Discussion
The condition for two distinct quantum mechanical states, represented by wave functions \(\psi_{1}(x)\) and \(\psi_{2}(x)\) respectively, to be orthogonal is :
Answer: B
The states are orthogonal when \(\int_{-\infty}^{\infty}\psi_{1}^{*}(x)\psi_{2}(x)\,dx = 0\).
Orthogonality means the inner product of the two wave functions vanishes. For functions of x the inner product is the overlap integral of the complex conjugate of one with the other, so the condition is that this integral is zero. Eigenfunctions of a Hermitian operator belonging to different eigenvalues always satisfy it.
An integral equal to 1 is the normalization condition, which applies when the two functions are the same normalized state. An infinite overlap signals non-normalizable functions, and the symmetric sum set equal to 1 is not a condition of orthogonality at all.
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