Continuous Orthonormality For a discrete basis: (using Kronecker Delta) ⟨x∣x′⟩≡⟨xk∣xk′⟩\braket{x|x'}\equiv \braket{x_k|x_{k'}}⟨x∣x′⟩≡⟨xk∣xk′⟩ =⟨k∣k′⟩Δx=δkk′1Δx={0ifx≠x′k≠k′1/Δxifx=x′k=k=\frac{\braket{k|k'}}{\Delta x}=\delta_{kk'}\frac{1}{\Delta x}=\begin{cases}0& if\quad x\neq x'\quad k\neq k'\\ 1/\Delta x\quad& if \quad x=x'\quad k=k\end{cases}=Δx⟨k∣k′⟩=δkk′Δx1={01/Δxifx=x′k=k′ifx=x′k=k For a continuous basis: (using Dirac Delta) ∫⟨x∣x′⟩dx≡∑xk=−LL⟨x∣xk′⟩Δx\int\braket{x|x'}dx\equiv \sum_{x_k=-L}^L \braket{x|x_{k'}}\Delta x∫⟨x∣x′⟩dx≡xk=−L∑L⟨x∣xk′⟩Δx =∑k=−NN⟨k∣k′⟩=δ(k−k′)=1=\sum_{k=-N}^N\braket{k|k'}=\delta(k-k')=1=k=−N∑N⟨k∣k′⟩=δ(k−k′)=1