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We will use the variational method to approximate the ground-state energy of a particle in a 1D box with the trial wavefunction: \[ \psi(x) = A x (L - x) \] where \( A \) is the normalization constant. --- ### Step 1: Normalize the Wavefunction The wavefunction must satisfy the normalization condition: \[ \int_0^L |\psi(x)|^2 dx = 1 \] Substituting \( \psi(x) \): \[ \int_0^L A^2 x^2 (L - x)^2 dx = 1 \] Expanding \( (L - x)^2 \): \[ (L - x)^2 __________ ____ ________ ______ ______ _____ _________ _____ _______ ___.
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