What is the value of the normalization constant A if the wave function $\Psi$(x)= A(x-2)² for x between 0 and 1,and otherwise,$\Psi$(x)= 0,is to represent a particle in 1 dimension? A)sqrt(5/31) B)sqrt(5/33) C)sqrt(3/7) D)sqrt(1/3) E)None are correct.

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Quizplus · Accepted Answer

The wave function must be normalized, so that the integral of |$\Psi$(x)|^2 over all space is 1. In this case, we have:$$\int_{-\infty}^{\infty}|\Psi(x)|^2dx = \int_0^1 A^2(x-2)^4dx = A^2\int_0^1 (x^4 - 8x^3 + 24x^2 - 32x + 16)dx$$$$= A^2\left[\frac{x^5}{5} - 2x^4 + 8x^3 - 16x^2 + 16x\right]_0^1 = \frac{31A^2}{5}$$Setting this equal to 1, we get:$$A^2 = \frac{5}{31}$$$$A = \sqrt{\frac{5}{31}}$$

What Is the Value of the Normalization Constant a If Ψ\PsiΨ

What Is the Value of the Normalization Constant a If $\Psi$