Consider the normalized wave function ? =square root of (15/16)(1-x^2) This wave

Question

Consider the normalized wave function

?=square root of (15/16)(1-x^2)

This wave function is non-zero when -1 < X > 1. This means the wave function is zero at 1 and -1.

a) Show that this wave function find is normalized. Hint, integrate it between -1 and 1.

b) Using this wave function find the average value of X.

Consider the normalized wave function ?=square root of (15/16)(1-x^2) This wave function is non-zero when -1 1. This means the wave function is zero at 1 and -1. Show that this wave function find is normalized. Hint, integrate it between -1 and 1. Using this wave function find the average value of X.

Explanation / Answer

a) integral of psi^2 dx from -1 to 1 = 15/16 integral of (1-x^2)^2 dx from -1 to 1=15/16 integral of 1 - 2 x^2 + x^4 from -1 to 1

= 30/16*integral of 1 - 2 x^2 + x^4 from 0 to 1

= 30/16* (1 - 2/3 + 1/5)=1

b) average of x = integral ofx psi^2 dx from -1 to 1 = 15/16 integral of x (1-x^2)^2 dx from -1 to 1

integreal is odd

so average of x = 0

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