Question

Problem 4 Suppose we know that a particle of mass is stuck on the x-axis, confined to the region -1<x< 1. Its wavefunction is given -x) -1 << < 1 < -1 or 2 > 1 where A is a real, as-yet-undetermined constant. We'll assume that all numbers are in Sl units, without actually writing the units down. a) Draw a set of graph axes below like the one below and draw a sketch of this wavefunction on the axes. Include appropriate labels on the vertical axis (-axis). b) What is 4* (2), the complex conjugate of (x)? (Hint: does the wavefunction include any imaginary numbers?] According to the second postulate, the product (2)2 = * (2)/(x) is equal to P(x), the probability density of the wavefunction. This function gives us the likelihood of the particle being at any position. c) Write down the probability density function P(x) for the given wavefunction. d) Draw a set of graph axes below like the one below and sketch the function P(x). Include appropriate labels on the vertical axis (P-axis). P=yv The second postulate tells us that the integral ſ P(x)dx gives the probability of finding the particle in some range of positions. Since the particle must be somewhere, we must have P(x2)d:c = 1. This is the normalization condition. e) Apply the normalization condition to the function P(x) that you wrote down in part cand determine the value of the constant A. This is called the normalization constant, and the wavefunction with this value of A is said to be normalized.

Answer 1

so I have solved it in parts one by one please have a look at it. The wave function does not have any imaginary part.

X e xcid,x>1) a) anasangramenet -2 -1 (no imaginary bart] W*(x) = 4(x) = A (1-x²) c) P(x) = 44+ - A² (1-x²)? -2 E s PCx) dx - A SC 4" 1 x + + - 231 - AP 2+2 ) - ! A² ( 30 + 12 - 40)=1 A² x 2 = 1 15

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