- #1
bowlbase
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Homework Statement
Wave function of an electron:
##\frac{1}{N}∫_a^b[e^{ik_0x}(1+\frac{1}{2}e^{i\frac{0.1}{2}x}+\frac{1}{2}e^{-i\frac{0.1}{2}x})]dx##
The integrand becomes zero both to the left and right of x = 0 . Let a be the first time it hits zero to the left and b the first time it hits zero to the right. What value of N is required to normalize this wave-function?
What is the probability that the electron is between (−1, 1) ?
What is the expected value of the electron location?
What value of α minimizes the variance of the electron?
Homework Equations
The Attempt at a Solution
I'm pretty lost on this question. I have that the zero points as +-2(π)/0.1. I'm not sure how to get N from that.
From what I've read online (I have no notes describing this from class) I should take the integral for the squared function with the limits -1 to 1. I don't even know where to start on this integral. The first exponential resembles one I've seen (it was given) before that was δ(k)2π when solved with limits of +-infinity. From what I understand it should look something like:
##\frac{1}{N}∫_a^b[e^{ik_0x}(1+\frac{1}{2}e^{i\frac{0.1}{2}x}+\frac{1}{2}e^{-i\frac{0.1}{2}x})]^2dx=1##
Surely there must be a way to simplify this right?
For the last problem I was told I have to multiply the inside of the integral with x so that it looks like
##\frac{1}{N}∫_a^b[e^{ik_0x}(1+\frac{1}{2}e^{i\frac{0.1}{2}x}+\frac{1}{2}e^{-i\frac{0.1}{2}x})]^2x dx##
But I don't know what the limits should be, still -1,1?
It's the first homework of the class and I'm immediately overwhelmed and confused. Thanks for any help!