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Chapter 6: Problem 8

Consider a fictitious one-dimensional system with one electron. The wave function for the electron, drawn below, is \(\psi(x)=\sin x\) from \(x=0\) to $x=2 \pi .\( (a) Sketch the probability density, \)\psi^{2}(x),\( from \)x=0\( to \)x=2 \pi .(\mathbf{b})\( At what value or values of \)x$ will there be the greatest probability of finding the electron? (c) What is the probability that the electron will be found at \(x=\pi ?\) What is such a point in a wave function called? [Section 6.5\(]\)

Short Answer

Expert verified
(a) The probability density function is \(\psi^2(x) = \sin^2 x\). (b) The greatest probability of finding the electron is at \(x=\frac{\pi}{2}\) and \(x=\frac{3\pi}{2}\). (c) The probability of finding the electron at \(x=\pi\) is 0, and such a point is called a "node."

Step by step solution

01

(a) Sketching the Probability Density Function

To sketch the probability density function \(\psi^2(x)\), simply square the given wave function \(\psi(x)\) and plot it on the range from \(0\) to \(2\pi\). Given \(\psi(x)=\sin x\), the probability density function is: \[\psi^2(x)=(\sin x)^2=\sin^2 x\] Now, plot \(\sin^2 x\) on the range \([0, 2\pi]\).
02

(b) Finding Maximum Probability

To find the values of \(x\) with the greatest probability of finding the electron, we must look for the maximum points in the probability density function \(\psi^2(x)=\sin^2 x\). To find maxima, take the derivative of the function and set it to 0: \[\frac{d(\sin^2 x)}{dx}=0\] Using the chain rule, we get: \[\frac{d(\sin^2 x)}{dx}=2\sin x\cdot \cos x=0\] Solve for \(x\) and make sure to verify that they are maxima: \[2\sin x\cdot \cos x=0 \implies x=\{0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\}\] After checking for maxima, the greatest probability of finding the electron is at \(x=\frac{\pi}{2}\) and \(x=\frac{3\pi}{2}\).
03

(c) Probability at \(x=\pi\) and Terminology

To find the probability of finding the electron at \(x=\pi\), we need to evaluate the probability density function at this point: \[\psi^2(\pi)=\sin^2 \pi =0\] A point in a wave function where the probability of finding the electron is 0 is called a "node." Thus, at \(x=\pi\), the wave function \(\psi(x)=\sin x\) has a node.

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