Chapter 14, Problem 86E

(a):

Interpretation Introduction

To determine:

Explanation:

Given The wave function for the electron in one dimensional system is,

$\Psi \left(x\right)=\sin x$

The probability density curve for the function ${\Psi }^2\left(x\right)={\sin }^{2}x$ contains all the positive values of the given function over the whole range. Therefore, the probability density curve for the given function is,

Figure 1

(b)

Explanation:

Given The wave function for the electron in one dimensional system is,

$\Psi \left(x\right)=\sin x$

The probability of finding electron for the given function is maximum on the values of $x$ where the probability density curve has the maximum value. For the given function the value of $\sin x$ is maximum at the values $x=\frac{\pi }{2}$ and $x=\frac{3\pi }{2}$ . Therefore, the probability density curve for the given function has a peak at these values of $x$ where probability of finding an electron is maximum.

(c)

Explanation: The wave function for the electron in one dimensional system is,

$\Psi \left(x\right)=\sin x$

The probability of finding electron for the given function is minimum on the values of $x$ where the probability density curve has the minimum value. For the given function the value of $\sin x$ is zero at the value of $x=\pi$ . Therefore, the probability density curve for the given function has a node at this value of $x$ where probability of finding electron is nil.

Conclusion:

(a) The probability density curve for the given function is as follows:

(b) The values of $x$ is maximum at $x=\frac{\pi }{2}$ and $x=\frac{3\pi }{2}$ .
(c) The probability of finding an electron at $x=\pi$ is zero and this point is called node.