7.1 Wave Functions

If $a = 3 + 4 i$, what is the product $a^{\star} a$?

Suppose that a particle with energy E is moving along the x-axis and is confined in the region between 0 and L. One possible wave function is

Determine the normalization constant.

For the particle in the above example, find the probability of locating it between positions 0 and L/4

What is the physical unit of a wave function, $\Psi \left(x , t\right) ?$ What is the physical unit of the square of this wave function?

What kind of physical quantity does a wave function of an electron represent?

What is the meaning of the expression “expectation value?” Explain.

Compute $\left|\Psi \left(x , t\right)\right|^{\textrm{ } 2}$ for the function $\Psi \left(x , t\right) = \psi \left(x\right) \text{sin} \omega \textrm{ } \textrm{ } t$, where $\omega$ is a real constant.

Which one of the following functions, and why, qualifies to be a wave function of a particle that can move along the entire real axis? (a) $\psi \left(x\right) = A e^{− x^{\textrm{ } 2}}$;
(b) $\psi \left(x\right) = A e^{− x}$; (c) $\psi \left(x\right) = A \text{tan} x$;
(d) $\psi \left(x\right) = A \left(\right. \text{sin} x \left.\right) / x$; (e) $\psi \left(x\right) = A e^{− \left|x\right|}$.

A wave function of a particle with mass m is given by

$\psi \left(x\right) = \left\{\right. \begin{aligned}A \text{cos} \alpha x , & - \frac{\pi}{2 \alpha} \leq x \leq + \frac{\pi}{2 \alpha} , \\ 0 , & \text{otherwise},\end{aligned}$

where $\alpha = 1.00 \times 10^{10} / \text{m}$. (a) Find the normalization constant. (b) Find the probability that the particle can be found on the interval $0 \leq x \leq 0.5 \times 10^{−10} \text{m}$. (c) Find the particle’s average position. (d) Find its average momentum. (e) Find its average kinetic energy.