7.1 Wave functions  (Page 7/22)

What is the physical unit of a wave function, $\text{Ψ}(x,t)?$ What is the physical unit of the square of this wave function?

Can the magnitude of a wave function $({\text{Ψ}}^{*}\left(x,t\right)\text{Ψ}(x,t))$ be a negative number? Explain.

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

What is the physical meaning of a wave function of a particle?

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

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

Given the complex-valued function $f(x,y)=(x-iy)\text{/}(x+iy)$ , calculate $|f(x,y){|}^2$ .

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) $\psi (x)=A{e}^{\text{−}{x}^2}$ ;
(b) $\psi (x)=A{e}^{\text{−}x}$ ; (c) $\psi (x)=A\text{tan}x$ ;
(d) $\psi (x)=A(\text{sin}x)\text{/}x$ ; (e) $\psi (x)=A{e}^{\text{−}\left|x\right|}$ .

A particle with mass m moving along the x -axis and its quantum state is represented by the following wave function:

$\text{Ψ}(x,t)=\{\begin{array}{cc}\hfill 0,& x<0,\hfill \\ \hfill Ax{e}^{\text{−}\alpha x}{e}^{\text{−}iEt\text{/}\hslash },& x\ge 0\text{,}\hfill \end{array}$

where $\alpha =2.0\times {10}^{10}{\text{m}}^{\mathrm{-1}}$ . (a) Find the normalization constant. (b) Find the probability that the particle can be found on the interval $0\le x\le L$ . (c) Find the expectation value of position. (d) Find the expectation value of kinetic energy.

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

$\psi (x)=\{\begin{array}{cc}\hfill A\text{cos}\alpha x,& -\frac{\pi }{2\alpha }\le x\le +\frac{\pi }{2\alpha },\hfill \\ \hfill 0,\hfill & \text{otherwise,}\hfill \end{array}$

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