91Ó°ÊÓ

Can we simultaneously measure position and energy of a quantum oscillator? Why? Why not?

When an electron and a proton of the same kinetic energy encounter a potential barrier of the same height and width, which one of them will tunnel through the barrier more easily? Why?

What decreases the tunneling probability most: doubling the barrier width or halving the kinetic energy of the incident particle?

Explain the difference between a box-potential and a potential of a quantum dot.

Can a quantum particle 'escape' from an infinite potential well like that in a box? Why? Why not?

A tunnel diode and a resonant-tunneling diode both utilize the same physics principle of quantum tunneling. In what important way are they different?

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

Given the complex-valued function \(f(x, y)=(x-i y) /(x+i y),\) 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) \(\psi(x)=A e^{-x^{2}}\) (b) \(\psi(x)=A e^{-x} ;\) (c) \(\psi(x)=A \tan x\) (d) \(\psi(x)=A(\sin x) / x ;\) (e) \(\psi(x)=A e^{-|x|}\)

A particle with mass \(m\) moving along the \(x\) -axis and its quantum state is represented by the following wave function: $$ \Psi(x, t)=\left\\{\begin{aligned} 0, & x<0 \\ A x e^{-\alpha x} e^{-i E t / \hbar}, & x \geq 0 \end{aligned}\right. $$ where \(\alpha=2.0 \times 10^{10} \mathrm{m}^{-1} .\) (a) Find the normalization constant. (b) Find the probability that the particle can be found on the interval \(0 \leq x \leq L\). (c) Find the expectation value of position. (d) Find the expectation value of kinetic energy.