# 7.6 The quantum tunneling of particles through potential barriers  (Page 10/22)

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## Conceptual questions

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?

doubling the barrier width

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?

No, the restoring force on the particle at the walls of an infinite square well is infinity.

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

## Problems

A complex function of the form, $A{e}^{i\varphi }$ , satisfies Schrӧdinger’s time-independent equation. The operators for kinetic and total energy are linear, so any linear combination of such wave functions is also a valid solution to Schrӧdinger’s equation. Therefore, we conclude that [link] satisfies [link] , and [link] satisfies [link] .

A 6.0-eV electron impacts on a barrier with height 11.0 eV. Find the probability of the electron to tunnel through the barrier if the barrier width is (a) 0.80 nm and (b) 0.40 nm.

A 5.0-eV electron impacts on a barrier of with 0.60 nm. Find the probability of the electron to tunnel through the barrier if the barrier height is (a) 7.0 eV; (b) 9.0 eV; and (c) 13.0 eV.

a. 4.21%; b. 0.84%; c. 0.06%

A 12.0-eV electron encounters a barrier of height 15.0 eV. If the probability of the electron tunneling through the barrier is 2.5 %, find its width.

A quantum particle with initial kinetic energy 32.0 eV encounters a square barrier with height 41.0 eV and width 0.25 nm. Find probability that the particle tunnels through this barrier if the particle is (a) an electron and, (b) a proton.

a. 0.13%; b. close to 0%

A simple model of a radioactive nuclear decay assumes that $\text{α}$ -particles are trapped inside a well of nuclear potential that walls are the barriers of a finite width 2.0 fm and height 30.0 MeV. Find the tunneling probability across the potential barrier of the wall for $\text{α}$ -particles having kinetic energy (a) 29.0 MeV and (b) 20.0 MeV. The mass of the $\text{α}$ -particle is $m=6.64\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-27}\phantom{\rule{0.2em}{0ex}}\text{kg}$ .

A muon, a quantum particle with a mass approximately 200 times that of an electron, is incident on a potential barrier of height 10.0 eV. The kinetic energy of the impacting muon is 5.5 eV and only about 0.10% of the squared amplitude of its incoming wave function filters through the barrier. What is the barrier’s width?

0.38 nm

A grain of sand with mass 1.0 mg and kinetic energy 1.0 J is incident on a potential energy barrier with height 1.000001 J and width 2500 nm. How many grains of sand have to fall on this barrier before, on the average, one passes through?

Show that if the uncertainty in the position of a particle is on the order of its de Broglie’s wavelength, then the uncertainty in its momentum is on the order of the value of its momentum.

proof

fundamental note of a vibrating string
what are matter waves? Give some examples
according to de Broglie any matter particles by attaining the higher velocity as compared to light'ill show the wave nature and equation of wave will applicable on it but in practical life people see it is impossible however it is practicaly true and possible while looking at the earth matter at far
Manikant
Mathematical expression of principle of relativity
given that the velocity v of wave depends on the tension f in the spring, it's length 'I' and it's mass 'm'. derive using dimension the equation of the wave
What is the importance of de-broglie's wavelength?
he related wave to matter
Zahid
at subatomic level wave and matter are associated. this refering to mass energy equivalence
Zahid
it is key of quantum
Manikant
how those weight effect a stable motion at equilibrium
how do I differentiate this equation- A sinwt with respect to t
just use the chain rule : let u =wt , the dy/dt = dy/du × du/dt : wA × cos(wt)
Jerry
I see my message got garbled , anyway use the chain rule with u= wt , etc...
Jerry
de broglie wave equation
vy beautiful equation
chandrasekhar
what is electro statics
when you consider systems consisting of fixed charges
Sherly
Diagram of the derive rotational analog equation of v= u+at
what is carat
a unit of weight for precious stones and pearls, now equivalent to 200 milligrams.
LoNE
a science that deals with the composition, structure, and properties of substances and with the transformations that they undergo.
LoNE
what is chemistry
what chemistry ?
Abakar
where are the mcq
ok
Giorgi
acids and bases
Navya
How does unpolarized light have electric vector randomly oriented in all directions.
unpolarized light refers to a wave collection which has an equal distribution of electric field orientations for all directions
pro
In a grating, the angle of diffraction for second order maximum is 30°.When light of wavelength 5*10^-10cm is used. Calculate the number of lines per cm of the grating.
OK I can solve that for you using Bragg's equation 2dsin0over lander
ossy