# 7.6 The quantum tunneling of particles through potential barriers  (Page 11/22)

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The mass of a $\rho$ -meson is measured to be $770\phantom{\rule{0.2em}{0ex}}\text{MeV}\text{/}{c}^{2}$ with an uncertainty of $100\phantom{\rule{0.2em}{0ex}}\text{MeV}\text{/}{c}^{2}$ . Estimate the lifetime of this meson.

A particle of mass m is confined to a box of width L . If the particle is in the first excited state, what are the probabilities of finding the particle in a region of width 0.020 L around the given point x : (a) $x=0.25L$ ; (b) $x=0.40L$ ; (c) $x=0.75L$ ; and (d) $x=0.90L$ .

a. 4.0 %; b. 1.4 %; c. 4.0%; d. 1.4%

A particle in a box [0; L ] is in the third excited state. What are its most probable positions?

A 0.20-kg billiard ball bounces back and forth without losing its energy between the cushions of a 1.5 m long table. (a) If the ball is in its ground state, how many years does it need to get from one cushion to the other? You may compare this time interval to the age of the universe. (b) How much energy is required to make the ball go from its ground state to its first excited state? Compare it with the kinetic energy of the ball moving at 2.0 m/s.

a. $t=m{L}^{2}\text{/}h=2.15\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{26}\text{years}$ ; b. ${E}_{1}=1.46\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-66}\text{J,}K=0.4\phantom{\rule{0.2em}{0ex}}\text{J}$

Find the expectation value of the position squared when the particle in the box is in its third excited state and the length of the box is L .

Consider an infinite square well with wall boundaries $x=0$ and $x=L.$ Show that the function $\psi \left(x\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}kx$ is the solution to the stationary Schrӧdinger equation for the particle in a box only if $k=\sqrt{2mE}\text{/}\hslash .$ Explain why this is an acceptable wave function only if k is an integer multiple of $\pi \text{/}L.$

proof

Consider an infinite square well with wall boundaries $x=0$ and $x=L.$ Explain why the function $\psi \left(x\right)=A\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}kx$ is not a solution to the stationary Schrӧdinger equation for the particle in a box.

Atoms in a crystal lattice vibrate in simple harmonic motion. Assuming a lattice atom has a mass of $9.4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-26}\phantom{\rule{0.2em}{0ex}}\text{kg}$ , what is the force constant of the lattice if a lattice atom makes a transition from the ground state to first excited state when it absorbs a $\text{525-}µ\text{m}$ photon?

1.2 N/m

A diatomic molecule behaves like a quantum harmonic oscillator with the force constant 12.0 N/m and mass $5.60\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-26}\phantom{\rule{0.2em}{0ex}}\text{kg}$ . (a) What is the wavelength of the emitted photon when the molecule makes the transition from the third excited state to the second excited state? (b) Find the ground state energy of vibrations for this diatomic molecule.

An electron with kinetic energy 2.0 MeV encounters a potential energy barrier of height 16.0 MeV and width 2.00 nm. What is the probability that the electron emerges on the other side of the barrier?

0

A beam of mono-energetic protons with energy 2.0 MeV falls on a potential energy barrier of height 20.0 MeV and of width 1.5 fm. What percentage of the beam is transmitted through the barrier?

## Challenge problems

An electron in a long, organic molecule used in a dye laser behaves approximately like a quantum particle in a box with width 4.18 nm. Find the emitted photon when the electron makes a transition from the first excited state to the ground state and from the second excited state to the first excited state.

$19.2\phantom{\rule{0.2em}{0ex}}µ\text{m;}\phantom{\rule{0.2em}{0ex}}11.5\phantom{\rule{0.2em}{0ex}}µ\text{m}$

In STM, an elevation of the tip above the surface being scanned can be determined with a great precision, because the tunneling-electron current between surface atoms and the atoms of the tip is extremely sensitive to the variation of the separation gap between them from point to point along the surface. Assuming that the tunneling-electron current is in direct proportion to the tunneling probability and that the tunneling probability is to a good approximation expressed by the exponential function ${e}^{-2\beta \phantom{\rule{0.2em}{0ex}}L}$ with $\beta =10.0\text{/nm}$ , determine the ratio of the tunneling current when the tip is 0.500 nm above the surface to the current when the tip is 0.515 nm above the surface.

If STM is to detect surface features with local heights of about 0.00200 nm, what percent change in tunneling-electron current must the STM electronics be able to detect? Assume that the tunneling-electron current has characteristics given in the preceding problem.

3.92%

Use Heisenberg’s uncertainty principle to estimate the ground state energy of a particle oscillating on an spring with angular frequency, $\omega =\sqrt{k\text{/}m}$ , where k is the spring constant and m is the mass.

Suppose an infinite square well extends from $\text{−}L\text{/}2$ to $+L\text{/}2$ . Solve the time-independent Schrӧdinger’s equation to find the allowed energies and stationary states of a particle with mass m that is confined to this well. Then show that these solutions can be obtained by making the coordinate transformation $x\text{′}=x-L\text{/}2$ for the solutions obtained for the well extending between 0 and L .

proof

A particle of mass m confined to a box of width L is in its first excited state ${\psi }_{2}\left(x\right)$ . (a) Find its average position (which is the expectation value of the position). (b) Where is the particle most likely to be found?

#### Questions & Answers

If you lie on a beach looking at the water with your head tipped slightly sideways, your polarized sunglasses do not work very well.Why not?
it has everything to do with the angle the UV sunlight strikes your sunglasses.
Jallal
this is known as optical physics. it describes how visible light, ultraviolet light and infrared light interact when they come into contact with physical matter. usually the photons or light upon interaction result in either reflection refraction diffraction or interference of the light.
Jallal
I hope I'm clear if I'm not please tell me to clarify further or rephrase
Jallal
what is bohrs model for hydrogen atom
hi
Tr
Hello
Youte
Hi
Nwangwu-ike
hi
Siddiquee
hi
Omar
helo
Mcjoi
what is the value of speed of light
1.79×10_¹⁹ km per hour
Swagatika
what r dwarf planet
what is energy
কাজের একক কী
Jasim
কাজের একক কী
Jasim
friction ka direction Kaise pata karte hai
friction is always in the opposite of the direction of moving object
Punia
A twin paradox in the special theory of relativity arises due to.....? a) asymmetric of time only b) symmetric of time only c) only time
b) symmetric of time only
Swagatika
fundamental note of a vibrating string
every matter made up of particles and particles are also subdivided which are themselves subdivided and so on ,and the basic and smallest smallest smallest division is energy which vibrates to become particles and thats why particles have wave nature
Alvin
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
a centeral part of theory of quantum mechanics example:just like a beam of light or a water wave
Swagatika
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