# 7.6 The quantum tunneling of particles through potential barriers  (Page 9/22)

 Page 9 / 22

## Summary

• A quantum particle that is incident on a potential barrier of a finite width and height may cross the barrier and appear on its other side. This phenomenon is called ‘quantum tunneling.’ It does not have a classical analog.
• To find the probability of quantum tunneling, we assume the energy of an incident particle and solve the stationary Schrӧdinger equation to find wave functions inside and outside the barrier. The tunneling probability is a ratio of squared amplitudes of the wave past the barrier to the incident wave.
• The tunneling probability depends on the energy of the incident particle relative to the height of the barrier and on the width of the barrier. It is strongly affected by the width of the barrier in a nonlinear, exponential way so that a small change in the barrier width causes a disproportionately large change in the transmission probability.
• Quantum-tunneling phenomena govern radioactive nuclear decays. They are utilized in many modern technologies such as STM and nano-electronics. STM allows us to see individual atoms on metal surfaces. Electron-tunneling devices have revolutionized electronics and allow us to build fast electronic devices of miniature sizes.

## Key equations

 Normalization condition in one dimension $P\left(x=\text{−}\infty ,\text{+}\infty \right)=\underset{\text{−}\infty }{\overset{\infty }{\int }}|\text{Ψ}\left(x,t\right){|}^{2}dx=1$ Probability of finding a particle in a narrow interval of position in one dimension $\left(x,x+dx\right)$ $P\left(x,x+dx\right)={\text{Ψ}}^{*}\left(x,t\right)\text{Ψ}\left(x,t\right)dx$ Expectation value of position in one dimension $⟨x⟩=\underset{\text{−}\infty }{\overset{\infty }{\int }}{\text{Ψ}}^{*}\left(x,t\right)x\text{Ψ}\left(x,t\right)dx$ Heisenberg’s position-momentum uncertainty principle $\text{Δ}x\text{Δ}p\ge \frac{\hslash }{2}$ Heisenberg’s energy-time uncertainty principle $\text{Δ}E\text{Δ}t\ge \frac{\hslash }{2}$ Schrӧdinger’s time-dependent equation $-\frac{{\hslash }^{2}}{2m}\phantom{\rule{0.2em}{0ex}}\frac{{\partial }^{2}\text{Ψ}\left(x,t\right)}{\partial {x}^{2}}+U\left(x,t\right)\text{Ψ}\left(x,t\right)=i\hslash \frac{{\partial }^{2}\text{Ψ}\left(x,t\right)}{\partial t}$ General form of the wave function for a time-independent potential in one dimension $\text{Ψ}\left(x,t\right)=\psi \left(x\right){e}^{\text{−}i\omega t}$ Schrӧdinger’s time-independent equation $-\frac{{\hslash }^{2}}{2m}\phantom{\rule{0.2em}{0ex}}\frac{{d}^{2}\psi \left(x\right)}{d{x}^{2}}+U\left(x\right)\psi \left(x\right)=E\psi \left(x\right)$ Schrӧdinger’s equation (free particle) $-\frac{{\hslash }^{2}}{2m}\phantom{\rule{0.2em}{0ex}}\frac{{\partial }^{2}\psi \left(x\right)}{\partial {x}^{2}}=E\psi \left(x\right)$ Allowed energies (particle in box of length L ) ${E}_{n}={n}^{2}\frac{{\pi }^{2}{\hslash }^{2}}{2m{L}^{2}},n=1,2,3,...$ Stationary states (particle in a box of length L ) ${\psi }_{n}\left(x\right)=\sqrt{\frac{2}{L}}\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\frac{n\pi x}{L},n=1,2,3,...$ Potential-energy function of a harmonic oscillator $U\left(x\right)=\frac{1}{2}m{\omega }^{2}{x}^{2}$ Stationary Schrӧdinger equation $-\frac{\hslash }{2m}\phantom{\rule{0.2em}{0ex}}\frac{{d}^{2}\psi \left(x\right)}{d{x}^{2}}+\frac{1}{2}m{\omega }^{2}{x}^{2}\psi \left(x\right)=E\psi \left(x\right)$ The energy spectrum ${E}_{n}=\left(n+\frac{1}{2}\right)\hslash \omega ,n=0,1,2,3,...$ The energy wave functions ${\psi }_{n}\left(x\right)={N}_{n}{e}^{\text{−}{\beta }^{2}{x}^{2}\text{/}2}{H}_{n}\left(\beta x\right),n=0,1,2,3,...$ Potential barrier $U\left(x\right)=\left\{\begin{array}{cc}\hfill 0,& \text{when}\phantom{\rule{0.2em}{0ex}}x<0\hfill \\ \hfill {U}_{0},& \text{when}\phantom{\rule{0.2em}{0ex}}0\le x\le L\hfill \\ \hfill 0,& \text{when}\phantom{\rule{0.2em}{0ex}}x>L\hfill \end{array}$ Definition of the transmission coefficient $T\left(L,E\right)=\frac{|{\psi }_{\text{tra}}\left(x\right){|}^{2}}{|{\psi }_{\text{in}}\left(x\right){|}^{2}}$ A parameter in the transmission coefficient ${\beta }^{2}=\frac{2m}{{\hslash }^{2}}\left({U}_{0}-E\right)$ Transmission coefficient, exact $T\left(L,E\right)=\frac{1}{{\text{cosh}}^{2}\beta L+{\left(\gamma \text{/}2\right)}^{2}{\text{sinh}}^{2}\beta L}$ Transmission coefficient, approximate $T\left(L,E\right)=16\frac{E}{{U}_{0}}\left(1-\frac{E}{{U}_{0}}\right){e}^{\text{−}2\beta \phantom{\rule{0.2em}{0ex}}L}$

#### Questions & Answers

A Pb wire wound in a tight solenoid of diameter of 4.0 mm is cooled to a temperature of 5.0 K. The wire is connected in series with a 50-Ωresistor and a variable source of emf. As the emf is increased, what value does it have when the superconductivity of the wire is destroyed?
how does colour appear in thin films
in the wave equation y=Asin(kx-wt+¢) what does k and w stand for.
derivation of lateral shieft
hi
Imran
total binding energy of ionic crystal at equilibrium is
How does, ray of light coming form focus, behaves in concave mirror after refraction?
Refraction does not occur in concave mirror. If refraction occurs then I don't know about this.
Sushant
What is motion
Anything which changes itself with respect to time or surrounding
Sushant
good
Chemist
and what's time? is time everywhere same
Chemist
No
Sushant
how can u say that
Chemist
do u know about black hole
Chemist
Not so more
Sushant
DHEERAJ
These substance create harmful radiation like alpha particle radiation, beta particle radiation, gamma particle radiation
Sushant
But ask anything changes itself with respect to time or surrounding A Not any harmful radiation
DHEERAJ
explain cavendish experiment to determine the value of gravitational concept.
Cavendish Experiment to Measure Gravitational Constant. ... This experiment used a torsion balance device to attract lead balls together, measuring the torque on a wire and equating it to the gravitational force between the balls. Then by a complex derivation, the value of G was determined.
Triio
For the question about the scuba instructor's head above the pool, how did you arrive at this answer? What is the process?
as a free falling object increases speed what is happening to the acceleration
of course g is constant
Alwielland
acceleration also inc
Usman
which paper will be subjective and which one objective
jay
normal distributiin of errors report
Dennis
normal distribution of errors
Dennis
acceleration also increases
Jay
there are two correct answers depending on whether air resistance is considered. none of those answers have acceleration increasing.
Michael
Acceleration is the change in velocity over time, hence it's the derivative of the velocity with respect to time. So this case would depend on the velocity. More specifically the change in velocity in the system.
Big
photo electrons doesn't emmit when electrons are free to move on surface of metal why?
What would be the minimum work function of a metal have to be for visible light(400-700)nm to ejected photoelectrons?
give any fix value to wave length
Rafi
40 cm into change mm
40cm=40.0×10^-2m =400.0×10^-3m =400mm. that cap(^) I have used above is to the power.
Prema
i.e. 10to the power -2 in the first line and 10 to the power -3 in the the second line.
Prema
there is mistake in my first msg correction is 40cm=40.0×10^-2m =400.0×10^-3m =400mm. sorry for the mistake friends.
Prema
40cm=40.0×10^-2m =400.0×10^-3m =400mm.
Prema
this msg is out of mistake. sorry friends​.
Prema
what is physics?
why we have physics
because is the study of mater and natural world
John
because physics is nature. it explains the laws of nature. some laws already discovered. some laws yet to be discovered.
Yoblaze
physics is the study of non living things if we added it with biology it becomes biophysics and bio is the study of living things tell me please what is this?
tahreem
physics is the study of matter,energy and their interactions
Buvanes
all living things are matter
Buvanes
why rolling friction is less than sliding friction
tahreem
thanks buvanas
tahreem