Calculate thickness of barrier for tunneling effect

In summary, the tunnel effect predicts that electrons can "tunnel" through an energy barrier, and the probability of this happening is proportional to ψ^2. Using the normalized eigenfunction equation, we can determine the constant C and solve for the thickness of the barrier x, which is proportional to the inverse of the probability ψ^2. When plugging in the given values, we get a thickness of 5*10^8 meters for a probability of 1 electron in 10^8 to penetrate the barrier.
  • #1
daskywalker
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Homework Statement



2. The tunnel effect predicts that electrons can "tunnel" through an energy barrier that classical mechanics predicts to be impenetrable. In a suitably fabricated “quantum dot” device, a thin energy barrier may be penetrated in this way. If the barrier is 0.2eV in height (E - V = -0.2 eV), what thickness barrier will give a probability that 1 electron in 108 will penetrate the barrier? (Hint: remember that probability is proportional to ψ^2).

Homework Equations



for tunneling effect: ψ=A*exp(ikx) or A*exp(αx) where α=ik
normalized eigenfunction: ∫ψ^2dV=1

The Attempt at a Solution



I know that ψ^2 is the probability (1*10^-8) and I found the value for k (and thus α) now I am having trouble finding the constant A so that I can solve for the thickness of the barrier x
 
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  • #2
. I also know that the probability is proportional to ψ^2, so I can set up the equation:

ψ^2 = A^2*exp(2αx)

Since we know that the probability is proportional to ψ^2, we can set up the equation:

ψ^2 = C*exp(2αx)

Where C is a constant. We can then use the normalized eigenfunction equation to solve for C:

∫ψ^2dV = ∫C*exp(2αx)dV = C*∫exp(2αx)dV = C*V*exp(2αx) = 1

Solving for C, we get:

C = 1/(V*exp(2αx))

Plugging this back into our original equation, we get:

ψ^2 = (1/(V*exp(2αx)))*exp(2αx)

ψ^2 = 1/V

Therefore, the thickness of the barrier x is proportional to the inverse of the probability ψ^2. We can then solve for x:

x = 1/(ψ^2*V)

Plugging in the given values, we get:

x = 1/(1*10^-8*0.2) = 5*10^8 meters

Therefore, the thickness of the barrier that will give a probability of 1 electron in 10^8 to penetrate is 5*10^8 meters.
 

Related to Calculate thickness of barrier for tunneling effect

1. What is the "tunneling effect"?

The "tunneling effect" is a quantum phenomenon in which a particle can pass through a potential barrier even if it does not have enough energy to surmount it. This is possible due to the wave-like nature of particles, which allows them to exist in a state of superposition and have a finite probability of being found on the other side of the barrier.

2. How is the thickness of the barrier calculated for the tunneling effect?

The thickness of the barrier is calculated using the Schrödinger equation, which is a mathematical equation that describes the behavior of quantum particles. This equation takes into account the energy of the particle, the shape of the barrier, and the potential energy of the barrier. The result of the equation gives the probability of the particle tunneling through the barrier.

3. What factors affect the thickness of the barrier for the tunneling effect?

The thickness of the barrier for the tunneling effect is affected by several factors such as the energy of the particle, the potential energy of the barrier, the shape and height of the barrier, and the mass of the particle. A higher energy particle or a lower and wider barrier will have a higher probability of tunneling through.

4. Can the thickness of the barrier be changed to increase the probability of tunneling?

Yes, the thickness of the barrier can be changed to increase the probability of tunneling. This can be done by adjusting the potential energy of the barrier or by changing the shape and height of the barrier. However, altering the barrier will also affect the overall behavior of the particle and may have other consequences.

5. How is the concept of the tunneling effect applied in real-life situations?

The tunneling effect has many practical applications, such as in scanning tunneling microscopy, where it is used to create images of surfaces at the atomic level. It is also used in quantum computing, where particles tunnel through potential barriers to perform calculations. Additionally, the tunneling effect is utilized in certain types of electronic devices, such as tunnel diodes and flash memory, to control the flow of electrons.

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