# Quantum transmission resonances

• b2386
In summary, the conversation is discussing how to show that B=0 when a beam of particles is incident at the first transmission resonance in an E>U_0 potential barrier. The solution involves using continuity conditions and simplifying equations using the known values for k and k'. An easier method is suggested which involves plugging in the value for k' at the transmission resonance and using algebra to show B=0.
b2386
Hi all,

Here is my question:

In the $$E>U_0$$ potential barrier, there should be no reflected wave when the incident wave is at one of the transmisson resonances. Assuming that a beam of particles is incident at the first transmission resonance, $$E=U_0+(\frac{\pi^2 h^2}{2mL^2})$$, combine the continuity conditions to show that B=0. Here are the continuity conditions:

1st $$A+B=C+D$$

2nd $$k(A- B)=k^{'}(C-D)$$

3rd $$Ce^{ik^{'}L}+De^{-ik^{'}L}=Fe^{ikL}$$

4th $$k^{'}(Ce^{ik^{'}L}-De^{-ik^{'}L})=kFe^{ikL}$$

A couple more equations that we already know are $$k=\sqrt{\frac{2mE}{h^2}}$$ and $$k^'=\sqrt{\frac{2m(E-U_0)}{h^2}$$

Here is my attempted solution:

I divided the 4th equation by K and then set equation 3 and 4 equal to each other. I then used the new equation to solve for C in terms of D giving me

$$C=De^{-2i \pi}\frac{k^{'}+k}{k^{'}-K}$$ where I substituted $$\frac{\pi}{L}$$ in for k'.

I substituted this result into the first equation to now give me

$$A+B={De^{-2i \pi}\frac{k^{'}+k}{k^{'}-K} + D$$.

Solving for D gives me

$$\frac{A+B}{e^{-2i \pi}\frac{k^{'}+k}{k^{'}-K} + 1} = D$$

Now, plugging in our solutions for D and C into the 2nd equation

$$\frac{(A+B)e^{-2i \pi}\frac{k^{'}+k}{k^{'}-K}}{e^{-2i \pi}\frac{k^{'}+k}{k^{'}-K}+1}-\frac{A+B}{e^{-2i \pi}\frac{k^{'}+k}{k^{'}-K}+1}=\frac{k}{k^{'}}(A-B})$$

At this point, it seems impossible to simplify the equation to a point where it is obvious that B = 0. Am I on the right track or is there an easier way?

There is an easier way.

I see that you've already evaluated k' at the transmission resonance. Plug this value into your 3rd and 4th equations and get rid of the exponentials on the left side. You should see that the left side of these two now look similar to the right side of your first two equations, and a little algebra should give you B=0.

## 1. What is quantum transmission resonance?

Quantum transmission resonance is a phenomenon in which particles or waves of quantum matter are able to pass through a barrier or potential well with nearly 100% probability. This is due to the particles' or waves' energy matching the energy of the barrier, resulting in a resonant interaction that allows for efficient transmission.

## 2. How is quantum transmission resonance different from classical transmission?

In classical transmission, particles or waves must have enough energy to overcome a barrier in order to pass through it. However, in quantum transmission resonance, particles or waves can pass through the barrier even if they do not have enough energy, as long as their energy is close enough to the energy of the barrier.

## 3. What are some real-world applications of quantum transmission resonance?

Quantum transmission resonance has potential applications in various fields, such as quantum computing, quantum sensing, and quantum communication. It can also be used in nanotechnology to enhance the efficiency of energy transfer and in quantum tunneling for efficient energy harvesting.

## 4. How is quantum transmission resonance studied?

Quantum transmission resonance is typically studied through theoretical calculations and experiments using specialized equipment, such as quantum simulators and scanning tunneling microscopes. Scientists also use mathematical models and simulations to understand and predict the behavior of quantum systems exhibiting transmission resonance.

## 5. Can quantum transmission resonance be controlled?

Yes, quantum transmission resonance can be controlled by adjusting the energy levels of the particle or wave and the barrier. This can be achieved through various techniques, such as manipulating the external environment, applying electric or magnetic fields, or using specific materials with desired properties.

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