Transmission amplitude using path-integrals

[gulsen]

Hello,

As a path-integral newbie, I've been trying to calculate the amplitude for an electron which enters a box (potential within the box is given) at a point to emerge the other edge of the box (it doesn't matter when it exits). For simplicity, I first tried to work out the problem in one dimension, and in discrete space-time. To simplify it even further, I tried with a constant (but non-zero) potential.

I worked out the kernel $$K(b,a)$$, but it -naturally- depends on time spent in the "box". But I don't care when will exit, I care only whether if it can or can not penetrate through.

I have $$\psi(b,t_b) = \int K(b,a) \psi(a,t_a) da$$, and transmission amplitude at $$t_b$$ would be "inner product" of wavefunctions at $$t_b$$ and $$t_a$$ (but well, how? They don't have a variable in common at all! Do I get to expand the wavefunction in eigenstates of position?). So I guess, to get the total amplitiude, I get to compute the amplitude for all times after $$t_a$$, and add them all. Sounds plausible to me, but how would I do an inner product with $$\psi(b,t_b)$$ and $$\psi(a,t_a)$$? Or am I quite off?

 

[BigJDubs]

Thanks in advance!</code>The amplitude for the electron to penetrate through the box is determined by the probability of it being in the box at any given time. This is related to the kernel K(b,a), which is a measure of the probability of the electron travelling from point a to point b in a given amount of time. To calculate the total amplitude, you need to sum up the amplitudes from all times after t_a (since the electron may have entered the box at any time). The inner product between the wavefunctions at t_b and t_a can be done using the Fourier transform. By taking the Fourier transform of both wavefunctions, you can identify the coefficients of each mode of the wavefunction and multiply them together in order to calculate the inner product. Once you have calculated the inner product for all modes, you can then sum them all up to get the total amplitude.

 

[charng]

Hello,

Thank you for your question about calculating the transmission amplitude using path-integrals. This is a complex and interesting problem that requires a deep understanding of quantum mechanics and mathematical techniques. I will try my best to provide a response that can help you in your calculations.

Firstly, it is important to note that path-integrals are a powerful mathematical tool used in quantum mechanics to calculate the probability amplitudes of a particle moving from one point to another. In your case, you are interested in the transmission amplitude, which is the probability of the electron passing through the box. This can be calculated using the Feynman path-integral formulation, where the amplitude is given by the integral over all possible paths that the electron can take from the initial point to the final point.

In your problem, the box represents a potential barrier that the electron must pass through. The potential within the box is given, and you are interested in the case where the potential is constant but non-zero. To simplify the problem, you have considered the case of one dimension and discrete space-time. This is a common approach in quantum mechanics calculations, and it allows for easier visualization and understanding of the problem.

You mention that you have worked out the kernel K(b,a), which represents the amplitude for the electron to travel from point a to point b. However, this kernel depends on the time spent in the box, which is not of interest to you. This is where the Feynman path-integral formulation becomes very useful. The beauty of this approach is that it allows you to sum over all possible paths, including those that spend different amounts of time in the box. This means that the time spent in the box will be accounted for in the calculation of the transmission amplitude.

To calculate the total transmission amplitude, you are correct in thinking that you need to compute the amplitude for all times after t_a and add them together. This is because the electron could potentially exit the box at any time after t_a, and each of these possibilities needs to be taken into account.

As for the inner product of the wavefunctions at t_b and t_a, you are correct that you need to expand the wavefunction in eigenstates of position. This is because the wavefunction at t_a and t_b are in different bases, and you need to transform one into the other to perform the inner product. This is a standard procedure in quantum mechanics and can be found in many textbooks and online resources.

In summary,