# Solving Dirac Delta Potential: Reflection & Transmission Coefficients

• kcirick
In summary, to obtain the reflection and transmission coefficients for a particle of mass m in a 1D potential with a repulsive potential, one must integrate the Schrodinger equation over a small interval, take the limit as the interval approaches zero, and use the assumption of continuity at x = 0 to find a relationship between the wave function coefficients. This will give the ratios \frac{D}{C} and \frac{E}{C} which can then be used to calculate the reflection and transmission coefficients, R and T. The hint given in the question suggests integrating the Schrodinger equation over the interval (-\epsilon, \epsilon) to obtain this relationship.
kcirick
Question:
Consider the motion of a particle of mass m in a 1D potential $V(x) = \lambda \delta (x)$. For $\lambda > 0$ (repulsive potential), obtain the reflection R and transmission T coefficients.

[Hint] Integrate the Schordinger equation from $-\eta$ to $\eta$ i.e.
$$\Psi^{'}(x=\epsilon )-\Psi^{'}(x=-\epsilon )=\frac{2m}{\hbar^{2}}\lambda\int^{\epsilon}_{-\epsilon}\delta (x)\Psi (x)dx = \frac{2m}{\hbar^{2}}\lambda\Psi (x > 0)$$

What I have so far:
Inside the barrier, the wave function is:

$$\psi (x)= Ae^{\kappa x}+Be^{-\kappa x}$$

where:

$$\kappa = \sqrt{\frac{2m}{\hbar^{2}}\left(V-E\right)}$$

Outside we have wave function in the form of:

$$\psi (x) = Ce^{ikx}+De^{-ikx} x < 0$$
$$\psi (x) = Ee^{ikx} x > a$$

and $R = \frac{|D|^2}{|C|^2}$ and $T = \frac{|E|^2}{|C|^2}$.

I have in my notes how to get the ratio $\frac{D}{C}$ and $\frac{E}{C}$, but how does the hint that was given to me used for? where does the delta function come in play?

I don't really get the hint itself either. How does integrating Schrodinger Equation give me that relation in the hint? I am very lost...

Inside the barrier

There is no "inside the barrier," since a delta function is a width-zero barrier.

1) Write down Schrodinger's equation.

2) Integrate it term-by-term over the interval $(-\epsilon, \epsilon)$.

3) Take the limit as $\epsilon \rightarrow 0$.

Outside we have wave function in the form of:

$$\psi (x) = Ce^{ikx}+De^{-ikx} x < 0$$
$$\psi (x) = Ee^{ikx} x > 0$$

Assume $\psi$ is continuous at $x = 0$. This gives you a relationship between the three coefficients.

## What is the Dirac Delta Potential?

The Dirac Delta Potential is a mathematical construct used to model a point-like potential in quantum mechanics. It is often used to describe the behavior of a particle at the boundary between two regions with different potential energies.

## How is the Dirac Delta Potential solved?

The Dirac Delta Potential is solved using the reflection and transmission coefficients. These coefficients describe the probability of a particle being reflected or transmitted when encountering the potential barrier.

## What are the factors that affect the reflection and transmission coefficients?

The reflection and transmission coefficients are affected by the energy of the particle, the height and width of the potential barrier, and the mass of the particle. Additionally, the angle of incidence and the potential profile also play a role.

## What is the significance of the reflection and transmission coefficients in quantum mechanics?

The reflection and transmission coefficients play a crucial role in understanding the behavior of particles at potential barriers. They allow us to predict the probability of a particle being reflected or transmitted, and thus help us understand the behavior of quantum systems.

## How are the reflection and transmission coefficients experimentally determined?

The reflection and transmission coefficients can be experimentally determined by measuring the intensity of the reflected and transmitted particles. By comparing the measured intensities with the predicted values using the coefficients, we can confirm the validity of our calculations.

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