# Quantum mechanics

# Ramsauer effect

Take 15 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel free to ask them on the forum.

## Question

Schrodinger's time-independent equation is, where is the probability amplitude :

Solve this equation in the case of a quantum particle of mass , placed in an infinite potential well of width , in order to find the results of the lesson.

### Hint

### Solution

See the lesson : "Particle in an infinite well"

## Question

In 1921, physicist Ramsauer noticed that, for some values of the incidental energy, gases like Helium were perfectly transparent to electron beams of low energy.

Let us consider a stationary solution to Schrodinger's equation, of positive energy for an electron of mass in the potential described by the following picture.

### Hint

### Solution

The expressions of and are :

## Question

Write the four equations linking these constants of integration.

### Hint

### Solution

The continuity of the wave function and its derivative in and lead to :

## Question

A calculation, which is not asked, leads to the expression of the probability amplitude ratio :

Determine the expression of , the probability of current reflection coefficient of the well of potential that is studied.

What is the value of , the probability of current transmission coefficient ?

### Hint

### Solution

The coefficient of reflection is :

Coefficient is simply equal to :

## Question

How can the Ramsauer effect be interpreted ?

For Helium, the minimum incidental energy for which Ramsauer effect happens is .

The radius of the Helium atom is .

Compute the value of the depth of the well inside the atom.

*Data : *

and (mass of the electron)

### Hint

### Solution

Helium is transparent to the electron if (or ) : the electron does not see the potential well.

But is equivalent to :

The De Broglie wavelength of the electron inside the well is :

All reflected waves interfere in a destructive manner.

The well comes transparent for the incidental wave.

The reflected wave in , which does not enter the well, interferes in a destructive way with the waves resulting from multiple reflections.

The energy associated to mode is :

For , we find .

## Question

Can you give an analogous phenomena to the Ramsauer effect in other subjects of physics ?

### Hint

### Solution

Gel used in ultrasound scan for a better transmission

Anti-noise walls

Anti-reflection glasses