ISEM3 le 20 Octobre 1998

DEVOIR SURVEILLE

Durée: 2h

Sans Document

Avec Calculatrice

Exercice 1: GP Thomson experiment ( 3 points)

Figure 1: GP Thomson Experimental Apparatus and Results

The interference rings in Figure 1 were produced by sending X-rays of wavelength Å through a polycrystalline thin film of copper (interatomic spacing 2.55 Å) of thickness t=1µm (10^-4cm). To produce the same set of rings but with electrons () instead of X-rays, what kinetic energy (in eV) should the electrons in the incoming beam have?

Planck's constant: h=6.62x10^-34Js.

From MIT (Massachusetts Institute of Technology - spring 1997)

Exercice 2: Representations ( 8 points)

Background: In this problem we will consider the physics of particles which remain bound to the well in Figure 2, which binds only two pure states of energy, and . The zero of energy is chosen so that the energies of these two states are and , respectively.

Figure 2: Potential Well Binding Two States

In addition to the energy E, we are told that there are two other physical observables for particles bound to the well, which we will call F and G. Measurements of these observables also always only produce the two values for particles bound in the well. The pure states associated with these values of F and G are given in terms of the pure states of energy in Table 1. The aim of this exercise is to show that you are able to make predictions on the probability distribution for measurements on G given only information about the distributions for measurements of E and F!

Observed value	Wavefunction of pure state (properly normalized)

Table 1: Table of pure states of the new observables F and G

1. Probabilities in the E representation

Energy measurements are performed on systems in each of the six states in Table 1. Use the Principle of Quantum Superposition to complete the table by giving the probabilities of finding and in measurements performed on systems in each of the six states.

2. G representation

Show how the states , , and may be written as a superposition of the two pure states of G.

For a particle in the state , what is the probability of finding the value in an experiment measuring the observable G?

3. (*) Specification of the state from experimental information

Now, you are given a new state . Repeated measurements yield that and , but and . Use this information to show

1) that c₊ and c_- must have the form:

2) that

Hints: i) Write as a superposition of and .

ii) Use the fact that .

4. Physical Prediction

Given systems in the state from 3., show that the probability is given by

From MIT (Massachusetts Institute of Technology - spring 1996)

Exercice 3: Opérateur parité. ( 9 points)

On considère l'opérateur parité qui agit sur les solutions de l'équation de Schrödinger indépendante du temps (à une dimension) de la manière suivante:

avec pour tout x :

1. L'opérateur parité est-il hermitique?

2. Quelles sont les valeurs propres et les fonctions propres associées de ?

1. Quelle propriété doit avoir l'énergie potentielle V(x) pour que l'opérateur Hamiltonien et l'opérateur parité commutent?

4. Dans le cas où cette condition est satisfaite, montrer (en utilisant le fait que qu'on peut toujours imposer aux fonctions propres de l'Hamiltonien d'être soit symétrique soit antisymétrique (pour une valeur d'énergie non dégénérée).