Question 1. Consider the three-dimensional wave function

ψ(x, y, z) = Ne-[|x|/2a + |b|/2b + |z|/2x]

where a, b and c are three positive lengths.

a. Calculate the constant N which normalizes ψ.

b. Calculate the probability that a measurement of X will yield a result included between 0 and a.

c. Calculate the probability that simultaneous measurements of Y and Z will yield results included respectively between - b and + b, and - c and + c.

d. Calculate the probability that a measurement of the momentum will yield a result included in the element dp_x, dp_y, dp_z centered at the point p_x = p_y = 0; p_z = h/c.

Question 2. Let ψ (x, y, z) = ψ(r) be the normalized wave function of a particle. Express in terms of ψ(r) the probability for:

a. a measurement of the abscissa X, to yield a result included between x₁ and x₂;

b. a measurement of the component P_x of the momentum, to yield a result included between p₁ and p₂;

c. simultaneous measurements of X and P_z, to yield :

x1 ≤ x ≤ x2

p_z ≥ 0

d. simultaneous measurements of P_x, P_y, P_z, to yield :

P1 ≤ Px ≤ P2

P3 ≤ Py ≤ P4

P5 ≤ Pz ≤ P6

Show that this probability is equal to the result of b when p₃, p₅ → ∞; Pi, P6 ---4" + X ;

e. A measurement of the component U = -1/√3 (X + Y + Z) of the position, to yield a result included between u₁ and u₂.

Question 3. Let J(r) be the probability current associated with a wave function ψ(r) describing the state of a particle of mass m [chap. III, relations (D-17) and (D-19)].

a. Show that:

m ∫d³r J(r) = < P >

where < P) is the mean value of the momentum.

b. Consider the operator L (orbital angular momentum) defined by L = R x P. Are the three components of L Hermitian operators? Establish the relation:

m ∫d³r [r x J(r)] = < L>.

Question 4. One wants to show that the physical state of a (spinless) particle is completely defined by specifying the probability density p(r) = Ψ(r)² and the probability current J(r).

a. Assume the function Ψ(r) known and let ξ(r) be its argument:

Ψ(r) = √ρ(r)^eiξ(r)

Show that :

J(r) = h/mp(r)∇ξ(r)

Deduce that two wave functions leading to the same density p(r) and current J(r) can differ only by a global phase factor.

b. Given arbitrary functions p(r) and J(r), show that a quantum state Ψ(r) can be associated with them only if ∇ x v(r) = 0, where v(r) = J(r)/p(r) is the velocity associated with the probability fluid.

c. Now assume that the particle is submitted to a magnetic field B(r) = ∇ x A(r) [see chap. III, definition (D-20) of the probability current in this case]. Show that:

J = ρ(r)/m [h∇ξ(r) - qA(r)]

and:

∇ x v(r) = -q/m B(r)

Question 5. Virial theorem

a. In a one-dimensional problem, consider a particle with the Hamiltonian:

H = p²/2m + V(X)

where:

V(X) = λXⁿ

Calculate the commutator [H, XP]. If there exists one or several stationary states | φ > in the potential V, show that the mean values < T> and < V> of the kinetic and potential energies in these states satisfy the relation : 2 < T) = n< V>.

b. In a three-dimensional problem, H is written:

H = p²/2m + V(R)

Calculate the commutator [H, R.P]. Assume that V(R) is a homogeneous function of nth order in the variables X, Y, Z. What relation necessarily exists between the mean kinetic energy and the mean potential energy of the particle in a stationary state?

Apply this to a particle moving in the potential V(r) = - e²/r (hydrogen atom).

Recall that a homogeneous function V of n^th degree in the variables x, y and z by definition satisfies the relation:

V(αx, αy, αz) = αⁿV(x, y, z)

and satisfies Euler's identity :

x ∂V/∂x + y∂V/∂y + z∂V/∂z = nV(x,y,z).

c. Consider a system of N particles of positions R_i and momenta P_i (i = 1, 2, ... N). When their potential energy is a homogeneous (n^th degree) function of the set of components X_i, Y_i, Z_i, can the results obtained above be generalized?

An application of this can be made to the study of an arbitrary molecule formed of nuclei of charges - Zi_q and electrons of charge q. All these particles interact by pairs through Coulomb forces. In a stationary state of the molecule, what relation exists between the kinetic energy of the system of particles and their energy of mutual interaction?