Question 1. Two-particle wave function
In a one-dimensional problem, consider a system of two particles (1) and (2) with which is associated the wave function ψ (x1, x2).
a. What is the probability of finding, in a measurement of the positions X1 and X2 of the two particles, a result such that:
x ≤ x1 ≤ x + dx
α ≤ x2 ≤ β
b. What is the probability of finding particle (1) between x and x + dx [when no observations are made on particle (2)] ?
c. Give the probability of finding at least one of the particles between α and β.
d. Give the probability of finding one and only one particle between α and β.
e. What is the probability of finding the momentum of particle (1) included between p1 and p2 and the position of particle (2) between α and β.
f. The momenta P1 and P2 of the two particles are measured; what is the probability of finding p' ≤ p1 ≤ p"; p"' ≤ p2 ≤ p""?
g. The only quantity measured is the momentum P1 of the first particle. Calculate, first from the results of e and then from those of .1, the probability of finding this momentum included between p' and p". Compare the two results obtained.
h. The algebraic distance X1 - X2 between the two particles is measured; what is the probability of finding a result included between - d and + d? What is the mean value of this distance?
Question 2. Infinite one-dimensional well
Consider a particle of mass m submitted to the potential:
V(x) = 0 if 0 ≤ x ≤ a.
V(x)= +∞ if x < 0 or x > a.
| φn > are the eigenstates of the Hamiltonian H of the system, and their eigenvalues are En = n2Π2h2/2ma2 (cf. complement H1). The state of the particle at the instant t = 0 is :
|Ψ(0) > = a1 (Ψ1 > + a2 |(Ψ2 > + a3 l (Ψ3 > + a4|(Ψ4 >
a. What is the probability, when the energy of the particle in the state |Ψ (0)> is measured, of finding a value smaller than 3Π2h2/ma2?
b. What is the mean value and what is the root-mean-square deviation of the energy of the particle in the state |Ψ(t)>?
c. Calculate the state vector |Ψ(t)> at the instant t. Do the results found in a and b at the instant t = 0 remain valid at an arbitrary time t?
d. When the energy is measured, the result 8Π2h2/ma2 is found. After the measurement, what is the state of the system? What is the result if the energy is measured again?
Question 3. Infinite two-dimensional well (cf. complement GII,)
In a two-dimensional problem, consider a particle of mass m; its Hamiltonian H is written :
H = Hx + Hy
with:
Hx = px2/2m + V(X) Hy = px2/2m + V(Y)
The potential energy V(x) [or V(y)] is zero when x (or y) is included in the interval [0, a] and is infinite everywhere else.
a. Of the following sets of operators, which form a C.S.C.O. ?
{ H },{Hx}, {Hx, Hy} {H, Hx}
b. Consider a particle whose wave function is:
Ψ(x, y) = N cos Πx/a.cos Πy/a.sin2Πx/a.sin2Πy/a
when 0 < x < a and 0 < y < a, and is zero everywhere else (where N is a constant).
α. What is the mean value < H > of the energy of the particle? If the energy H is measured, what results can be found, and with what probabilities ?
β. The observable Hx is measured; what results can be found, and with what probabilities? If this measurement yields the result Π2h2/2ma2, what will be the results of a subsequent measurement of Hy, and with what probabilities?
γ. Instead of performing the preceding measurements, one now performs a simultaneous measurement of Hx and Py. What are the probabilities of finding:
Ex = 9Π2h2/2ma2
and :
Po ≤ Py ≤ po + dp ?
Question 4. Consider a physical system whose state space, which is three-dimensional, is spanned by the orthonormal basis formed by the three kets |u1 >, |u2 >, | u3 >. In this basis, the Hamiltonian operator H of the system and the two observables A and B are written:
where ωo, a and b are positive real constants.
The physical system at time t = 0 is in the state:
|Ψ(0) > = 1/√2 |μ1| > + 1/2 |μ2> + 1/2|μ3>
a. At time t = 0, the energy of the system is measured. What values can be found, and with what probabilities? Calculate, for the system in the state 0(0) >, the mean value < H> and the root-mean-square deviation ΔH.
b. Instead of measuring H at time t = 0, one measures A; what results can be found, and with what probabilities? What is the state vector immediately after the measurement?
c. Calculate the state vector |ψ(t)> of the system at time t.
d. Calculate the mean values < A > (t) and < B > (t) of A and B at time t. What comments can be made?
e. What results are obtained if the observable A is measured at time t? Same question for the observable B. Interpret.
Question 5. Interaction picture
(It is recommended that complement FIII and perhaps complement GIII be read before this exercise is undertaken.)
Consider an arbitrary physical system. Denote its Hamiltonian by Ho(t) and the corresponding evolution operator by Uo(t, t'):
Now assume that the system is perturbed in such a way that its Hamiltonian becomes :
H(t) = Ho(t) + W(t)
The state vector of the system in the "interaction picture", |ΨI(t)> is defined from the state vector |ΨI(t) > in the Schrodinger picture by :
|ψ(t)> = Uo+(t, to)W(t)U0(t,t0)
a. Show that the evolution of I 1P/(t)> is given by :
ihd/dt|ψI(t)>| = WI(t)|ψI(t)>
where W1(t) is the transform operator of W(t) under the unitary transformation associated with U0t(t, to)
W1(t) = U0t(t, to),W(t) Uo(t, to)
Explain qualitatively why, when the perturbation W(t) is much smaller than Ho(t), the motion of the vector |ψs(t)>. is much slower than that of |ψs(t)>.
b. Show that the preceding differential equation is equivalent to the integral equation :
|ψI(t)> = |ψI(to)> + 1/ihto∫tdt'WI(t')|ψI(t'))>
where : |ψI(to) > = |ψs(t)>.
c. Solving this integral equation by iteration, show that the ket |ψI(t)> can be expanded in a power series in W of the form:
