Evolution of probabilities

Problem:

Consider two-flavor neutrino mixing. 
The energy (or mass) eigenstates
image
are related to the flavor eigenstates
image
as  
image

The energy eigenvalues are E1 and E2, with E1 ≠ E2
Assume the system starts at time t = 0 in the electron-neutrino state

 image

What is the probabilities that at time t the system will be found in the μ-neutrino state

 image ?

Solution:

Problem:

A box containing a particle is divided into right and left compartments by a thin partition.  If the particle is known to be on the right or left sides with certainty, the state is represented by the eigenkets |R> and |L>, respectively.  The particle can tunnel though the partition; this tunneling effect is characterized by the Hamiltonian
H = ∆(|L><R|+|R><L|),
where ∆ is a real number with the dimension of energy. 
Suppose that at time t = 0 the particle is on the right side with certainty. 
What is the probability for observing the particle on the left side as a function of time?

Solution:

Problem:

A system is described by a Hamiltonian whose matrix is

image.
in the {|1>, |2>} basis.  Here a = |a|exp(iφ).
(a)  Find the eigenvalues and normalized eigenvectors of this Hamiltonian.
(b)  Calculate the probability of finding the system in the state |1> at time t, if it was in the state |2> at t = 0.

Solution:

Problem:

Consider a three-state quantum mechanical system with an orthonormal ‘color' basis {|R>, |G>, |B>} (‘red,' ‘blue,' and ‘green' respectively).  Its evolution is governed by the Hamiltonian
H = E0(2|R><R| + 2|B><B| + 2|G><G| - |G><B| - |B><G|).
(a)  Construct the matrix representation of this Hamiltonian using the {|R>, |G>, |B>} basis.
(b)  Find the energy eigenvalues and normalized eigenstates of the system.  Express the latter as linear combinations of |R>, |G>, |B>.
(c)  At time t = 0 the state vector is |ψ(0)> = |G>.  Find the state vector |ψ(t)> at an arbitrary time t.
(d)  After starting from the initial conditions of (c), the ‘color' is measured at time t = t0 and found to be green.  What are the probabilities for the color to be measured as red, green, or blue at time t = 2t0?

Solution:

Problem:

A quantum system can exist in two states |ψ1> and |ψ2>, which are eigenstates of the Hamiltonian with eigenvalues E1 and E2
An observable A has eigenvalues ±1 and eigenstates |ψ±>= (1/√2)(|ψ1> ± |ψ2>).
This observable is measured at times t = 0, T, 2T, ... .  The normalized state of the system at t = 0, just before the first measurement, is c11>+  c22>. 
(a)  What is the probability of measuring A = 1 at t = 0?
(b)  If Pn denotes the probability that the measurement at t = nT gives the result A = 1, show that Pn+1 = ½(1 - cosα) + Pn cosα, where
α/T = (E1 - E2)/ħ>
(c)  Deduce that  Pn = ½ (1 - cosnα) + ½|c1 + c2|2 cosnα.
(d)  What happens in the limit as n --> ∞ with nT = t fixed?

Solution:

Problem:

A particle in a potential well U(x) is initially in a state whose wave function is an equal-weight superposition of the ground state and first excited state wave functions
Ψ(x,0) = C[ψ1(x) + ψ2(x)],
where C is a constant and ψ1(x) and ψ2(x) are normalized solutions to the time-independent Schrödinger equation with energies E1 and E2
(a)  Show that the value C = 1/√2  normalizes Ψ(x,0).
(b)  Determine Ψ(x,t) at any later time t.
(c)  Show that the average energy <E> for Ψ(x,t) is the arithmetic mean of the energies E1 and E2
(d)  Determine the uncertainty ∆E of the energy for Ψ(x,t).

Solution:

Problem:

(a)  Define what is meant by the term “stationary state” in quantum mechanics, and explain the distinction between the time-dependent and time-independent Schrödinger equation.
(b)  At time t = 0, the wave function of a particle in one dimension is ψ(x) = (u1(x) + u2(x))/√2, where u1(x) and  u2(x) are two solutions of the time-independent Schrödinger equation.  
For this particle, how does the probability density change with time?

Solution:

Problem:

Consider a quantum particle for which the Hamiltonian operator is H.  Denote by |Φn> the eigenvectors of H, corresponding to the energy eigenvalues En.  Assume that the |Φn> form a complete, orthonormal basis.  At time t = 0 the particle is in the state

|Ψ> = A[|Φ0> + 2i|Φ1> + 4|Φ3> - 2|Φ4>].

(a)  Normalize |Ψ>.
(b)  Imagine that you perform an experiment to measure the energy of the system.  What is the most likely result of this measurement and what is the probability of getting this result?
(c)  What is the expectation value of the energy <H>?
(d)  Find |Ψ(t)>.

Solution: