In the present study, a balanced steady-state free precession pulse sequence combined with compressed sensing was applied to hyperpolarized (129) Xe lung imaging in spontaneously breathing mice. With the aid of fast imaging techniques, the temporal resolution was markedly improved in the resulting images. Using these protocols and respiratory gating, (129) Xe lung images in end-inspiratory and end-expiratory phases were obtained successfully. The application of these techniques for pulmonary functional imaging made it possible to simultaneously evaluate regional ventilation and gas exchange in the same animal. A comparative study between healthy and elastase-induced mouse models of emphysema showed abnormal ventilation as well as gas exchange in elastase-treated mice.

We have solved the two-dimensional time-dependent Schrödinger equation for a single particle in the presence of a nonuniform magnetic field for initial speeds of 10-100 m/s. By linear extrapolation, it is shown that the variance, or the uncertainty, in position would reach the square of the interparticle separation n −2/3 with a number density of n = 10 20 m −3 in a time interval of the order of 10 −4 sec. After this time the wavefunctions of neighboring particles would overlap, as a result the conventional classical analysis may lose its validity: Plasmas may behave more-or-less like extremely-low-density liquids, not gases, since the size of each particle is of the same order of the interparticle separation.

We have solved the two-dimensional time-dependent Schrödinger equation for a single particle in the presence of a non-uniform magnetic field for initial speed of 10-100 m/s, mass of the particle at 1-10 m p , where m p is the mass of a proton. Magnetic field at the origin of 5-10 T, charge of 1-4 e, where e is the charge of the particle and gradient scale length of 2.610 × 10 −5 -5.219 m. It was numerically found that the variance, or the uncertainty, in position can be expressed as dσ 2 r /dt = 4.1 v 0 /qB 0 L B , where m is the mass of the particle, q is the charge, v 0 is the initial speed of the corresponding classical particle, B 0 is the magnetic field at the origin and L B is the gradient scale length of the magnetic field. In this expression, we found out that mass, m does not affect our newly developed expression.

We have solved the two-dimensional time-dependent Schödinger equation for a magnetized proton in the presence of a fixed field particle with an electric charge of 2×10−5 e, where e is the elementary electric charge, and of a uniform megnetic field of B = 10 T. In the relatively high-speed case of v 0 = 100 m/s, behaviors are similar to those of classical ones. However, in the low-speed case of v 0 = 30 m/s, the magnitudes both in momentum mv = |mu|, where m is the mass and u is the velocity of the particle, and position r = |r| are appreciably decreasing with time. However, the kinetic energy K = m u 2 /2 and the potential energy U = qV , where q is the electric charge of the particle and V is the scalar potential, do not show appreciable changes. This is because of the increasing variances, i.e. uncertainty, both in momentum and position. The increment in variance of momentum corresponds to the decrement in the magnitude of momentum: Part of energy is transfered from the directional (the kinetic) energy to the uncertainty (the zero-point) energy.

We have solved the two-dimensional time-dependent Schrödinger equation for a single particle in the presence of a non-uniform magnetic field for initial speed of 8 -100 m/s, mass of the particle at 1 -10 m p , where m p is the mass of a proton. Magnetic field at the origin of 5 -10 T, charge of 1 -4 e, where e is the charge of the particle and gradient scale length of 2.610 × 10 −5 -5.219 m. Previously, we found out that the variance, or the uncertainty, in position can be expressed as dσwhere m is the mass of the particle, q is the charge, v 0 is the initial speed of the corresponding classical particle, B 0 is the magnetic field at the origin and L B is the gradient scale length of the magnetic field. In this research, it was numerically found that the variance, or the uncertainty, in total momentum can be expressed as dσ 2 P /dt = 0.57 qB 0 v 0 /L B . In this expression, we found out that mass, m does not affect both our newly developed expression for uncertainty in position and total momentum.