Question

1. Which couple of assertions relates to classical statistics? and why
   (a) Probability density function is a function of coordinates and momenta of all particles; infinite time is needed to measure positions of all energy levels.
   (b) Initial conditions cannot be determined simultaneously for all particles; system’s state is described by coordinates and momenta (in phase space).
   (c) It is impossible to determine the quantum states with certain energy (stationary levels); system’s state is set by the value of system’s energy, but not by coordinates and momenta.
   (d) Number of states is proportional to the volume of phase space; probability of state is proportional to the diagonal element of the density matrix.

Answer 1

(a) Given :Probability density function is a function of coordinates and momenta of all particles; infinite time is needed to measure positions of all energy levels. Consider  "Probability density function is a function of coordinates and momenta of all particles". In quantum statistics, we cannot specify position and momenta of particle simultaneously with accuracy as it follows Heisenberg uncertainty principle. Hence we cannot represent a particle as a point in phase space. And hence Probability density function cannot be a function of coordinates and momenta of all particles in quantum statistics. But in classical statistics, we need position and momenta of particle to know the dynamics of the particle. It can be represented in phase space. Hence to find probability  density function, we should know both position and momenta of particle. Consider 'Infinite time is needed to measure positions of all energy levels." This is true for quantum mechanical system as they obey Heisenberg uncertainty principle. According to that principle, we cannot know energy and time of particle in particular state simultaneously with accuracy. So if we need all energy accurately, uncertainty in time should be infinite. So that uncertainty in energy be zero.

Classical statistics : Probability density function is a function of coordinates and momenta of all particles.

(b) Given : Initial conditions cannot be determined simultaneously for all particles; system’s state is described by coordinates and momenta (in phase space). Consider  " Initial conditions cannot be determined simultaneously for all particles." This is true in quantum systems due to inherent uncertainty accompanied during measurement of conjugate pairs simultaneously with accuracy. Conjugate position and momentum also follow this principle called  Heisenberg uncertainty principle. In classical we can find position and momentum without any uncertainty. Consider  "System’s state is described by coordinates and momenta (in phase space)." Phase space is the space which specifies all possible states of a particle. To know dynamics of particle, initial position and momentum are necessary. Hence in classical physics, state of a particle is represented by it's position and momentum.

Classical statistics : System’s state is described by coordinates and momenta (in phase space).

(c) Given : It is impossible to determine the quantum states with certain energy (stationary levels); system’s state is set by the value of system’s energy, but not by coordinates and momenta. Consider  "  It is impossible to determine the quantum states with certain energy (stationary levels)." Classically it's not possible to determine the quantum state energy. But quantum mechanically we can associate certain energy with quantum states by solving Schrodinger equation. Consider  "System’s state is set by the value of system’s energy, but not by coordinates and momenta". Quantum mechanical system's cannot be set by value of coordinates and momenta because of associated uncertainty. Hence we can associate it with hamiltonian of the system.

Classical statistics :  It is impossible to determine the quantum states with certain energy (stationary levels).

(d) Given :  Number of states is proportional to the volume of phase space; probability of state is proportional to the diagonal element of the density matrix. Consider  " Number of states is proportional to the volume of phase space." In phase space, state of a system is represented by it's position and momentum. So each point on the phase space can be considered as micro states. So total number of states is ratio of volume of phase space to the volume of each micro states. In classical statistics, number of states is proportional to the volume of phase space. Consider  " Probability of state is proportional to the diagonal element of the density matrix." This is for  quantum mechanical system as the state of the system can be represented by matrix using Heisenberg matrix formulation. For normalized states, the diagonal element of the density matrix gives the probability of a particle in particular state. Ad particle can be in mixed states.

Classical statistics :Number of states is proportional to the volume of phase space.