Introduction:
The percentage of quantum states occupied up to the Fermi level at 0 Kelvin (0 deg K) can be determined using the concept of Fermi-Dirac statistics. This statistical distribution describes the behavior of fermions, such as electrons, in a system at thermal equilibrium.

Explanation:
At absolute zero temperature (0 deg K), the system is in its ground state, and all thermal motion ceases. In this state, the electrons occupy the lowest energy levels, starting from the lowest energy level and filling up successively higher energy levels until the Fermi level.

The Fermi level represents the highest energy level occupied by an electron at 0 Kelvin. Electrons fill up the available energy levels in a system according to the Pauli exclusion principle, which states that no two electrons can occupy the same quantum state simultaneously.

Percentage of quantum states occupied:
At 0 Kelvin, the Fermi-Dirac distribution function can be simplified to a step function. This step function is equal to 1 for energy levels below the Fermi level and equal to 0 for energy levels above the Fermi level.

Since the step function is equal to 1 for energy levels below the Fermi level, it implies that all quantum states up to the Fermi level are occupied by electrons. Therefore, the percentage of quantum states occupied up to the Fermi level at 0 Kelvin is 100% (option D).

Conclusion:
At 0 Kelvin, all quantum states up to the Fermi level are occupied by electrons according to the Fermi-Dirac distribution. Therefore, the correct answer is option D, which states that the percentage of quantum states occupied up to the Fermi level at 0 deg K is 100%.