Introduction:
The 3-dimensional quantum mechanical harmonic oscillator is described by the energy levels given by the equation E = (n + 3/2)hw, where n is the principal quantum number and w is the angular frequency. In a particular state, if the energy is given as 3.5hw, we need to determine the degree of degeneracy, which represents the number of different states with the same energy.

Calculating the principal quantum number:
To determine the degree of degeneracy, we first need to find the principal quantum number for the given energy level. We use the equation E = (n + 3/2)hw and rearrange it to solve for n:
n = (E/hw) - 3/2
Substituting the given energy value of 3.5hw into the equation, we have:
n = (3.5hw/hw) - 3/2
n = 3.5 - 3/2
n = 2

Determining the degeneracy:
The degeneracy, denoted by g, is the number of states with the same energy. In the case of the harmonic oscillator, the degeneracy is given by the formula:
g = (n + 1)(n + 2)/2

Substituting the value of n = 2 into the equation, we have:
g = (2 + 1)(2 + 2)/2
g = 3 * 4 / 2
g = 12 / 2
g = 6

Explanation:
The degree of degeneracy for a 3-dimensional quantum mechanical harmonic oscillator with an energy of 3.5hw in a particular state is 6. This means that there are 6 different states that have the same energy value of 3.5hw. The degeneracy arises due to the different combinations of quantum numbers that can result in the same energy level. In this case, there are 6 different states corresponding to the principal quantum number n = 2, each with a different combination of the other quantum numbers.

Therefore, the correct answer is '6'.