Consider the simplest case in quantum statistical mechanics, where we find the density of states in the case of a cuboidal 3 dimensional box. In the derivation we take only those states which are product seperable into wavefunctions along the three directions i.e. can be denoted by three quantum numbers (n1, n2, n3) henceforth written as |n1,n2,n3> . However I feel that even states which are not product seperable should be considered. For example a particle in the system could be in the state |1,0,0>+|0,1,0> with appropriate normalisation. This will alter the counting of number of states. Why are such states excluded?

Consider the simplest case in quantum statistical mechanics, where we find the density of states in the case of a cuboidal 3 dimensional box. In the derivation we take only those states which are product seperable into wavefunctions along the three directions i.e. can be denoted by three quantum numbers $(n_1, n_2, n_3)$ henceforth written as $|n_1,n_2,n_3\rangle$ . However I feel that even states which are not product seperable should be considered. For example a particle in the system could be in the state $\frac{|1,0,0\rangle+|0,1,0\rangle}{\sqrt{2}}$. This will alter the counting of number of states. Why are such states excluded?