the quantum states of the entire system in terms of the quantum states of each such container In particular, lets pretend that we know the quantum states that pertain to N molecules in a container of volume V as shown in Fig. 7. 2, and let's label these states by an index J. That is J=1 labels the first energy state of n molecules in the container of volume V, J=2 labels the second such state, and so on. I understand that it may seem daunting to think of how one actually finds these N-molecule eigenstates. However, we are just deriving a general framework that gives the probabilities of being in each such state. In so doing, we are allowed to pretend that we know these states. In any actual application, we will, of course, have to use approximate expressions for such energies An energy labeling for states of the entire collection of M containers can be realized by giving the number of containers that exist in each single-container J-state This is possible because the energy of each M-container state is a sum of the energies of the M single-container states that comprise that M-container state. For example, if M=9, the label 1, 1, 2, 2, 1, 3, 4, 1, 2 specifies the energy of this 9-container state in terms of the energies (E) of the states of the 9 containers: E=461+3E+E+E4. Notice that this 9-container state has the same energy as several other 9-container states; for example, 1 2, 1, 2, 1, 3, 4, 1, 2 and 4, 1,3, 1, 2, 2, 1, 1, 2 have the same energy although they are different individual states. What differs among these distinct states is which box occupies which single-box quantum state The above example illustrates that an energy level of the M-container system can have a high degree of degeneracy because its total energy can be achieved by having the various single-container states appear in various orders. That is, which container is in page 8PAGE 8 the quantum states of the entire system in terms of the quantum states of each such container. In particular, let’s pretend that we know the quantum states that pertain to N molecules in a container of volume V as shown in Fig. 7.2, and let’s label these states by an index J. That is J=1 labels the first energy state of N molecules in the container of volume V, J=2 labels the second such state, and so on. I understand that it may seem daunting to think of how one actually finds these N-molecule eigenstates. However, we are just deriving a general framework that gives the probabilities of being in each such state. In so doing, we are allowed to pretend that we know these states. In any actual application, we will, of course, have to use approximate expressions for such energies. An energy labeling for states of the entire collection of M containers can be realized by giving the number of containers that exist in each single-container J-state. This is possible because the energy of each M-container state is a sum of the energies of the M single-container states that comprise that M-container state. For example, if M= 9, the label 1, 1, 2, 2, 1, 3, 4, 1, 2 specifies the energy of this 9-container state in terms of the energies {ej} of the states of the 9 containers: E = 4 e1 + 3 e2 + e3 + e4 . Notice that this 9-container state has the same energy as several other 9-container states; for example, 1, 2, 1, 2, 1, 3, 4, 1, 2 and 4, 1, 3, 1, 2, 2, 1, 1, 2 have the same energy although they are different individual states. What differs among these distinct states is which box occupies which single-box quantum state. The above example illustrates that an energy level of the M-container system can have a high degree of degeneracy because its total energy can be achieved by having the various single-container states appear in various orders. That is, which container is in