Problem Set 4 Hint 1

Problem Set 4 Hint 1

To deal with the fact that the variables y_i interact with each other, transform to new variables,

w_i = y_i - y_i-1

The constraint on the endpoints then becomes

L
∑
i=1 w_i = Y

If you now write the Hamiltonian in terms of the w_i, the partition function would factor into a product of independent pieces, if it were not for the above constraint on the sum of the w_i.

If you can't figure out how to handle the constraint, here is another hint.

(The above transformation works well for part (b) also.)