I tried messing around with the formulas, but honestly I don't really know what I'm doing. I tried to find P(a_jp_j^2,b_jq_j^2) from P(q,p), but I don't know what to do with the d^{sN}q\,d^{sN}p left over... and then, assuming I do manage to find the correct distribution formula, I don't know...
Homework Statement
For a Hamiltonian of the form H=\sum_{j=1}^m a_j p_j^2 + \sum_{j=1}^n b_j q_j^2 + H'(q_{n+1}, \dots, q_{sN}, p_{m+1}, \dots, p_{sN}) (for a system with n coordinates and m momenta, s degrees of freedom and N particles), show that \overline{a_jp_j^2}=\frac{kT}{2}, for...
Is "order" = "size"?
I'm really confused... we refer to the "size" of a group as the "order"... But are they really equivalent? Can we prove that simply?
Example:
Say we have a group G and a subgroup H. Then take one of the cosets of H, say Hx... It has order = o(G)/o(G/H). But does this...
Homework Statement
a) Let H be a normal subgroup of G. If the index of H in G is n, show that y^n \in H for all y \in G.
b) Let \varphi : G \rightarrow G' be a homomorphism and suppose that x \in G has order n. Prove that the order of \varphi(x) (in the group G') divides n. (Suggestion: Use...
Homework Statement
Consider the angular momentum operator \vec{J_{y}} in the subspace for which j=1. Write down the matrix for this operator in the usual basis (where J^{2} and J_{z} are diagonal). Diagonalize the matrix and find the eigenvalues and orthonormal eigenvectors.
Homework...
Homework Statement
Show that the inner product of the Pauli matrices, σ, and the momentum operator, \vec{p}, is given by:
σ \cdot \vec{p} = \frac{1}{r^{2}} (σ \cdot \vec{r} )(\frac{\hbar}{i} r \frac{\partial}{\partial r} + iσ \cdot \vec{L}),
where \vec{L} is the angular momentum operator and...