Yes, you use the chain rule for both derivatives, but you take the derivative once with R constant, and again with Ω constant. You should now be left with two expressions: one is imaginary, and one is real. Drop the i, and set the two equal together and manipulate to get the answer. Sorry if I...
I was originally correct, but my logic was flawed. One should take the derivative of the functional with respect to z. Then, by the product rule, you have two terms. When R is constant -- the tangential derivative -- one term is eliminated and vice versa. Set the two derivatives equal to each...
Homework Statement
Using f(z) = f(re^iθ) = R(r,θ)e^iΩ(r,θ), show that the Cauchy-Riemann conditions in polar coordinates become
∂R/∂r = (R/r)∂Ω/∂θ
Homework Equations
Cauchy-Riemann in polar coordinates
Hint: Set up the derivative first with dz radial and then with dz tangential...
It would seem that you are correct. Since the spatial equation cancels to zero at the ground state, thus making psi forbidden at that state, the lowest energy is with one fermion in n = 1 and the other in n = 2. Thanks for your cryptic yet insightful response.
That's your old buddy, the chain rule...
Hurky is telling you what to do. Try applying the operators to the psi ket vector to show that they give the same result.
Thanks for the response, but I'm afraid I still don't understand. If I don't know the spin state, how do I know whether to use a symmetric or anti-symmetric spatial equation? Do I have to use them all, multiplying by appropriate spatial equation to make the product anti-symmetric, and then sum...
Homework Statement
Two identical non-interacting spin 1/2 particles are in the one-dimensional simple harmonic oscillator potential V(x) = kx2/2. The particles are in the lowest-energy triplet state.
a. Write down the normalized space part of the wave function.
b. Calculate the energy of...
Homework Statement
Suppose that a Hermitian operator A, representing measurable a, has eigenvectors |A1>, |A2>, and |A3> such that A|Ak> = ak|Ak>. The system is at state:
|psi> = ((3)^(-1/2))|A1> + 2((3)^(-1/2))|A2> + ((5/3)^(1/2))|A3>.
Provide the possible measured values of a and...