This is from Modern Cosmology, Scott Dodelson, Chapter 6.
For the part "Show that its energy density dilutes as ##a^{−3}##", following is my attempt:
In the equation ##\frac{\partial \rho}{\partial t} = -3H(P+\rho)##, put ##P = \frac{1}{2} \dot{\phi}^2-V(\phi)## and ##\rho=\frac{1}{2}...
Initially '0' is the upper limit and ##a = \frac{Ze^2}{E}## is the lower limit. With change of variable ##x = \frac{Er}{Ze^2}##, for ##r=0##, ##x=0##, and for ##r=\frac{Ze^2}{E}##, ##x=1##, so 1 should be the lower limit. However, he takes 1 as the upper limit, and without a minus sign. Why is...
(This is from W. Greiner Quantum Mechanics, p. 293 from the topic of Ritz Variational Method)
1) Are ##\frac{\delta}{\delta \psi^{*}}## derivatives in equations 11.35a and 11.35b? If this is so, we can differentiate under the integral sign to get ##\int d^3x (\hat{H}\psi)## in equation 11.35a...
What are ##|\xi>## and ##|\eta>##? Are these single particle states? I don't know where to start since I don't understand the problem statement. The only thing I know is how to write symmetric and antisymmetric wave function. What follows the problem statement in the OP is my attempt.
Problem: A system contains two identical spinless particles. The one particle states are spanned by an orthonormal system ##|\phi_k>##. Suppose that particle states are ##|\phi_i>## and ##|\phi_j>## (##i \neq j##). (a) Find the probability of finding the particle in the state ##|\xi>## and...
In the boxed equation, how would you get the right hand side from the left hand side? We know that ##H(1,2) = H(2,1)##, but we first have to apply ##H(1,2)## to ##\psi(1,2)##, and then we would apply ##\hat{P}_{12}##; the result would not be ##H(2,1) \psi(2,1)##. ##\hat{P}_{12}## is the exchange...
The plot shows how ##P(\omega)##, i.e., the probability of transition from a fixed state ##a## to a fixed state ##b## changes with ##\omega##, the driving frequency. If that is correct, then it should be possible for a given driving frequency ##\omega < \omega_0## to provoke a transition between...