Homework Statement
Prove, for the matrix ##S = exp
\bigg(-\frac{i}{\hbar}\mathbf{\hat{n}}\cdot \mathbf{\hat{S}}\bigg)## (spin-rotation matrix), and for an arbitary vector ##\mathbf{a}## that:
$$ S^{-1} \mathbf{a} \cdot \mathbf{\hat{x}} S = a(-\theta) \cdot \mathbf{x} = a \cdot...
Problem solved!
The point is that at matter radiation equality, we must have that:
$$ \frac{\rho_{M}}{\rho_{R}} = 1 $$
This does NOT mean that:
$$ \frac{\Omega_{M}}{\Omega_{R}} = 1 $$ Since :
$$ \frac{\rho_{M}}{\rho_{R}} = \frac{\frac{\Omega_{M}}{a^{3}}}{\frac{\Omega_{R}}{a^{4}}} =...
Homework Statement
Show that the time of matter-radiation equality, t_{eq} can be written:
$$ t_{eq} =\frac{a_{eq}^{\frac{3}{2}}}{H_{0}\sqrt{\Omega_{m}}} \int_{0}^{1} \frac{x}{\sqrt{x+1}} dx $$
Homework Equations
$$ t = \int_{0}^{t} dt = \int_{0}^{a} \frac{1}{H(a)} \frac{da}{a} $$ [Given]...
Physically, it doesn't necessarily make sense that there isn't an extra state generated by this process?
Mathematically
- The basis ## \{ |\phi_{1}\rangle,|\phi_{2}\rangle \} ## is a valid basis for any state in a Hilbert space ## H ## of dimensionality 2. Therefore, so long as I know that...
Homework Statement
So we have a two state system, with unperturbed eigenstates ## |\phi_{1}\rangle##, ## |\phi_{2}\rangle ##, and Hamiltonian ## \mathbf{\hat{H_{0}}}## - i.e ##\mathbf{\hat{H_{0}}}|\phi_{1}\rangle = E_{1}|\phi_{1}\rangle##
We shine some z-polarized light on the system. This...
Sorry for the slow reply - I understand what you are saying for the definition of equilibrium - what you say seems intuitively sensible. However, is it not a result that for a given microstate ##j## in the Boltzmann distribution, we have ##p_{j} = \frac{e^{-\beta j}}{Z} ## - so how can the...
Homework Statement
Are the Gibbs and Boltzmann entropies always equivalent?
Homework Equations
$$ S=k_{B}ln\Omega $$ [Boltzmann entropy, where ##\Omega## is the number of available microstates
$$ S=-k_{B}\sum_{i}p_{i} ln(p_{i}) $$ [Gibbs entropy, where ##p_{i}## is the probability of a...
Ah,okay, so lifetimes are generally defined to only involve the Einstein ##A## coefficients.
So I can just ignore completely ##\psi_{A} \rightarrow \psi_{B}##? I wasn't sure the question implied that.... I guess if it does:
$$ \frac{dN_{A}}{dt} = -A_{ac}N_{A} $$
$$ \frac{dN_{B}}{dt} = -A_{bc}...
Unfortunately, it turns out I got that result as a given from another question- I'm not sure where they derived it from exactly! I've double checked, and they are the same formula, so I'm not sure what to do.
Homework Statement
Homework Equations
The Attempt at a Solution
Very confused by this problem. For one thing, it doesn't specify if there is or isn't any light present to drive the stimulated emission/absorbtion. I guess there's no reason to assume that there is no light - but since the...
Homework Statement [/B]
For a general operator ## \hat{O}##, let ##\hat{O}_{mn}(t)## be defined as:
$$ \hat{O_{mn}}(t) = \int u^{*}_{m}(x,t) \hat{O} u_{n}(x,t) $$
and
$$ \hat{O_{mn}} = \int u^{*}_{m}(x) \hat{O} u_{n}(x) $$
##u_{m}## and ##u_{n}## are energy eigenstates with corresponding...