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  1. bananabandana

    Rotation of spin projection different to general rotation?

    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...
  2. bananabandana

    Estimate of time of the universe

    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}}} =...
  3. bananabandana

    Estimate of time of the universe

    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]...
  4. bananabandana

    Why are unperturbed states valid basis for perturbed system?

    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...
  5. bananabandana

    Why are unperturbed states valid basis for perturbed system?

    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...
  6. bananabandana

    Relativistic Lagrangian

    Homework Statement Show that $$ \mathcal{L} = -\frac{1}{4}F^{\mu \nu}F_{\mu \nu} = - \frac{1}{2}\partial^{\mu}A^{\nu}(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}) $$ Where $$ F^{\mu \nu} = \partial^{\mu}A^{\nu} - \partial^{\nu}A^{\mu} $$ Homework Equations The Attempt at a Solution $$...
  7. bananabandana

    Are the Gibbs and Boltzmann forms of Entropy equivalent?

    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...
  8. bananabandana

    Are the Gibbs and Boltzmann forms of Entropy equivalent?

    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...
  9. bananabandana

    Einstein A & B Coefficients

    Oh dear, that's embarrassing.. Ah well, many thanks for the help, much appreciated!
  10. bananabandana

    Einstein A & B Coefficients

    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}...
  11. bananabandana

    Time Evolution of Operators

    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.
  12. bananabandana

    Einstein A & B Coefficients

    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...
  13. bananabandana

    Time Evolution of Operators

    Ah, ## [\hat{A},\hat{H}]## should in fact read ## [\hat{A}, \hat{H}]_{mn}## But otherwise: $$ [\hat{A},\hat{H}]_{mn} = exp\bigg( i \frac{(E_{m} -E_{n})t}{\hbar} \bigg) \int u^{*}_{m}(x) [\hat{A},\hat{H}] u_{n}(x) \ dx $$ $$ = exp\bigg( i \frac{(E_{m} -E_{n})t}{\hbar} \bigg) (E_{n}-E_{m}) \int...
  14. bananabandana

    Time Evolution of Operators

    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...
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