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  1. H

    B Is gravity a force?

    So a photon on the Earth surface experiences an acceleration of 9.81 ##ms^{-2}## too? So its radius of curvature ##r=\frac{c^2}{9.81}\approx10^{16}## m? But what if it travels straight down towards the Earth? It cannot move faster than ##c##, so its acceleration would have to be zero. So given...
  2. H

    B Is gravity a force?

    Could you explain further? If light is massless, then its motion should be unaffected by Newtonian gravity.
  3. H

    B Is gravity a force?

    Gravity can be described not as a force but a curvature of spacetime. I assume this can’t be done to the other 3 fundamental forces. If so, then we cannot treat gravity in a way similar to the other forces. Why then does QFT postulate the existence of gravitons? Why does it attempt to treat...
  4. H

    I Why is p^4 not Hermitian?

    My conclusion is because I use ##p^4=p^2p^2## to arrive at the error expression for ##p^4##, I cannot later use that expression to argue or conclude that ##p^4\neq p^2p^2##. The two terms in the middle simplifies to ##r^2(2-\frac{r}{a})e^{-3r/2a}##, ignoring any constant factor. I used...
  5. H

    I Why is p^4 not Hermitian?

    This is weird because I get the error expression for ##p^4## in post #7 by assuming ##p^4=(p^2)^2##. Ignoring the constant factor ##-4\pi\hbar##, ##\langle f|p^2(p^2g) \rangle=\left.\left( r^2 f\dfrac{d(\hat{p}^2g)}{dr} - r^2\hat{p}^2g\dfrac{df}{dr} \right)\right|_0^\infty+\langle p^2f|p^2g...
  6. H

    I Why is p^4 not Hermitian?

    For ##g=\psi_{100}## and ##f=\psi_{200}##, ##\left.\left( r^2 f\dfrac{dg}{dr} - r^2g\dfrac{df}{dr} \right)\right|_0^\infty=\left.\left(\frac{1}{8\sqrt{2}\pi a^5}r^3e^{-3r/2a}\right)\right|_0^\infty=0##, where ##a## is a constant.
  7. H

    I Why is p^4 not Hermitian?

    Does this mean that a measurement of ##p^2## is only approximately real valued? What does that even mean?
  8. H

    I Why is p^4 not Hermitian?

    ##p^2## is hermitian then. The contradiction still persists.
  9. H

    I Why is p^4 not Hermitian?

    All operators for observables must be hermitian. If ##\hat{p}^4## is not hermitian, then what would you obtain when you measure ##p^4## or ##E^2##? Would you get complex-valued measurements? What would it mean?
  10. H

    I Why is p^4 not Hermitian?

    c(r) has the factor ##r^2##, so it is exactly 0 at r=0.
  11. H

    I Why is p^4 not Hermitian?

    Everything involved is always differentiable. Taking ##f=\psi_{200}=(2-\frac{r}{a})e^{-r/2a}## and ignoring all constant factors, $$r^2\frac{df}{dr}=(4r^2-\frac{r^3}{a})e^{-r/2a}$$ $$\frac{1}{r^2}\frac{d}{dr}r^2\frac{df}{dr}=(\frac{r}{a}-\frac{2}{a}+\frac{16}{r})e^{-r/2a}$$
  12. H

    I Why is p^4 not Hermitian?

    Yes the boundary term does not vanish, indeed. But how can this be? All the hydrogen radial wave functions are infinitely differentiable. Doesn't it contradict the theorem?
  13. H

    I Why is p^4 not Hermitian?

    It is, because no matter how many times you differentiate ##e^{-r}##, multiplying and dividing by ##r^2##, in whatever order of these 3 operations, the result is still proportional to ##e^{-r}##.
  14. H

    I Why is p^4 not Hermitian?

    Yes they are since hydrogen radial wave functions all have the ##e^{-r}## term.
  15. H

    I Why is p^4 not Hermitian?

    ##p^2## is hermitian does indeed hold for all states, including ##l=0##, but not ##p^4##.
  16. H

    I Why is p^4 not Hermitian?

    ##p## is the momentum operator as seen from [6.52]. ##p^4=(p^2)^2## and ##p^2## is hermitian as seen from [6.51].
  17. H

    I Why is p^4 not Hermitian?

    He is saying ##p^2## is hermitian but ##p^4## is not for the same pair of states, f and g, both with ##l=0##.
  18. H

    I Why is p^4 not Hermitian?

    Intro to QM, David Griffiths, p269
  19. H

    I Why is p^4 not Hermitian?

    Why is ##p^4## not hermitian for hydrogen states with ##l=0## when ##p^2## is? Doesn't this contradict the following theorem?
  20. H

    I How to collapse a water wave through a double slit into particle behaviour?

    I would have thought all measurements, in one way or another, involve microscopic particles and are hence quantum in nature. If there are two kinds of measurements—one, quantum and the other, classical—then how do you tell them apart properly?
  21. H

    I Inadequate proof of Bloch's theorem?

    I guess you meant this part: All these I understand, but it does not mention how the condition ##\psi(x+a)=e^{iKa}\psi(x)## is motivated. Be aware that it is only by first assuming this condition to be true that we could get ##\psi(\vec{x}+N a)=e^{iKNa}\psi(\vec{x})##. If we start by not...
  22. H

    I How to collapse a water wave through a double slit into particle behaviour?

    So is it just pure coincidence that the double-slit interference for light and electrons are described by the same equation as the classical water-wave interference?
  23. H

    I How to collapse a water wave through a double slit into particle behaviour?

    Could you explain quantum coherence in simple terms? A water wave of wavelength ##\lambda## should behave as a particle with momentum ##p=\frac{h}{\lambda}##. How can we observe this particle? And how can we collapse the wave function of this particle so that it passes through either the left...
  24. H

    I How to collapse a water wave through a double slit into particle behaviour?

    Electrons passing through a double slit is in a superposition of passing through the left slit and the right slit, thereby producing an interference pattern on the screen. But when a detector is placed to detect which slit the electrons pass through, the interference pattern is destroyed. How...
  25. H

    I Inadequate proof of Bloch's theorem?

    Since you understood the argument with the Wronskian, could you explain to PeterDonis that it doesn't impose the following condition on ##K##: ##e^{i K a} = e^{- i K a}##? (Note that for the case of the free particle, K=k. K is defined by the Bloch's condition [5.49], while k is defined by the...
  26. H

    I Inadequate proof of Bloch's theorem?

    It is not a stationary state. But this is not a counter example because [5.48] is the time-independent Schrodinger solution. There exists an allowed value of k such that a solution to the time-independent Schrodinger equation is not a solution to the Bloch's condition. Could you explain how the...
  27. H

    I Inadequate proof of Bloch's theorem?

    There exists an allowed value of k such that a solution to the time-independent Schrodinger equation [5.48] is not a solution to the Bloch's condition [5.49], but Griffiths said (according to @PeterDonis) this is impossible: all solutions to the Schrodinger equation is a solution to the Bloch's...
  28. H

    I Inadequate proof of Bloch's theorem?

    ##\psi_{\lambda_1}## and ##\psi_{\lambda_2}## have the same E but different ##\lambda##, same eigenvalue for H but different ones for D! (Different ones in general: ##\lambda=\pm1## are the exceptions.) This is proven by B&J: YES!!! YES!!! YES!!! This is what I have been saying several times...
  29. H

    I Inadequate proof of Bloch's theorem?

    Ok, let's hear the opinion of more people first. [5.48] is Schrodinger equation. [5.49] is Bloch's theorem or Bloch's condition. [5.62] is in post #65: You are right that we need to take into account the Dirac-delta potential at ##x=0## and ##x=a##. But it turns out that [5.59] is still the...
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