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  1. M

    Question about spacetime quantization

    If three-dimensional space were quantized then you would still need three numbers to describe a location in it. Essentially you would be changing your space from \mathbb{R}^3 (triplets of real numbers) to \mathbb{Z}^3 (triplets of integers). In some sense you are right that there are fewer...
  2. M

    Limit as distinguishable->indistinguishable

    I have a question that I sort of feel should be easy to answer, but I haven't figured out how yet... Hopefully someone can either show me the easy answer or tell me that it's a little more subtle :) Consider a system of two particles, distinguishable by some continuous parameter, which can...
  3. M

    Light and Mass

    Yes, the energy in a photon does curve spacetime, although really not very much :). The" [Broken] on the right-hand side of Einstein's equation G_{\mu\nu}=8\pi T_{\mu\nu} describes how the existence and flow of both matter and energy curve...
  4. M

    Solving for spacetime

    I have restated a more specific version of the problem in post #10... Does it make the problem I little more clear? I'm a bit confused by this - don't the paths of photons in a flat spacetime have to be straight lines (in the Euclidean sense) in both 3 and 4 dimensions? This is the...
  5. M

    Solving for spacetime

    That's the original reason for my post - I first thought it would be easy, then impossible, then wasn't sure so came here :) . That is essentially what I want to do. Parametrizing a curve in 4D isn't a problem, but requiring the arbitrary curve to be a geodesic is the part at which I got a...
  6. M

    Dirac Proves 0 =1

    It does appear to be a proof by contradiction, at least for observables. If A is not Hermitean then \langle a|A=(A^\dagger|a\rangle)^\dagger\neq (A|a\rangle)^\dagger, so acting A to the left in the term \langle a|AB|a\rangle doesn't yield a\langle a|B|a\rangle, as required to obtain the...
  7. M

    Solving for spacetime

    Pervect I agree with you in general, but what is bothering Hurkyl was what was also bothering me as I posted :smile: . My original idea was to find a way to say, for example, that light was to follow a parabola. I would then have a curve in 3D like \gamma(\lambda)=(\lambda^2,0,0) and want to...
  8. M

    Solving for spacetime

    Hi all, I would like to write down the arc-length parametrization of a curve in 3-dimensional Euclidean space, \gamma(\lambda), then specify that in a certain spacetime this path is a null geodesic z^\mu and solve for the metric of that spacetime. My first question is, does this even make...
  9. M

    Dirac Proves 0 =1

    Except you can always find such A and B, so you can always find 0=1... :) It's just the definition of the commutator and linearity of the inner product: \langle a | [A,B] | a\rangle = \langle a | (AB-BA) | a \rangle = \langle a | AB | a \rangle - \langle a | BA | a \rangle. Physics...
  10. M

    Are stationery states eigenstates?

    George, do you know of any good reference books (or articles or websites) on this topic that would be accessible at the early graduate level? I'm interested in learning about Gelfand triples in general from a mathematical point of view, and specifically how they allow us to use delta...
  11. M

    Electric field operator (and the HOM dip)

    I'm trying to gain a better understanding of how the electric field operator is used and what it can do. I know that calculating its expectation value tells you that a coherent state is the 'most classical' quantum state of light, and the number states have zero average electric field. The...
  12. M

    Four-vectors, Minkowsky spacetime.

    These basically apply to three different possibilities for the spacetime distance between two events: positive, negative or zero. Lightlike (or null separated) events have zero distance between them; how timelike and spacelike match up with positive and negative depends on the signature of your...
  13. M

    1st & 2nd Postulates

    However, the point of a postulate is that it must be true by assumption (at least so long as you are working within the system it is helping to define). If you take it as true, then it turns out (as you'll learn in your course) that what isn't true is you're assumption that the speed of the...
  14. M

    How is relativity theory applied to gps?

    Yes, time dilation causes the satellites' clocks to lose about 7us per day, and gravitational redshift causes them to gain about 45us. The net effect is a gain of 38us per day which, as Russ said, is accounted for by adjusting the frequency of the clocks when they are on the ground. Check...
  15. M

    Relativity in rotating frames

    In case anyone's still interested in this... I tracked down the two papers above in the campus library and will attempt a quick summary here. If there's interest I could post more complete derivations or possibly scan the papers... Takeno uses a group-theoretic approach based on a few...
  16. M

    Fermi-Walker transport - problem with a minus sign!

    True enough... So I guess that solves my immediate problem, thanks! Seems to introduce a potential future one though... This implies that the DE for Fermi-Walker transport which I've seen many places online, and in MTW, is dependent on the metric of the lab frame, is that right? So if I...
  17. M

    How I would go about becoming a Quantum Physicist

    In general, you aren't going to specialize to anything beyond 'physics' at the undergraduate level. You can choose option courses at the senior level that will be more towards the area you are interested in, but you'll end up with the same B.Sc. in Physics as people who want to do general...
  18. M

    Maple 10, have you tired it yet?

    I have used Maple 10 briefly on my university's servers. I switched back to using 9.5 after only a couple of attempts at 10, which was unbearably slow. I'm sure this is at least partly due to the fact that I'm running it through X tunneling over our university LAN, but it is significantly...
  19. M

    Fermi-Walker transport - problem with a minus sign!

    I am looking at the Fermi-Walker transport of a tetrad transported by an observer in circular motion in Minkowski space. The 0-component of the tetrad should be the 4-velocity of the observer, which should therefore satisfy the FWT DE, but I'm finding that it is equal to the negative of what it...
  20. M

    A question about Dirac's quantization

    This is essentially what I said in my very first post in this thread, without using the language of 'constraint surfaces' - that while the functions \phi_m are required to be zero at the actual constraints, they are not everywhere zero so their derivatives may be non-zero and must therefore be...
  21. M

    A question about Dirac's quantization

    Oops, I did at that... Although, I did only divide by a weak zero :) The example did help, thanks. I think I now know where Daniel was coming from. I'm still not sure about the original question though, as to how/why Dirac used the weak inequality. This is because from your example it...
  22. M

    A question about Dirac's quantization

    I accept that you've given a definition of a weak equality... I just still am not sure that it is the same as the one Dirac uses in his book. He specifically states that the notation is solely as a reminder, not some mathematical formalism. This is the part that confuses me - I thought that...
  23. M

    A question about Dirac's quantization

    That doesn't seem to be exactly what Dirac is getting at in his book. He says: So he actually has the constraints weakly equal to zero, not functions evaluated on surfaces of all constraints. He also explicity says that while the constraints are equal to zero, this substitution must not be...
  24. M

    A question about Dirac's quantization

    I've been slightly confused by this too, but my understanding of it is that while the \phi_m may equal zero, their derivatives perhaps don't... For this reason then, the quantity \left[g,\phi_m\right] (e.g. in equations 1-15, 1-17 to 1-19) would be trivially zero if you plug in \phi_m=0...
  25. M

    Fermi normal coordinates

    I'm following the derivation of Fermi coordinates in MTW, section 13.6. Equation 13.60 states \mathbf{\nabla_ue}_{\hat{\alpha}} = -\mathbf{\Omega\cdot e}_{\hat{\alpha}} where \Omega^{\mu\nu} is antisymmetric (and \hat{\alpha} is the tetrad label). My question is, over which index is the...
  26. M

    Reduced Density Matrix

    I agree with this if someone makes a measurement on B but whoever is examining A is unaware of the result (or fact that measurement was made). That way Alice can't obtain any information from Bob when he makes a measurement on B far away from A. However, if you know the result then you know...
  27. M

    Reduced Density Matrix

    No worries Ron, glad to be able to help... Now, from a physical interpretation (rather than purely mathematical since Daniel already provided that) the reduced density matrix is really just the density matrix for the subsystem you are looking at, given that you don't know anything specific...
  28. M

    Relativity in rotating frames

    I think you've misunderstood the point of my question... I'm trying to examine a system in a rotating frame from the assumption that GR is correct, not attempting to disprove the theory. I'm hoping to find a coordinate transformation from a Minkowski spacetime to a spacetime rotating about...
  29. M

    Relativity in rotating frames

    I've looked at making that substitution and arrived at the same conclusion... The substitution seems to assume rigid rotation though, and since there isn't really any absolute concept of rigidity in GR I thought there might be some other method, perhaps a retarded rotation along the lines of...
  30. M

    Relativity in rotating frames

    Perhaps the gamma factor is too closely linked to special relativity to make sense in all non-inertial frames, but doesn't GR say that all reference frames, inertial or otherwise, are equally valid? That c is constant in all of them, with nothing able to exceed that speed in any frame?
  31. M

    Relativity in rotating frames

    Suppose there is an inertial frame S in which there exists some object A at rest, located at (x,y,z)=(10^8,0,0). Now consider the non-inertial frame S' whose axes are coincident with those of S at t=0, but which is rotating about the common z-axis with constant angular frequency w. If S' has a...
  32. M

    Electrical charge and Mass

    No, the neutron has roughly the same mass as the proton but zero charge...
  33. M

    Maxwell's Equations

    The relevant Maxwell's equations in a vacuum are: \nabla\times \mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t} (Faraday's law) \nabla\times\mathbf{B} = \mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t} (Ampere's law) So from Faraday's law you can see how a changing magnetic...
  34. M

    Scalar Operator

    The mathematical statement is that if \hat{T} is an operator such that [\hat{T},\hat{L}_x]=[\hat{T},\hat{L}_y]=[\hat{T},\hat{L}_z]=0 then \hat{T} is a scalar operator. So to show that an operator is a scalar operator, you just have to show that those three commutators are zero. Mike
  35. M

    Observables in Quanum Mechanics

    I was pretty sure... I know that in the Schroedinger picture using wave mechanics the position operator is equal to x. I know that the time derivative of x is equal to dx/dt. Why then would the time derivative of the position operator not also be dx/dt? Which part should I reconsider...
  36. M

    Observables in Quanum Mechanics

    Yes, there's no reason you can't differentiate an operator. For example, in position space the position operator is just \hat{X}=x. So then you can find the time derivative of it as: \frac{d\hat{X}}{dt} = \frac{dx}{dt} = v.
  37. M

    Yeah I'm a dummy I know

    There are many different ways for two systems A and B to be entangled, but suppose we have them so that when A is spin up, B will be spin down. At first you don't actually know anything about the state of either one, and they could each be in any superposition of up and down (i.e. some percent...
  38. M

    Commutation (Ehrenfest?) relations

    I think I have figured out what is happening... sort of. Much earlier in the text (p22) the authors state that \mathcal{U}^\dagger\left[\alpha_j, \alpha_k\right]\mathcal{U} = \mathcal{U}^\dagger i\hbar\left[\alpha_j, \alpha_k\right]_{\text{QPB}}\mathcal{U} where QPB means Quantum Poisson...
  39. M

    Commutation (Ehrenfest?) relations

    This section is in Chapter 2, Geometrical Symmetries in Ordinary Quantum Mechanics. Unfortunately there are references for the chapter for sections 2.2 (Transformation Laws of Coordinates and Momenta in Nonrelativistic Classical Mechanics), 2.4 (Space Rotations) and 2.6 (Space Reflection) but...
  40. M

    Can the history of a particle be traced in principal?

    If I give you three particles and tell you that two of them are entangled, I don't think that there is any experiment you can perform which will tell you with certainty which two they are. It would seem to me then that similarly if you had a collection of atoms and a photon emerged from them...
  41. M

    Commutation (Ehrenfest?) relations

    I'm following a derivation (p85 of Symmetry Principles in Quantum Physics by Fonda & Ghirardi, for anyone who has it) in which the following assertion is made: "...we have \left[\mathcal{G}_p,\mathbf{r}_i\right] &=& \mathbf{v}_0t\mathcal{G}_p, \left[\mathcal{G}_r,\mathbf{p}_i\right] &=&...
  42. M

    Lorentz force in two frames

    If all you have is a charged particle and an external magnetic field, the particle will move in a circle (or along a helix, which looks like a circle when viewed along its axis) if it has a non-zero velocity in a direction other than along a magnetic field line. However, if you now add some...
  43. M

    Help with this partial derivative problem

    Your problem seems to be the fact that \frac{\partial^2z}{\partial x\partial y} \neq \frac{\partial z}{\partial x}\frac{\partial z}{\partial y}. On the left-hand side (what you want to find), you are taking the partial derivative with respect to x of the partial derivative of z with respect...
  44. M

    Squeezed field modes

    Hey Phil, I've been trying to read about this too and have had the following books recommended: Birrell & Davies, Quantum fields in curved space Fulling, Aspects of quantum field theory in curved space-time Bjorken & Drell, Relativistic quantum mechanics Bjorken & Drell, Relativistic...
  45. M

    Proof by logic

    HallsOfIvy's method gives you: p = n^2 is divisible by 3 q = n is divisible by 3 if NOT q then NOT p It doesn't, but it proves if p then q... Which is what you originally wanted, no?
  46. M

    Need some help with basic complex variables (no proofs)

    Arg z is defined to be the angle between -pi and pi which is equivalent to the actual argument of z. So if z=e^{i\pi}\implies z^2=e^{2i\pi} then Arg z = arg z = pi, but arg z² = 2pi so Arg z² = 0. If I recall correctly a domain has to be connected, so something like...
  47. M

    Balls - Projectile

    The balls are released from rest, which tells you that their initial velocty V_{y_0} is zero. If you plug in t=0 you just get y=y_0, or 5.0 m, which just tells you that the balls do in fact start where they start (always a good check :smile: ). What you want to do to solve the problem is...
  48. M

    Interesting proof

    The link Daniel provided states that R_{n\ell}(r) = r^\ell \exp\left(\frac{zr}{na}\right) \sum_{j=0}^{n-\ell-1} b_jr^j. The sum is simply the definition of a polynomial of degree n-\ell-1, which the fundamental theorem of algebra guarantees will have exactly n-\ell-1 complex roots (not...
  49. M

    Basic qm derivation

    Try simply making the substitution x\to-x in the SWE, then using the fact that V(-x)=V(x). The new form should then show directly that \psi(-x) is a solution as well, since it satisfies the wave equation.
  50. M

    Converging and diverging lens

    After the rays pass through the lens, do they meet at a single point or spread away from each other? In general, a convex lens will cause them to converge to a point on the opposite side of the lens as the object, whereas a concave lens will cause them to diverge away from a point on the same...