How to apply an algebraic operator expression to a ket found in Dirac’s QM book?

I’ve been trying to learn quantum mechanics from a formal point of view, so I picked up Dirac’s book. In the fourth edition, 33rd page, starting from this:ξ|ξ′⟩=ξ′|ξ′⟩ (Where ξ is a linear operator and all the other ξ′‘s are eigen-(value|ket)s.) ,and this:ϕ(ξ)=a1ξn+a2ξn−1⋯an=0 (where ϕ is an algebraic expression) He has deduced ϕ(ξ)|ξ′⟩=ϕ(ξ′)|ξ′⟩ I understand … Read more

Contradiction of a scalar product

Can anyone resolve this contradiction: →r⋅˙→r=12ddt(→r2)=12ddt(|→r|2)≡12ddt(r2)=r˙r,r=|→r|. But the velocity →v=˙→r has not to be parallel to →r, so actually: →r⋅˙→r=r˙rcos∠(→r,˙→r) What am I doing wrong? Has anyone an idea? P.S. I have this problem from the book “Electromagnetic Theory” from Ferraro (p. 543). Answer The issue here is that ˙r is not the magnitude of … Read more

Lorentz algebra and its generators

I’m reading Maggiore’s book A Modern Introduction to Quantum Field Theory and I’m getting a bit confused when he writes about Lorentz algebra: Ki=Ji0, Ji=12ϵijkJjk, [Ji,Jj]=iϵijkJk, [Ji,Kj]=iϵijkKk. Then he states that Ki is a spatial vector due to the last commutation relation. Is that the way a spatial vector transform under the SO(3) algebra? If … Read more

Normal Vectors to these Hypersurfaces on a Lorentzian Manifold

With respect to the coordinates (x0,x1,x2,x3)=(v,r,θ,ϕ), we have the following components of the metric tensor: [g00g01g02g03g10g11g12g13g20g21g22g23g30g31g32g33] =[1−2Mr100100000r20000r2sin2θ] Suppose we’ve got a family of hypersurfaces defined by v=constant. I’ve been asked to characterize the normal vectors to these hypersurfaces (whether they are timelike, spacelike or null). If I would only know the form of the normal … Read more

Coulomb’s law with an r3r^3, not r2r^2, in the denominator [duplicate]

This question already has answers here: Why do we say that in Coulomb’s law the force is proportional to 1r2 and not 1r3? (2 answers) Closed 6 years ago. I am reading an older physics book that my professor gave me. It is going over Coulomb’s law and Gauss’ theorem. However, the book gives both … Read more

Why are triangles drawn like so when working with gravity on an inclined plane?

This is my first year as a physics student, and I’ve never learned about vectors past a basic level, so this is confusing me. When we have gravity on an inclined plane, we separate it into two components, which I understand. However, consider the image below, and there’s a box at point A. When separating … Read more

Generalization of F=mvdvdx=m2ddx(v2)F=mv\frac{dv}{dx}=\frac{m}{2}\frac{d}{dx}(v^2) to 3-dimensions in a compact notation

Starting from F=ma=mdvdt, in 1-dimension, it is easy to show that F=mvdvdx=m2ddx(v2). Can we generalize this formula in 3-dimensions? In 3D, F=mdvdt ⇒F=m[∂v∂x˙x+∂v∂y˙y+∂v∂z˙z] Is it possible to write (2) in a more compact notation using vector identities? Answer Isn’t this written in hydrodynamics as →F=m(→v⋅∇)→v ? AttributionSource : Link , Question Author : Solidification , … Read more

Functions versus Vectors in Quantum Mechanics

In the beginning quantum mechanics is introduced by representing the states as cute little complex vectors, for example: |a⟩=a+|a+⟩+a−|a−⟩ this is a complex vector representing a state that can collapse in two possible states, with corrisponding probabilities |a+|2,|a−|2. On the other hand observables are represented by hermitian operators, the eigenvalues of those operators are the … Read more

Motion with constant speed and constant acceleration magnitude

I was reading this and this posts. From what I gather In 2D: Constant speed ||˙x||=const and constant positive magnitude of the acceleration ||¨x||=const imply circular motion. In 3D: The same assumptions can give rise to helical motion. Hence my questions: Q1: Suppose x∈Rn depends on time t∈R+. Given the constraints ‖ and \|\ddot x\|=const\neq … Read more

Point charge potential (sign problem)

I’m a bit embarrassed, but I’m not able to compute the electric potential at point P (at a distance R from the origin) generated by a positive unitary point charge in the origin with the right sign. Simply use the definition V(P)=−∫P∞→E⋅d→l, forgetting the constant and choosing a straight line to integrate from infinity (so … Read more