Download A Guided Tour of Mathematical Physics by Roel Snieder PDF

Mathematical Physics

By Roel Snieder

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However, particles with a magnetic charge (usually called “magnetic monopoles”) have not been found in nature despite extensive searches. 13) but we should remember that this divergence is zero because of the observational absence of magnetic monopoles rather than a vanishing coupling constant. 4 The divergence in cylinder coordinates In the previous analysis we have only used the expression of the divergence is Cartesian coordinates: ∇ · v = ∂x vx + ∂y vy + ∂z vz . As you have (hopefully) discovered, the use of other coordinate systems such as cylinder coordinates or spherical coordinates can make life much simpler.

5) for the definition of the geometric variables. 5: Definition of the geometric variables for the computation of the curl in spherical coordinates. 5) the little surface. When we carry out the line integral along the surface we integrate in the direction shown in the figure. The reason for this is that the azimuth ϕ increases when we move into the figure, hence ϕ ˆ point into the figure. Following the rules of a right-handed screw this corresponds with the indicated sense of integration. The area enclosed by the contour is given by rdθdr.

More accurately, one can formulate this as the flux of the probability density current. 15) that the probability density current J is given by: J= i¯h (ψ∇ψ∗ − ψ∗ ∇ψ) 2m Pay in particular attention to the sign of the terms in this expression. 4. 17) where k is the wavevector and A an unspecified constant. Problem f: Show that the wavelength λ is related to the wavevector by the relation λ = 2π/ |k|. In which direction does the wave propagate? Problem g: Show that the probability density current J for this wavefunction satisfies: J= ¯hk 2 |ψ| .

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