130_notes.dvi

As an example, this has been used to compute the cross section forscattering from a spherical potential well (See section 15.7) ...

Here 0< ηℓ<1, with 0 represent complete absorption of the partial wave and 1 representing purely elastic scattering. An in ...

per square centimeter per second to get a cross section. To do this we set the volume to beV= (1 cm^2 )(vrel)(1 second). The rel ...

= Z 1 Z 2 e^2 2 π −i∆ [ − e−( (^1) a+i∆)r 1 a+i∆ + e−( (^1) a−i∆)r 1 a−i∆ ]∞ 0 = Z 1 Z 2 e^2 2 π −i∆ [ 1 1 a+i∆ − 1 1 a−i∆ ] = Z ...

δ 0 (k) =−kr 0 dσ dΩ = 1 k^2 ∣ ∣ ∣eiδℓ(k)sin(δℓ(k))P 0 (cosθ) ∣ ∣ ∣ 2 dσ dΩ = 1 k^2 ∣∣ e−ikr^0 sin(kr 0 ) ∣∣ 2 dσ dΩ = sin^2 (kr ...

31 Classical Scalar Fields Thenon-relativistic quantum mechanicsthat we have studied so far developed largely between 1923 and 1 ...

where theqiare the coordinates of the particle. This equation is derivable from theprinciple of least action. δ ∫t^2 t 1 L(qi,q ...

∂L ∂(∂η/∂t) = μη ̇ −Y ∂^2 η ∂x^2 +μ ̈η+ 0 = 0 η ̈ = Y μ ∂^2 η ∂x^2 This is thewave equation for the string. There are easier way ...

complicated. Theihere should never really be used to multiply aniin the complex wave function, but, everything will work out so ...

This means that ittransforms like a vector. Compare it to our original transformation formula forxμ. x′μ=aμνxν We may safely ass ...

the Schr ̈odinger equation. Writing that relation in the order termsappear in the Klein-Gordon equation above we get (lettingc= ...

We nowtake the transform of both sidesof the equation. ∇^2 φ−μ^2 φ = Gδ^3 (~x) 1 (2π) (^32) ∫ d^3 x e−i ~k·~x (∇^2 φ−μ^2 φ) = 1 ...

the rest of the contour to give zero. That’s easy to do since the integrand goes to zero at infinity on the real axis and the ex ...

k = k′ − ∫∞ 0 k k^2 +μ^2 e−ikrdk+ ∫∞ 0 k k^2 +μ^2 eikrdk = πie−μr ∫∞ 0 k k^2 +μ^2 eikrdk− ∫∞ 0 k k^2 +μ^2 e−ikrdk = πie−μr ∫∞ 0 ...

We have developed a covariantclassical theory for a scalar field. The Lagrangian density is a Lorentz scalar function. We have i ...

32 Classical Maxwell Fields 32.1 Rationalized Heaviside-Lorentz Units TheSI unitsare based on a unit of length of the order of h ...

∇·~ E~ = 4πρ ∇×~ B~−^1 c ∂E ∂t = 4 π c ~j. TheLorentz Forceis F~=−e(E~+^1 c ~v×B~). When we change toRationalized Heaviside-Lore ...

Note that this is automatically antisymmetric under the interchangeof the indices. As before, the first two (sourceless) Maxwell ...

32.3 The Lagrangian for Electromagnetic Fields There are not many ways to make ascalar Lagrangian from the field tensor. We alre ...

Note that, since we have four independent components ofAμas independent fields, we have four equations; or one 4-vector equation ...