For a two-body collision, the angular distribution of one of the scattered particles is described using the differential
solid angle:
The scattered path of the particle is within a
cone of solid angle Ω, and its axis in the direction (θ, φ).
The differential cross-section dσ/dΩ for N identical scattered particles is given by:[1][2][3]
where
is measured rate of the paricles entering the cone described above, N is the number of particles
is the particle
flux (number of particles passing through unit area per unit time) corresponding to the
number densityn (number of particles per unit volume) and velocity v (in the rest frame of the collision point).
Integrating the equation gives the scattering cross-section:
Ricci calculus Information
show/hide box as a NavFrame Information
Application to one spin particle in three spatial dimensions
For a particle, with spin, in all three spatial dimensions, the wavefunction is
Ket Ψ, ket bases, and orthonormality
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Noether's theorem Information
A simple way to formulate Noether's theorem using Lagrangian mechanics is as follows.[4]
If there is quantity that is a constant of the motion, it means the Lagrangian can be parameterized by a continuous variable and still describe the same motion. Let the continuous parameter be s (such as length, angle of rotation, etc.). Then for L independent of s it immediately follows that:
and
where q is the solution of the Euler-Lagrange equation for s = 0, and Q for any s, that is; q(t) = Q(s = 0, t). Since the same L describes the same motion for all s. By the chain rule:
Using the Euler-Lagrange equation:
gives
and the definition of generalized momentum conjugate to the generalized coordinate Q:
we have
where C is a constant, and A is a quantity that is time independent, and the equation is evaluated at s = 0 for convenience. It follows that A is a conserved quantity, with respect to the parameter s.
In general, for n lots of s parameters, and N generalized coordinates, writing them as the vectors: s = (s1, s2, ... sn), q = (q1, q2, ... qN), Q = (Q1, Q2, ... QN) and also q(t) = Q(s = 0, t), the generalized version is:
where · denotes the
dot product, written explicitly:
i.e. for every continuous parameter that leaves the Lagrangian invariant, there is a component of the conserved quantity. For translational lengths sj = xj, it follows Aj is a component of the system's total linear momentum pj. For angles sj = θj, then Aj is a component of the system's total angular momentum Lj. If s = t, then A is the total energy E of the system.
The covariant Levi-Civita tensor in an n-D metric space may be defined as the unique (up to a sign) n-form (completely antisymmetric order-n covariant tensor) that obeys the relation
The choice of sign defines an orientation in the space.
The contravariant Levi-Civita tensor is an n-vector that may be defined by raising each of the indices of the corresponding covariant tensor in the normal fashion:
The Levi-Civita symbols are defined for an n-dimensional space as +1, −1 or 0 respectively when the indices are an even permutation of the indices of the ordered basis vectors, an odd permutation thereof, or neither.
These symbols do not form the components of a tensor, but are related to the components of the Levi-Civita tensor by a scalar:
In this article, variants of the epsilon glyph are used to distinguish the tensor from the symbol. This notation is not standard.
Recognising other physical quantities as tensors also simplifies their transformation laws. First note that the
velocity four-vectorUμ is given by
Recognising this, we can turn the awkward looking law about composition of velocities into a simple statement about transforming the velocity four-vector of one particle from one frame to another. Uμ also has an invariant form:
So all velocity four-vectors have a magnitude of c. This is an expression of the fact that there is no such thing as being at coordinate rest in relativity: at the least, you are always moving forward through time. The
acceleration 4-vector is given by . Given this, differentiating the above equation by τ produces
So in relativity, the acceleration four-vector and the velocity four-vector are orthogonal.
Momentum in 4D
The momentum and energy combine into a covariant 4-vector:
We can work out what this invariant is by first arguing that, since it is a scalar, it doesn't matter which reference frame we calculate it, and then by transforming to a frame where the total momentum is zero.
We see that the rest energy is an independent invariant. A rest energy can be calculated even for particles and systems in motion, by translating to a frame in which momentum is zero.
The rest energy is related to the mass according to the celebrated equation discussed above:
Note that the mass of systems measured in their center of momentum frame (where total momentum is zero) is given by the total energy of the system in this frame. It may not be equal to the sum of individual system masses measured in other frames.
Force in 4D
To use
Newton's third law of motion, both forces must be defined as the rate of change of momentum with respect to the same time coordinate. That is, it requires the 3D force defined above. Unfortunately, there is no tensor in 4D which contains the components of the 3D force vector among its components.
If a particle is not traveling at c, one can transform the 3D force from the particle's co-moving reference frame into the observer's reference frame. This yields a 4-vector called the
four-force. It is the rate of change of the above energy momentum
four-vector with respect to proper time. The covariant version of the four-force is:
where is the proper time.
In the rest frame of the object, the time component of the four force is zero unless the "
invariant mass" of the object is changing (this requires a non-closed system in which energy/mass is being directly added or removed from the object) in which case it is the negative of that rate of change of mass, times c. In general, though, the components of the four force are not equal to the components of the three-force, because the three force is defined by the rate of change of momentum with respect to coordinate time, i.e. while the four force is defined by the rate of change of momentum with respect to proper time, i.e. .
In a continuous medium, the 3D density of force combines with the density of power to form a covariant 4-vector. The spatial part is the result of dividing the force on a small cell (in 3-space) by the volume of that cell. The time component is −1/c times the power transferred to that cell divided by the volume of the cell. This will be used below in the section on electromagnetism.
EM in 4d spacetime
Maxwell's equations in the 3D form are already consistent with the physical content of special relativity. But we must rewrite them to make them manifestly invariant.[6]
Although there appear to be 64 equations here, it actually reduces to just four independent equations. Using the antisymmetry of the electromagnetic field one can either reduce to an identity (0=0) or render redundant all the equations except for those with λ,μ,ν = either 1,2,3 or 2,3,0 or 3,0,1 or 0,1,2.
The
electric displacement and the
magnetic field are now unified into the (rank 2 antisymmetric contravariant) electromagnetic displacement tensor: