University:
Santa Fe College
Course:
PHY 2004 | Applied Physics 1
Academic year:

2024
Views:

94

Pages:

5
Author:

Jabin F.

# Properties of CT: ∫ −∞ ∞ f(t)δ(t−b)dt=f(b) ∫ −∞ ∞ f(t)δ (n) (t−b)dt=(−1) n f (n) (b) ∫ −∞ ∞ f(t)δ(at−b)dt= ∣a∣ 1 f( a b ) [value of δ(t) doesn't change under CT] ∫ −∞ ∞ f(t)δ(at−b)δ(ct−d)dt= ∣ac∣ 1 f( a b , c d ) Restricted CT (t not explicitly involved in T-relation) Unrestricted CT (t' is present explicitly in T-relation) t ′ =t+h Jacobian for CT is one Jacobian, J: J = |∂t'/∂t| = 1 For spherical: x = r sinθ cosϕ y = r sinθ sinϕ z = r cosθ J = |∂(x,y,z)/∂(r,θ,ϕ)| = r^2 sinθ For cylindrical: x = r cosϕ y = r sinϕ z = z J = |∂(x,y,z)/∂(r,ϕ,z)| = r → In CT, volume/area of phase space remains invariant. → CT forms a group. Note: Any set of variables which satisfy the conditions of CT are called canonical or conjugate variable pairs. P-104 Consider a transformation from one set of generalized coordinates & momentum (q,p) to another set (Q,P) denoted by Q = Q(q,p); P = P(q,p) where α and β are constants. The transformation is canonical if: Solution: If this transformation is canonical, then it follows: {Q, P} = {q, p} ⇒ {λpβ^(-1)(Q), -β^(-1)q} = {q, p} ⇒ -λβ^(-1) = 1 For this to be satisfied: λ = -1, β = -1