342-PS5sol - Solutions to Problem Set 5 Physics 342 by:...

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Physics 342 by: Callum Quigley 1 The Virial Theorem For H = P 2 2 m - e 2 R , we note that [( R · P ) ,H ] = h R i , P 2 2 m i P i + R i h P i , - e 2 R i = i ~ ± P 2 m - e 2 R ² (1.1) and thus, for a stationary state 0 = h d d t ( R · P ) i = 1 i ~ ³ [( R · P ) ,H ] ´ = 2 h T i + h V i (1.2) which implies the desired result. 2 Pion Mass Estimate Given a decay length of 10 - 5 ˚ A, then by the uncertainty relation Δ E ~ c Δ X 2000eV ˚ A 10 - 5 ˚ A = 2 × 10 8 eV (2.1) This gives an estimate of m π 200 MeV, which compares well to the actual value of m π 140 MeV. 3 Magnetic Moments (1) Using the Projection Theorem from Homework 4.5, for = γ 1 ~ J 1 + γ 2 ~ J 2 h μ x,y i = h γ 1 ~ J 1 · ~ J + γ 2 ~ J 2 · ~ J i ~ 2 j ( j + 1) h J x,y i = 0 (3.1) since h J x i = h J y i = 0. However, h μ z i = h γ 1 ~ J 1 · ~ J + γ 2 ~ J 2 · ~ J i ~ 2 j ( j + 1) h J z i = m ~ j ( j + 1) µ γ 1 h J 2 1 i + 1 2 ( γ 1 + γ 2 ) h ( J 2 - J 2 1 - J 2 2 ) i + γ 2 h J 2 2 i (3.2) = m ~ j ( j + 1) µ γ 1 + γ 2 2 j ( j + 1) + γ 1 - γ 2 2 ( j 1 ( j 1 + 1)
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342-PS5sol - Solutions to Problem Set 5 Physics 342 by:...

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