A quantum mechanics primer by Daniel T. Gillespie

By Daniel T. Gillespie

During this self-contained and systematic improvement, the writer offers a transparent and concise account of formal quantum mechanics. by means of conscientiously simplifying the idea and mostly ignoring its extra tricky purposes, he conveys a significant standpoint of the quantum thought with out wasting rigour of remedy.

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8. Let |p1 , r1 ; p2 , r2 = c†r1 (p1 )c†r2 (p2 ) |0 be a two-particle state. Find the energy, charge and helicity of this state. Here r1,2 are helicities of one-particle states. 9. 13 satisfy the commutation relation: [Qa , Qb ] = i abc Qc . 10. 17 and calculate the commutators: (a) [Qa , Qb ] , (b) [Qb , π a (x)], [Qb , ψi (x)], [Qb , ψ¯i (x)] . 11. 20 we showed that the action for a massless Dirac field is invariant under dilatations. Find the conserved charge D = d3 xj 0 for this symmetry and show that the relation [D, P μ ] = iP μ , is satisfied.

I) Properties of the matrix T , are given in Chapter 4. One should not forget that time reversal includes complex conjugation: τ (c . )τ −1 = c∗ τ . . τ −1 . Chapter 8. Canonical quantization of the Dirac field 45 • The operator C generates charge conjugation in the space of spinors: Cψa (x)C −1 = (Cγ0T )ab ψb† (x) . J) Properties of the matrix C are given in Chapter 4. The charge conjugation transforms a particle into an antiparticle and vice–versa. • In this chapter we will very often use the identities: [AB, C] = A[B, C] + [A, C]B , [AB, C] = A{B, C} − {A, C}B .

7. The arbitrary state not containing transversal photons has the form |Φ = Cn |Φn , n where Cn are constants and n |Φn = d3 k1 . . d3 kn f (k1 , . . , kn ) (a†0 (ki ) − a†3 (ki )) |0 , i=1 where f (k1 , . . , kn ) are arbitrary functions. The state |Φ0 is a vacuum. (a) Prove that Φn |Φn = δn,0 . (b) Show that Φ| Aμ (x) |Φ is a pure gauge. 8. Let P μν = g μν − and P⊥μν = k μ k¯ν + k ν k¯μ , k · k¯ k μ k¯ν + k ν k¯μ , k · k¯ where k¯ μ = (k 0 , −k). ⊥ ⊥ Calculate: P μν Pνσ , P⊥μν Pνσ , P μν + P⊥μν , g μν Pμν , g μν Pμν , P μ ν P⊥νσ , if k 2 = 0.

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