# ps7 - Harvard University Physics 143b: Quantum Mechanics II...

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Harvard University Physics 143b: Quantum Mechanics II Problem Set 7 due Friday November 19 We use units with ¯ h = c = 1 below. 1. Consider the Dirac equation for a particle of mass m in one spatial dimension - ∂x + βm ! Ψ = i Ψ ∂t (1) (a) Show that there are solutions for Dirac conditions using 2 × 2 matrices α and β , so that Ψ is a two component wavefunction. Use the Pauli matrices as solutions β = σ z and α = σ x . (b) Find plane wave solutions with energy E = k 2 + m 2 with Ψ k, + ( x,t ) = c 1 c 2 ! e ikx - iEt (2) Determine c 1 and c 2 so that | c 1 | 2 + | c 2 | 2 = 1. (c) Similarly ﬁnd the solution Ψ k, - with energy E = - k 2 + m 2 . (d) Take the non-relativistic limit of the above solutions, | k | ± m . Show that only one component of the wavefunction is large in this limit. (e) Write down the form of the solutions in the massless limit, m = 0, where the particles travel at the velocity of light. (Neutrinos are examples of such (nearly) massless particles). 2. Electrons in graphene are described by the two-dimensional Dirac equation with mass

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## This note was uploaded on 02/15/2011 for the course PHYS 143B taught by Professor Unknow during the Spring '10 term at Harvard.

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ps7 - Harvard University Physics 143b: Quantum Mechanics II...

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