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PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · The formalism of quantum mechanics

The Structure of Quantum Mechanics, Ch Hilbert space and Dirac notation

Problem

In C3\mathbb{C}^3 define ∣u1⟩=12(1,i,0)T|u_1\rangle = \frac{1}{\sqrt2}(1,i,0)^T, ∣u2⟩=12(1,−i,0)T|u_2\rangle = \frac{1}{\sqrt2}(1,-i,0)^T, ∣u3⟩=(0,0,1)T|u_3\rangle = (0,0,1)^T. (a) Verify that {∣u1⟩,∣u2⟩,∣u3⟩}\{|u_1\rangle,|u_2\rangle,|u_3\rangle\} is an orthonormal basis of C3\mathbb{C}^3. (b) The normalised state ∣ψ⟩=13(1,1,1)T|\psi\rangle = \frac{1}{\sqrt3}(1,1,1)^T is given. Find cn=⟨un∣ψ⟩c_n=\langle u_n|\psi\rangle for n=1,2,3n=1,2,3. (c) Check Parseval's identity.

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