Consider a Hilbert space spanned by the following basis set: {l1), 12), 13), (4)}. The Hamiltonian of a system is given

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Consider A Hilbert Space Spanned By The Following Basis Set L1 12 13 4 The Hamiltonian Of A System Is Given 1 (41.24 KiB) Viewed 22 times

Consider a Hilbert space spanned by the following basis set: {l1), 12), 13), (4)}. The Hamiltonian of a system is given by II = 11)(1/+212) (31+23)(21-14)(4]. Let G be some physical observable with the corresponding operator G =i[1/(21 - i|2)(11 + 2/2)(21 - 1/2)(31+ i|3)(21. (a) Find the spectral decomposition of lll. (b) Find the projection operator that projects onto the subspace spanned by the positive energy states. Write it in matrix form. (C) Suppose |) is a normalized state in the above Hilbert space. Consider two operators A = 73 (+14%*), B = + (1 -14%*). Are A and B projection operators? (a) Suppose at time t=0 a measurement of G is performed and yields a negative value. After some time to this measurement is repeated. Will it yield the same value as in the first measurement? Explain why yes or why not.