20 1911)} Problem 4. An electron in the hydrogen atom is described by the following wave function: % = Ce são {2e holt)

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20 1911 Problem 4 An Electron In The Hydrogen Atom Is Described By The Following Wave Function Ce Sao 2e Holt 1 (156.99 KiB) Viewed 18 times

20 1911)} Problem 4. An electron in the hydrogen atom is described by the following wave function: % = Ce são {2e holt) – zao -sin o sin ao Here 1) and 1) denote spin states with the positive and negative projection on the z-axis, | 4Toha do = mee, and C is a constant. Determine which values and with what probabilities will be obtained if we measure the following quantities (L, S, and J are the orbital, spin, and total angular momentum, respectively): (a) Energy (b) L2 (c) LZ (e) Sz (f) J, (d) S2 Problem 5. Consider a Hilbert space spanned by the following basis set: {(1), (2),|3),4)}. The Hamiltonian of a system is given by ỘT = 11){1}+2(2)(31+2|3) (21 – 14) (4). Let G be some physical observable with the corresponding operator Ĝ = i|1)(21 - i|2) (1| + 2/2) (21 – 1|2) (3| +i|3) (21. (a) Find the spectral decomposition of ÂU. (b) Find the projection operator that projects onto the subspace spanned by the positive energy states. Write it in matrix form. (c) Suppose | 1) is a normalized state in the above Hilbert space. Consider two operators В (Î – | )() V2 Are  and Ể projection operators? (d) Suppose at time t=0 a measurement of G is performed and yields a negative value. After some time t >0 this measurement is repeated. Will it yield the same value as in the first measurement? Explain why yes or why not. À = (1+l"}(vl), V2