8. The overall wave function of the electron in a hydrogen atom is given by 2 3 |2) = ·|2,2,1)| ↑) + 口 |2,2,2)|↓↓) If one were to measure the total angular momentum squared J², what values might one get and what would be their associated probabilities?
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- An electron is in a 3p state in the hydrogen atom, given that the expectation value is 12.5a_0 What is the probability of finding the electron within +/- a_0 of your expectation value. (That is, in the range (r − a_0) < r < (r+a_0) where r is the expectation value from above. The answer should be 0.1991.2. A wave function is a linear combination of 1s, 2s, and 3s orbitals: y(r) = N(0.25 y,s + 0.50W2, +0.30W). Find the normalization constant N, knowing that 1s, 2s, and 3s orbitals are normalized.A Quantum system has a ground state with energy E0 = 100meV and a 3-fold degenerate excited state with energy E1 = 100meV . Calculate the probability of finding the system in its groud state when it is at T = 300 K? a) 0.94 b) 0.06 c) 1 d) 0.98
- 1) An electron is confined to a square box of length L, and the walls of that box are infinitely high. The zero-point energy (ZPE) is defined as the minimal energy that corresponds to the smallest quantum number n. What would be the length of the box L such that the ZPE of the electron located inside this box is equal to its rest mass energy mec2?A quantum system is described by a wave function (r) being a superposition of two states with different energies E1 and E2: (x) = c191(r)e iEit/h+ c292(x)e¯iE2t/h. where ci = 2icz and the real functions p1(x) and p2(r) have the following properties: vile)dz = ile)dz = 1, "0 = rp(x)T#(x)l& p1(x)92(x)dx% D0. Calculate: 1. Probabilities of measurement of energies E1 and E2 2. Expectation valuc of cnergy (E)4. Show that the wave functions for the ground state and first excited state of the simple harmonic oscillator, given by W0 (x) and W1 (x), are orthogonal, where %(x) = Aoe¬max² /2h 4 (x) = A1V m@ -mox² /2h -xe