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- The Hamiltonian operator y for the harmonic oscillator is given by  = h? d? + suw? â?, where l is the reduced mass and w = 27V is the angular 2µ dæ? P[-] pwx? frequency. Is the function f(x) = x exp an eigenfunction of that Hamiltonian? 2h O No. Yes, and the eigenvalue is ħw. 2 Yes, and the eigenvalue is hw. 3ħw Yes, and the eigenvalue is 4What is the eigen value when the eigen function e* is operated on the operator d" | dx" ? (А) а " (B) а" (C) a-*n (D) na1:51 PM Sun Oct 9 -1/2 5) For a particle on a ring with the wavefunction: () = π kinetic energy of the particle using the operator K = -1/2 Y(0) = IT 21T K=-ħ²d² 21d² 2 cos ($) = kdo T (17¹/2)² 211 - f = "Cos (0) . # de TỈ Cos (O). t d. TỈ Cosco) do = 21 0 21 cos() calculate the average cos(0). -ħ² d² do 21 do² 10:52 @ 79% + : 5 9 +
- 1-37. A proton and a negatively charged u meson (called a muon)can form a short-lived species called a mesonic atom. The charge of a muon is the same as that on an electron and the mass of a muon is 207 me. Assume that the Bohr theory can be applied to such a mesonic atom and calculate the ground-state energy, the radius of the first Bohr orbit, and the energy and frequency associated with the n = 1 to n = 2 transition in a mesonic atom.Find the uncertainty ox in x as a function of the uncertainties ou and oy in u and v for the following functions: (u+v) (а) х %3 2 (b) x =uv? (c) x= u? +v?(b) A system with chemical potential u = 12.5 meV contains identical particles each with integer spin and has a single quantum state at energy E, = 0 meV. The temperature of the system is fixed at 290 K. To three significant figures, what is the ratio Pg/P1, where P is the probability of finding a single particle in the state Eo, and P3 is the probability of finding 3 particles in this state? P3/P (Input your answer as a number.)
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- Which of the following functions can be normalized (in all cases the range for x is from x = −∞ to ∞, and a is a positive constant): (i) e-ax^2; (ii) e–ax. Which of these functions are acceptable as wavefunctions?4. Measurement momentum yield (A) +1 ħ. (C) 0 ħ. of the z component of angular for an electron in the 2px state will (B) (D) -1 ħ. ±1 h.Prove the following identities. (a) [t, p²] = 2ihp (b) [p, ²] = -2ihâ