5.1 Consider a one-dimensional bound particle. Show that if the particle is in a stationary state at a given time, then it will always remain in a stationary state.
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- Estimate the partition function for the hypothetical system represented in Figure 6.3 (attached). Then estimate the probability of this system being in its ground state.5.9. Show that if the operator Aop corresponding to the observable A is Hermitian then (4²) ≥ 02.4. A particle moves in an infinite cubic potential well described by: V (x1, x2) = {00 12= if 0 ≤ x1, x2 a otherwise 1/2(+1) (a) Write down the exact energy and wave-function of the ground state. (2) (b) Write down the exact energy and wavefunction of the first excited states and specify their degeneracies. Now add the following perturbation to the infinite cubic well: H' = 18(x₁-x2) (c) Calculate the ground state energy to the first order correction. (5) (d) Calculate the energy of the first order correction to the first excited degenerated state. (3) (e) Calculate the energy of the first order correction to the second non-degenerate excited state. (3) (f) Use degenerate perturbation theory to determine the first-order correction to the two initially degenerate eigenvalues (energies). (3)
- Plot the first three wavefunctions and the first three energies for the particle in a box of length L and infinite potential outside the box. Do these for n = 1, n = 2, and n = 3In the pair production process, photon energy gets converted to particle/antiparticle pairs. Imagine a single photon in free space, turning into one electron and one positron (antielectron), each with mass mec?. Assume for simplicity both particles move together with equal momenta in the same direction as the original photon as shown, and the photon disappears. Prove this can't happen!