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- 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?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 -xeDetermine the wavelength if the electron is excited n=2 for Helium (Z=2).( E=(-13.6ev)Z2/n2 and 1/mu=1/hc(En-En-1)
- 22 A particle is confined to the one-dimensional infinite poten- tial well of Fig. 39-2. If the particle is in its ground state, what is its probability of detection between (a) x = 0 and x = 0.30L. (b) x = 0.70L and x = L, and (c) x = 0.30L and x = 0.70L? U(x) Fig 39-2The energies in a 2D particle-in-a-box are given by h² 8mL 2 in which the box is a square enclosure with Lx = Ly = L, and nx, ny = 1, 2, 3,... . (a) If the particle is an electron and L = 300 pm (assume three significant figures), find the value of the lowest energy level in units of 10-18 J (that is, if the energy is 5.00 × 10-18 J, you would report it as "5.00"). E n, n (n₂ ² + n₂²) y x y4. Normalize the following wavefunctions 4 55 (a) v(x) = sin (#2); =sin(); for a particle in a 1D box of length L. (b) (2) = xe-z|2 (c) (x) = e(x²/a²)+(ikz) 5. In a region of space, a particle with mass m and with zero energy has a time- independent wave-function (x) = Ae-2/12, where A and L are constants. Use your knowledge of the Schrödinger equation to determine the potential energy V(x) of the particle. Plot the potential function? What is the minimum potential energy for the particle, if it is an electron and L = 1 fm? Is this potential repulsive or attractive?