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- What is the probability of finding a particle in the last 10% of a one-dimensional box for n = 1 state?
- Find the solution(s) to the differential equations: a. y''(x) + y'(x) - 6y(x) = 0 and b. y''(x) - 4y'(x) + 4y(x) = 0.
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- A particle of mass m is in a harmonic oscillator potential with angular frequency o. It is in the state u(x) = (3u{n=1}(x) + 4u{n=3}(x))/5). a. Give the result of a single measurement of the energy. b. Give the expectation value of the energy. c. Give the expectation value of the momentum. d. Give the result of a single measurement of the momentum. Hint: Average energy is shared equally between the kinetic and potential energy for a simple harmonic oscillator.3. a) According to the spherical shell model, predict J" for the ground state and first excitation of 'Be, ¹70 and 2¹Ne. b) Explain pros and cons of the shell model based on a harmonic oscillator potential of the type: Vo = 1/2 kr².The probability of finding a particle moves in the left quarter of a one-dimensional box of length a for energy level n =2 is: A. 15% В. 20% C. 25% D. 35% E. 50% A C B C c C D C E C
- a.Draw the wave function for a particle in a box at the n-3 energy level. b.Draw the probability distribution for a particle in a box at the n-3 energy level. c.A nanoparticle with mass equal to 15 x 10-27 g exists in a 10 nm one-dimensional box. What is the wavelength of radiation emitted when it decays from the n-3 level to the n- 2 level? For a 1 nm box?a. For the allowed energies of a particle in a box to be large, should the box be very big or very small? Explain.b. Which is likely to have larger values for the allowed energies: an atom in a molecule, an electron in an atom, or a proton in a nucleus? Explain.6. An electron in hydrogen atom is in initial state Þ(r, 0) = A(2410o + iÞ210 + 421–1 – 2ib211) where wnim are the eigenfunctions of the hydrogen atom a. Determine the constant A b. What is the probability of finding the electron in the first excited state? hw = - n2 c. Write the state Þ(r, t) at time t, using energy eigenvalues as En d. Find the expectation value of L in the state Þ(r,t e. Find the expectation values of Lx and Ly in the state (r, t f. If measurement of Lz led to the value –ħ what will be results of measurement of energy and the square of total orbital momentum immediately afterwards and what are their probabilities?
- A finite potential well has a potential of 0 between a = 0 and a = L. For an electron, the difference between the ground state and the outside potential is 3 eV. At what distance to the left of the well does the probability density of the electron drop to 0.0001 its value just inside? a. 0.52 nm b. 1.04 nm c. 0.26 nm d. It will never drop to 0.0001 no matter how far you go to the left of the well. e. The probability density is identically zero outside the well. Which statement regarding the well is true? a. The well has infinitely many energy levels. b. For a sufficiently small nonzero outside potential, there won't be any energy levels left. c. No matter how small the nonzero outside potential, there is at least one energy level in the well.A quantum system has a ground state with energy Eo = 0 meV and a 2-fold degenerate excited state with energy E₁ = 50 meV. E1 Calculate the probability of finding the system in its ground state when it is at T = 300 K. Select one: O a. 0.78 O b. 0.22 O c. 1 O d. 0.87Chapter 39, Problem 025 The two-dimensional, infinite corral of the figure is square, with edge length L = 160 pm. A square probe is centered at xy coordinates (0.100L, 0.900L) and has an x width of 3.00 pm and a y width of 3.00 pm. What is the probability of detection if the electron is in the E, 3 energy state? Probe Number Units
- 6. An electron in hydrogen atom is in initial state p(r, 0) = A(2µ100 + ¡Þ210 + 4Þ21–1 – 2i4211) where ynim are the eigenfunctions of the hydrogen atom a. Determine the constant A b. What is the probability of finding the electron in the first excited state? c. Write the state p(r, t) at time t, using energy eigenvalues as En = d. Find the expectation value of L in the state (r,t e. Find the expectation values of Lx and Ly in the state (r, t f. If measurement of L, led to the value -ħ what will be results of measurement of energy and the square of total orbital momentum immediately afterwards and what are their probabilities? hw n27. One electron is trapped in a one-dimensional square well potential with infinitely high sides. a. If you have a probe that has a width for electron detection Ax = 0.00350L in the x direction, for the first excited state ( n =2), what is the probability that the electron is found in the probe when it is centered at x = L/4, (hint: you can use an approximation for this - you do not need to do an integral)? b. What is the average number of electrons that you would detect using the probe described in part "b." centered at x = L/4, ifthe electron is in the first excited state (n = 2) for each experiment and you repeat the experiment N, =100,000 times?A system with 3 (three) indistinguishable particles has five energy states: E1 = 0, E2 = E, E3 2E, E4 = 3E, Es = 4E. Determine the = possible formation if it satisfies: a. Bose-Einstein stats?