1. A particle in an infinite well is in the ground state with energy 1.54 eV. How much energy must be added to reach the second excited state (n = 3)? The third excited state (n = 4)?
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- 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?3. An electron is trapped in an infinitely deep potential well 0.300 nm in width. a. If the electron is in its ground state, what is the probability of finding it within 0.100 nm of the left-hand wall? b. Repeat a. for an electron in the 99th excited state. c. Are your answers consistent with the correspondence principle?Q1 V(x) a: The figure to the right shows a potential well where V(X)=0 for 0An electron has a wavelength of 400 nm (the same as blue light). A. What is the electron’s momentum? B. What’s the minimum uncertainty in the electron’s position if its uncertainty in its momentum is 10-28 kg m/s? Compare this value to its wavelength.What constraints does the uncertainty principle place on our knowledge of the world? To get a sense of scale, we’ll look at the uncertainty in speed for a confined electron and for a confined dust particle.a. What range of velocities might an electron have if confined toa 0.30-nm-wide region, about the size of an atom?b. A 1.0@mm@diameter dust particle (m ≈ 10-15 kg) is confined within a 5-μm-long box. Can we know with certainty if the particle is at rest? If not, within what range is its velocity likely to be found?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 n2Q7. The partition function for the Maxwell Boltzmann distribution is given by Z = Li=1. Where N is the number of particles, g is the degeneracy, and ɛ is the energy. a. Write the expression of the parameter ß. b. A system consisting of 6 non-interacting particles possesses 3 energy levels E, = E, E, = 2 €, and Eg = 3 E with degeneracy 3, 2, and 1 respectively. If the system is kept at a constant temperature, write down the partition function of the system c. Given that e= kgT calculate the value of the partition function d. Determine the value of the Free Energy A. e. Determine the Entropy 'S' of the system.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.1. Particle in a Box. A particle of mass m that is confined in a one- dimensional box of length L, i.e. x € (0, L), is described by the wave function: (x, t) = A sin where En = nnx L expli- n²π²ħ² 2m [², Ent ħ where n E N where n E N. The wave function is zero outside the box. Calculate the normalization constant A and compute the uncertainty of position and momentum regardless of the quantum number n.1. Show that the function f(x) = cos(kx) satisfies the classical wave equation. %3! 2. a. Why must an acceptable wave function be single-valued? b. Why does an acceptable wave function have to have a continuous first derivative?2. An electron trapped in an infinitely deep well as a ground state energy of E = 9.0 ev. a. What frequency will be emitted if an electron transitions from the third excited state to its first excited state? b. What is the longest wavelength photon than an excited state of this system can emit? c. Describe the wavefunction of a particle in this infinite potential well. d. What is the width of the well?a. An electron is found to a wavelength of A = 0.65 nm. What is the velocity of the electron? m/s b. A photon has a wavelength of A = 0.65 nm. What is the energy of the photon? JoulesSEE MORE QUESTIONS