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Solve the Schrödinger equation for the potential V(x) = |x| and find the eigen values.
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- By taking the derivative of the first equation with respect to b, show that the second equation is true. Use this result to determine △x and △p for the ground state of the simple harmonic oscialltor.Solve the time-independent Schrödinger equation and determine the energy levels and the wave function of a particle in the potential a? V (x) = Vol a + 2r2 with a = const.Show that ? (x,t) = A cos (kx - ?t) is not a solution to the time-dependent Schroedinger equation for a free particle [U(x) = 0].
- You are given a free particle (no potential) Hamiltonian Ĥ dependent wave-functions = -it 2h7² m sin(2x) e = V₁(x, t) V₂(x, t) 2 sin(x)e -ithm + sin(2x)e¯ What would be results of kinetic energy measurements for these two wave-functions? Give only possible outcomes, for example, it is possible to get the following values 5, 6, and 7. No need to provide corresponding probabilities. ħ² d² 2m dx2 and two time- -it 2hr 2 mThe following Eigen function is a typical solution of the time-independent Schrödinger equation and satisfies boundary conditions for a particle in a confined space of a certain length. y(x) = sin (~77) (a) Plot the wave function as a function of x for L = 30 cm and n = 1, 2, 3 and 4. Note: You will need to have 4 plots in the same graph. (b) On a separate graph, plot the probability density (112) as a function of x using the conditions specified in part (a). Note: You will need to have 4 plots in the same graph. (c) Report your observations for parts (a) and (b)What is the solution of the time-dependent Schrödinger equation (x, t) for the total energy eigenfunction 4(x) = √2/a sin(3mx/a) for an electron in a one-dimensional box of length 1.00 x 10-10 m? Write explicitly in terms of the parameters of the problem. Give numerical values for the angular frequency and the wave- length of the particle.
- consider an infinite square well with sides at x= -L/2 and x = L/2 (centered at the origin). Then the potential energy is 0 for [x] L/2 Let E be the total energy of the particle. =0 (a) Solve the one-dimensional time-independent Schrodinger equation to find y(x) in each region. (b) Apply the boundary condition that must be continuous. (c) Apply the normalization condition. (d) Find the allowed values of E. (e) Sketch w(x) for the three lowest energy states. (f) Compare your results for (d) and (e) to the infinite square well (with sides at x=0 and x=L)A 1.2 eV electron has a 10^-4 probability of tunneling through a 2.3 eV potential barrier. What is the probability of a 1.2 eV proton tunneling through the same barrier?