Problem 2.8 A particle of mass m in the infinite square well (of width a) starts out in the left half of the well, and is (at t = 0) equally likely to be found at any point in that region. (a) What is its initial wave function, (x, 0)? (Assume it is real. Don't forget to normalize it.) (b) What is the probability that a measurement of the energy would yield the value ²ħ²/2ma²?
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Q: Problem 2.8 A particle of mass m in the infinite square well (of width a) starts out in the state…
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Q: Please, I want to solve the question correctly, clearly and concisely
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Q: Question 2 2.1 Consider an infinite well for which the bottom is not flat, as sketched here. If the…
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Q: 5.2 Consider a one-dimensional bound particle Show that fo ; * 4* (x, t) ¢ (x, t) dx = 0. dt need…
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- Problem 2.13 A particle in the harmonic oscillator potential starts out in the state ¥ (x. 0) = A[3¥o(x)+ 4¼1(x)]. (a) Find A. (b) Construct ¥ (x, t) and |¥(x. t)P. (c) Find (x) and (p). Don't get too excited if they oscillate at the classical frequency; what would it have been had I specified ¥2(x), instead of Vi(x)? Check that Ehrenfest's theorem (Equation 1.38) holds for this wave function. (d) If you measured the energy of this particle, what values might you get, and with what probabilities?By employing the prescribed definitions of the raising and lowering operators pertaining to the one-dimensional harmonic oscillator: x = ħ 2mω -(â+ + â_) hmw ê = i Compute the expectation values of the following quantities for the nth stationary staten. Keep in mind that the stationary states form an orthogonal set. 2 · (â+ − â_) [ pm 4ndx YmVndx = 8mn a. The position of particle (x) b. The momentum of the particle (p). c. (x²) d. (p²) e. Confirm that the uncertainty principle is satisfied for all values of n4.3 A particle with mass m and energy E is moving in one dimension from right to left. It is incident on the step potential V(x) = 0 for x 0, as shown on the diagram. The energy of the particle is E > Vo. = V(x) V = Vo V=0 x = 0 (a) Solve the Schrödinger equation to derive 4(x) for x 0. Express the solution in terms of a single unknown constant. (b) Calculate the value of the reflection coefficient R for the parti- cle.
- Exercise 6.4 Consider an anisotropic three-dimensional harmonic oscillator potential acy = { m (w² x ² + w} y² + w? 2²). V (x, y, z) = = m(o² x² + @z. (a) Evaluate the energy levels in terms of wx, @y, and (b) Calculate [Ĥ, Î₂]. Do you expect the wave functions to be eigenfunctions of 1²? (c) Find the three lowest levels for the case @x = @y= = 2002/3, and determine the degener- of each level.2.4. 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)(a) Write down the wave functions for the three regions of the potential energy barrier (Figure 5.25) for E < U₁. You will need six coefficients in all. Use complex exponential notation. (b) Use the boundary conditions at x = 0 and at x = L to find four relationships among the six coeffi- cients. (Do not try to solve these relationships.) (c) Sup- pose particles are incident on the barrier from the left. Which coefficient should be set to zero? Why?